#:include "common.fypp"
submodule(stdlib_lapack_eig_svd_lsq) stdlib_lapack_svd_comp
implicit none
contains
#:for ik,it,ii in LINALG_INT_KINDS_TYPES
pure module subroutine stdlib${ii}$_sgebrd( m, n, a, lda, d, e, tauq, taup, work, lwork,info )
!! SGEBRD reduces a general real M-by-N matrix A to upper or lower
!! bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
!! If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, lwork, m, n
! Array Arguments
real(sp), intent(inout) :: a(lda,*)
real(sp), intent(out) :: d(*), e(*), taup(*), tauq(*), work(*)
! =====================================================================
! Local Scalars
logical(lk) :: lquery
integer(${ik}$) :: i, iinfo, j, ldwrkx, ldwrky, lwkopt, minmn, nb, nbmin, nx, ws
! Intrinsic Functions
! Executable Statements
! test the input parameters
info = 0_${ik}$
nb = max( 1_${ik}$, stdlib${ii}$_ilaenv( 1_${ik}$, 'SGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ ) )
lwkopt = ( m+n )*nb
work( 1_${ik}$ ) = real( lwkopt,KIND=sp)
lquery = ( lwork==-1_${ik}$ )
if( m<0_${ik}$ ) then
info = -1_${ik}$
else if( n<0_${ik}$ ) then
info = -2_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -4_${ik}$
else if( lwork<max( 1_${ik}$, m, n ) .and. .not.lquery ) then
info = -10_${ik}$
end if
if( info<0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'SGEBRD', -info )
return
else if( lquery ) then
return
end if
! quick return if possible
minmn = min( m, n )
if( minmn==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
return
end if
ws = max( m, n )
ldwrkx = m
ldwrky = n
if( nb>1_${ik}$ .and. nb<minmn ) then
! set the crossover point nx.
nx = max( nb, stdlib${ii}$_ilaenv( 3_${ik}$, 'SGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ ) )
! determine when to switch from blocked to unblocked code.
if( nx<minmn ) then
ws = ( m+n )*nb
if( lwork<ws ) then
! not enough work space for the optimal nb, consider using
! a smaller block size.
nbmin = stdlib${ii}$_ilaenv( 2_${ik}$, 'SGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ )
if( lwork>=( m+n )*nbmin ) then
nb = lwork / ( m+n )
else
nb = 1_${ik}$
nx = minmn
end if
end if
end if
else
nx = minmn
end if
do i = 1, minmn - nx, nb
! reduce rows and columns i:i+nb-1 to bidiagonal form and return
! the matrices x and y which are needed to update the unreduced
! part of the matrix
call stdlib${ii}$_slabrd( m-i+1, n-i+1, nb, a( i, i ), lda, d( i ), e( i ),tauq( i ), &
taup( i ), work, ldwrkx,work( ldwrkx*nb+1 ), ldwrky )
! update the trailing submatrix a(i+nb:m,i+nb:n), using an update
! of the form a := a - v*y**t - x*u**t
call stdlib${ii}$_sgemm( 'NO TRANSPOSE', 'TRANSPOSE', m-i-nb+1, n-i-nb+1,nb, -one, a( i+&
nb, i ), lda,work( ldwrkx*nb+nb+1 ), ldwrky, one,a( i+nb, i+nb ), lda )
call stdlib${ii}$_sgemm( 'NO TRANSPOSE', 'NO TRANSPOSE', m-i-nb+1, n-i-nb+1,nb, -one, &
work( nb+1 ), ldwrkx, a( i, i+nb ), lda,one, a( i+nb, i+nb ), lda )
! copy diagonal and off-diagonal elements of b back into a
if( m>=n ) then
do j = i, i + nb - 1
a( j, j ) = d( j )
a( j, j+1 ) = e( j )
end do
else
do j = i, i + nb - 1
a( j, j ) = d( j )
a( j+1, j ) = e( j )
end do
end if
end do
! use unblocked code to reduce the remainder of the matrix
call stdlib${ii}$_sgebd2( m-i+1, n-i+1, a( i, i ), lda, d( i ), e( i ),tauq( i ), taup( i ), &
work, iinfo )
work( 1_${ik}$ ) = ws
return
end subroutine stdlib${ii}$_sgebrd
pure module subroutine stdlib${ii}$_dgebrd( m, n, a, lda, d, e, tauq, taup, work, lwork,info )
!! DGEBRD reduces a general real M-by-N matrix A to upper or lower
!! bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
!! If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, lwork, m, n
! Array Arguments
real(dp), intent(inout) :: a(lda,*)
real(dp), intent(out) :: d(*), e(*), taup(*), tauq(*), work(*)
! =====================================================================
! Local Scalars
logical(lk) :: lquery
integer(${ik}$) :: i, iinfo, j, ldwrkx, ldwrky, lwkopt, minmn, nb, nbmin, nx, ws
! Intrinsic Functions
! Executable Statements
! test the input parameters
info = 0_${ik}$
nb = max( 1_${ik}$, stdlib${ii}$_ilaenv( 1_${ik}$, 'DGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ ) )
lwkopt = ( m+n )*nb
work( 1_${ik}$ ) = real( lwkopt,KIND=dp)
lquery = ( lwork==-1_${ik}$ )
if( m<0_${ik}$ ) then
info = -1_${ik}$
else if( n<0_${ik}$ ) then
info = -2_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -4_${ik}$
else if( lwork<max( 1_${ik}$, m, n ) .and. .not.lquery ) then
info = -10_${ik}$
end if
if( info<0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DGEBRD', -info )
return
else if( lquery ) then
return
end if
! quick return if possible
minmn = min( m, n )
if( minmn==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
return
end if
ws = max( m, n )
ldwrkx = m
ldwrky = n
if( nb>1_${ik}$ .and. nb<minmn ) then
! set the crossover point nx.
nx = max( nb, stdlib${ii}$_ilaenv( 3_${ik}$, 'DGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ ) )
! determine when to switch from blocked to unblocked code.
if( nx<minmn ) then
ws = ( m+n )*nb
if( lwork<ws ) then
! not enough work space for the optimal nb, consider using
! a smaller block size.
nbmin = stdlib${ii}$_ilaenv( 2_${ik}$, 'DGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ )
if( lwork>=( m+n )*nbmin ) then
nb = lwork / ( m+n )
else
nb = 1_${ik}$
nx = minmn
end if
end if
end if
else
nx = minmn
end if
do i = 1, minmn - nx, nb
! reduce rows and columns i:i+nb-1 to bidiagonal form and return
! the matrices x and y which are needed to update the unreduced
! part of the matrix
call stdlib${ii}$_dlabrd( m-i+1, n-i+1, nb, a( i, i ), lda, d( i ), e( i ),tauq( i ), &
taup( i ), work, ldwrkx,work( ldwrkx*nb+1 ), ldwrky )
! update the trailing submatrix a(i+nb:m,i+nb:n), using an update
! of the form a := a - v*y**t - x*u**t
call stdlib${ii}$_dgemm( 'NO TRANSPOSE', 'TRANSPOSE', m-i-nb+1, n-i-nb+1,nb, -one, a( i+&
nb, i ), lda,work( ldwrkx*nb+nb+1 ), ldwrky, one,a( i+nb, i+nb ), lda )
call stdlib${ii}$_dgemm( 'NO TRANSPOSE', 'NO TRANSPOSE', m-i-nb+1, n-i-nb+1,nb, -one, &
work( nb+1 ), ldwrkx, a( i, i+nb ), lda,one, a( i+nb, i+nb ), lda )
! copy diagonal and off-diagonal elements of b back into a
if( m>=n ) then
do j = i, i + nb - 1
a( j, j ) = d( j )
a( j, j+1 ) = e( j )
end do
else
do j = i, i + nb - 1
a( j, j ) = d( j )
a( j+1, j ) = e( j )
end do
end if
end do
! use unblocked code to reduce the remainder of the matrix
call stdlib${ii}$_dgebd2( m-i+1, n-i+1, a( i, i ), lda, d( i ), e( i ),tauq( i ), taup( i ), &
work, iinfo )
work( 1_${ik}$ ) = ws
return
end subroutine stdlib${ii}$_dgebrd
#:for rk,rt,ri in REAL_KINDS_TYPES
#:if not rk in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ri}$gebrd( m, n, a, lda, d, e, tauq, taup, work, lwork,info )
!! DGEBRD: reduces a general real M-by-N matrix A to upper or lower
!! bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
!! If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${rk}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, lwork, m, n
! Array Arguments
real(${rk}$), intent(inout) :: a(lda,*)
real(${rk}$), intent(out) :: d(*), e(*), taup(*), tauq(*), work(*)
! =====================================================================
! Local Scalars
logical(lk) :: lquery
integer(${ik}$) :: i, iinfo, j, ldwrkx, ldwrky, lwkopt, minmn, nb, nbmin, nx, ws
! Intrinsic Functions
! Executable Statements
! test the input parameters
info = 0_${ik}$
nb = max( 1_${ik}$, stdlib${ii}$_ilaenv( 1_${ik}$, 'DGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ ) )
lwkopt = ( m+n )*nb
work( 1_${ik}$ ) = real( lwkopt,KIND=${rk}$)
lquery = ( lwork==-1_${ik}$ )
if( m<0_${ik}$ ) then
info = -1_${ik}$
else if( n<0_${ik}$ ) then
info = -2_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -4_${ik}$
else if( lwork<max( 1_${ik}$, m, n ) .and. .not.lquery ) then
info = -10_${ik}$
end if
if( info<0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DGEBRD', -info )
return
else if( lquery ) then
return
end if
! quick return if possible
minmn = min( m, n )
if( minmn==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
return
end if
ws = max( m, n )
ldwrkx = m
ldwrky = n
if( nb>1_${ik}$ .and. nb<minmn ) then
! set the crossover point nx.
nx = max( nb, stdlib${ii}$_ilaenv( 3_${ik}$, 'DGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ ) )
! determine when to switch from blocked to unblocked code.
if( nx<minmn ) then
ws = ( m+n )*nb
if( lwork<ws ) then
! not enough work space for the optimal nb, consider using
! a smaller block size.
nbmin = stdlib${ii}$_ilaenv( 2_${ik}$, 'DGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ )
if( lwork>=( m+n )*nbmin ) then
nb = lwork / ( m+n )
else
nb = 1_${ik}$
nx = minmn
end if
end if
end if
else
nx = minmn
end if
do i = 1, minmn - nx, nb
! reduce rows and columns i:i+nb-1 to bidiagonal form and return
! the matrices x and y which are needed to update the unreduced
! part of the matrix
call stdlib${ii}$_${ri}$labrd( m-i+1, n-i+1, nb, a( i, i ), lda, d( i ), e( i ),tauq( i ), &
taup( i ), work, ldwrkx,work( ldwrkx*nb+1 ), ldwrky )
! update the trailing submatrix a(i+nb:m,i+nb:n), using an update
! of the form a := a - v*y**t - x*u**t
call stdlib${ii}$_${ri}$gemm( 'NO TRANSPOSE', 'TRANSPOSE', m-i-nb+1, n-i-nb+1,nb, -one, a( i+&
nb, i ), lda,work( ldwrkx*nb+nb+1 ), ldwrky, one,a( i+nb, i+nb ), lda )
call stdlib${ii}$_${ri}$gemm( 'NO TRANSPOSE', 'NO TRANSPOSE', m-i-nb+1, n-i-nb+1,nb, -one, &
work( nb+1 ), ldwrkx, a( i, i+nb ), lda,one, a( i+nb, i+nb ), lda )
! copy diagonal and off-diagonal elements of b back into a
if( m>=n ) then
do j = i, i + nb - 1
a( j, j ) = d( j )
a( j, j+1 ) = e( j )
end do
else
do j = i, i + nb - 1
a( j, j ) = d( j )
a( j+1, j ) = e( j )
end do
end if
end do
! use unblocked code to reduce the remainder of the matrix
call stdlib${ii}$_${ri}$gebd2( m-i+1, n-i+1, a( i, i ), lda, d( i ), e( i ),tauq( i ), taup( i ), &
work, iinfo )
work( 1_${ik}$ ) = ws
return
end subroutine stdlib${ii}$_${ri}$gebrd
#:endif
#:endfor
pure module subroutine stdlib${ii}$_cgebrd( m, n, a, lda, d, e, tauq, taup, work, lwork,info )
!! CGEBRD reduces a general complex M-by-N matrix A to upper or lower
!! bidiagonal form B by a unitary transformation: Q**H * A * P = B.
!! If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, lwork, m, n
! Array Arguments
real(sp), intent(out) :: d(*), e(*)
complex(sp), intent(inout) :: a(lda,*)
complex(sp), intent(out) :: taup(*), tauq(*), work(*)
! =====================================================================
! Local Scalars
logical(lk) :: lquery
integer(${ik}$) :: i, iinfo, j, ldwrkx, ldwrky, lwkopt, minmn, nb, nbmin, nx, ws
! Intrinsic Functions
! Executable Statements
! test the input parameters
info = 0_${ik}$
nb = max( 1_${ik}$, stdlib${ii}$_ilaenv( 1_${ik}$, 'CGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ ) )
lwkopt = ( m+n )*nb
work( 1_${ik}$ ) = real( lwkopt,KIND=sp)
lquery = ( lwork==-1_${ik}$ )
if( m<0_${ik}$ ) then
info = -1_${ik}$
else if( n<0_${ik}$ ) then
info = -2_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -4_${ik}$
else if( lwork<max( 1_${ik}$, m, n ) .and. .not.lquery ) then
info = -10_${ik}$
end if
if( info<0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'CGEBRD', -info )
return
else if( lquery ) then
return
end if
! quick return if possible
minmn = min( m, n )
if( minmn==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
return
end if
ws = max( m, n )
ldwrkx = m
ldwrky = n
if( nb>1_${ik}$ .and. nb<minmn ) then
! set the crossover point nx.
nx = max( nb, stdlib${ii}$_ilaenv( 3_${ik}$, 'CGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ ) )
! determine when to switch from blocked to unblocked code.
if( nx<minmn ) then
ws = ( m+n )*nb
if( lwork<ws ) then
! not enough work space for the optimal nb, consider using
! a smaller block size.
nbmin = stdlib${ii}$_ilaenv( 2_${ik}$, 'CGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ )
if( lwork>=( m+n )*nbmin ) then
nb = lwork / ( m+n )
else
nb = 1_${ik}$
nx = minmn
end if
end if
end if
else
nx = minmn
end if
do i = 1, minmn - nx, nb
! reduce rows and columns i:i+ib-1 to bidiagonal form and return
! the matrices x and y which are needed to update the unreduced
! part of the matrix
call stdlib${ii}$_clabrd( m-i+1, n-i+1, nb, a( i, i ), lda, d( i ), e( i ),tauq( i ), &
taup( i ), work, ldwrkx,work( ldwrkx*nb+1 ), ldwrky )
! update the trailing submatrix a(i+ib:m,i+ib:n), using
! an update of the form a := a - v*y**h - x*u**h
call stdlib${ii}$_cgemm( 'NO TRANSPOSE', 'CONJUGATE TRANSPOSE', m-i-nb+1,n-i-nb+1, nb, -&
cone, a( i+nb, i ), lda,work( ldwrkx*nb+nb+1 ), ldwrky, cone,a( i+nb, i+nb ), lda )
call stdlib${ii}$_cgemm( 'NO TRANSPOSE', 'NO TRANSPOSE', m-i-nb+1, n-i-nb+1,nb, -cone, &
work( nb+1 ), ldwrkx, a( i, i+nb ), lda,cone, a( i+nb, i+nb ), lda )
! copy diagonal and off-diagonal elements of b back into a
if( m>=n ) then
do j = i, i + nb - 1
a( j, j ) = d( j )
a( j, j+1 ) = e( j )
end do
else
do j = i, i + nb - 1
a( j, j ) = d( j )
a( j+1, j ) = e( j )
end do
end if
end do
! use unblocked code to reduce the remainder of the matrix
call stdlib${ii}$_cgebd2( m-i+1, n-i+1, a( i, i ), lda, d( i ), e( i ),tauq( i ), taup( i ), &
work, iinfo )
work( 1_${ik}$ ) = ws
return
end subroutine stdlib${ii}$_cgebrd
pure module subroutine stdlib${ii}$_zgebrd( m, n, a, lda, d, e, tauq, taup, work, lwork,info )
!! ZGEBRD reduces a general complex M-by-N matrix A to upper or lower
!! bidiagonal form B by a unitary transformation: Q**H * A * P = B.
!! If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, lwork, m, n
! Array Arguments
real(dp), intent(out) :: d(*), e(*)
complex(dp), intent(inout) :: a(lda,*)
complex(dp), intent(out) :: taup(*), tauq(*), work(*)
! =====================================================================
! Local Scalars
logical(lk) :: lquery
integer(${ik}$) :: i, iinfo, j, ldwrkx, ldwrky, lwkopt, minmn, nb, nbmin, nx, ws
! Intrinsic Functions
! Executable Statements
! test the input parameters
info = 0_${ik}$
nb = max( 1_${ik}$, stdlib${ii}$_ilaenv( 1_${ik}$, 'ZGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ ) )
lwkopt = ( m+n )*nb
work( 1_${ik}$ ) = real( lwkopt,KIND=dp)
lquery = ( lwork==-1_${ik}$ )
if( m<0_${ik}$ ) then
info = -1_${ik}$
else if( n<0_${ik}$ ) then
info = -2_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -4_${ik}$
else if( lwork<max( 1_${ik}$, m, n ) .and. .not.lquery ) then
info = -10_${ik}$
end if
if( info<0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZGEBRD', -info )
return
else if( lquery ) then
return
end if
! quick return if possible
minmn = min( m, n )
if( minmn==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
return
end if
ws = max( m, n )
ldwrkx = m
ldwrky = n
if( nb>1_${ik}$ .and. nb<minmn ) then
! set the crossover point nx.
nx = max( nb, stdlib${ii}$_ilaenv( 3_${ik}$, 'ZGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ ) )
! determine when to switch from blocked to unblocked code.
if( nx<minmn ) then
ws = ( m+n )*nb
if( lwork<ws ) then
! not enough work space for the optimal nb, consider using
! a smaller block size.
nbmin = stdlib${ii}$_ilaenv( 2_${ik}$, 'ZGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ )
if( lwork>=( m+n )*nbmin ) then
nb = lwork / ( m+n )
else
nb = 1_${ik}$
nx = minmn
end if
end if
end if
else
nx = minmn
end if
do i = 1, minmn - nx, nb
! reduce rows and columns i:i+ib-1 to bidiagonal form and return
! the matrices x and y which are needed to update the unreduced
! part of the matrix
call stdlib${ii}$_zlabrd( m-i+1, n-i+1, nb, a( i, i ), lda, d( i ), e( i ),tauq( i ), &
taup( i ), work, ldwrkx,work( ldwrkx*nb+1 ), ldwrky )
! update the trailing submatrix a(i+ib:m,i+ib:n), using
! an update of the form a := a - v*y**h - x*u**h
call stdlib${ii}$_zgemm( 'NO TRANSPOSE', 'CONJUGATE TRANSPOSE', m-i-nb+1,n-i-nb+1, nb, -&
cone, a( i+nb, i ), lda,work( ldwrkx*nb+nb+1 ), ldwrky, cone,a( i+nb, i+nb ), lda )
call stdlib${ii}$_zgemm( 'NO TRANSPOSE', 'NO TRANSPOSE', m-i-nb+1, n-i-nb+1,nb, -cone, &
work( nb+1 ), ldwrkx, a( i, i+nb ), lda,cone, a( i+nb, i+nb ), lda )
! copy diagonal and off-diagonal elements of b back into a
if( m>=n ) then
do j = i, i + nb - 1
a( j, j ) = d( j )
a( j, j+1 ) = e( j )
end do
else
do j = i, i + nb - 1
a( j, j ) = d( j )
a( j+1, j ) = e( j )
end do
end if
end do
! use unblocked code to reduce the remainder of the matrix
call stdlib${ii}$_zgebd2( m-i+1, n-i+1, a( i, i ), lda, d( i ), e( i ),tauq( i ), taup( i ), &
work, iinfo )
work( 1_${ik}$ ) = ws
return
end subroutine stdlib${ii}$_zgebrd
#:for ck,ct,ci in CMPLX_KINDS_TYPES
#:if not ck in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ci}$gebrd( m, n, a, lda, d, e, tauq, taup, work, lwork,info )
!! ZGEBRD: reduces a general complex M-by-N matrix A to upper or lower
!! bidiagonal form B by a unitary transformation: Q**H * A * P = B.
!! If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${ck}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, lwork, m, n
! Array Arguments
real(${ck}$), intent(out) :: d(*), e(*)
complex(${ck}$), intent(inout) :: a(lda,*)
complex(${ck}$), intent(out) :: taup(*), tauq(*), work(*)
! =====================================================================
! Local Scalars
logical(lk) :: lquery
integer(${ik}$) :: i, iinfo, j, ldwrkx, ldwrky, lwkopt, minmn, nb, nbmin, nx, ws
! Intrinsic Functions
! Executable Statements
! test the input parameters
info = 0_${ik}$
nb = max( 1_${ik}$, stdlib${ii}$_ilaenv( 1_${ik}$, 'ZGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ ) )
lwkopt = ( m+n )*nb
work( 1_${ik}$ ) = real( lwkopt,KIND=${ck}$)
lquery = ( lwork==-1_${ik}$ )
if( m<0_${ik}$ ) then
info = -1_${ik}$
else if( n<0_${ik}$ ) then
info = -2_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -4_${ik}$
else if( lwork<max( 1_${ik}$, m, n ) .and. .not.lquery ) then
info = -10_${ik}$
end if
if( info<0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZGEBRD', -info )
return
else if( lquery ) then
return
end if
! quick return if possible
minmn = min( m, n )
if( minmn==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
return
end if
ws = max( m, n )
ldwrkx = m
ldwrky = n
if( nb>1_${ik}$ .and. nb<minmn ) then
! set the crossover point nx.
nx = max( nb, stdlib${ii}$_ilaenv( 3_${ik}$, 'ZGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ ) )
! determine when to switch from blocked to unblocked code.
if( nx<minmn ) then
ws = ( m+n )*nb
if( lwork<ws ) then
! not enough work space for the optimal nb, consider using
! a smaller block size.
nbmin = stdlib${ii}$_ilaenv( 2_${ik}$, 'ZGEBRD', ' ', m, n, -1_${ik}$, -1_${ik}$ )
if( lwork>=( m+n )*nbmin ) then
nb = lwork / ( m+n )
else
nb = 1_${ik}$
nx = minmn
end if
end if
end if
else
nx = minmn
end if
do i = 1, minmn - nx, nb
! reduce rows and columns i:i+ib-1 to bidiagonal form and return
! the matrices x and y which are needed to update the unreduced
! part of the matrix
call stdlib${ii}$_${ci}$labrd( m-i+1, n-i+1, nb, a( i, i ), lda, d( i ), e( i ),tauq( i ), &
taup( i ), work, ldwrkx,work( ldwrkx*nb+1 ), ldwrky )
! update the trailing submatrix a(i+ib:m,i+ib:n), using
! an update of the form a := a - v*y**h - x*u**h
call stdlib${ii}$_${ci}$gemm( 'NO TRANSPOSE', 'CONJUGATE TRANSPOSE', m-i-nb+1,n-i-nb+1, nb, -&
cone, a( i+nb, i ), lda,work( ldwrkx*nb+nb+1 ), ldwrky, cone,a( i+nb, i+nb ), lda )
call stdlib${ii}$_${ci}$gemm( 'NO TRANSPOSE', 'NO TRANSPOSE', m-i-nb+1, n-i-nb+1,nb, -cone, &
work( nb+1 ), ldwrkx, a( i, i+nb ), lda,cone, a( i+nb, i+nb ), lda )
! copy diagonal and off-diagonal elements of b back into a
if( m>=n ) then
do j = i, i + nb - 1
a( j, j ) = d( j )
a( j, j+1 ) = e( j )
end do
else
do j = i, i + nb - 1
a( j, j ) = d( j )
a( j+1, j ) = e( j )
end do
end if
end do
! use unblocked code to reduce the remainder of the matrix
call stdlib${ii}$_${ci}$gebd2( m-i+1, n-i+1, a( i, i ), lda, d( i ), e( i ),tauq( i ), taup( i ), &
work, iinfo )
work( 1_${ik}$ ) = ws
return
end subroutine stdlib${ii}$_${ci}$gebrd
#:endif
#:endfor
pure module subroutine stdlib${ii}$_sgebd2( m, n, a, lda, d, e, tauq, taup, work, info )
!! SGEBD2 reduces a real general m by n matrix A to upper or lower
!! bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
!! If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, m, n
! Array Arguments
real(sp), intent(inout) :: a(lda,*)
real(sp), intent(out) :: d(*), e(*), taup(*), tauq(*), work(*)
! =====================================================================
! Local Scalars
integer(${ik}$) :: i
! Intrinsic Functions
! Executable Statements
! test the input parameters
info = 0_${ik}$
if( m<0_${ik}$ ) then
info = -1_${ik}$
else if( n<0_${ik}$ ) then
info = -2_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -4_${ik}$
end if
if( info<0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'SGEBD2', -info )
return
end if
if( m>=n ) then
! reduce to upper bidiagonal form
do i = 1, n
! generate elementary reflector h(i) to annihilate a(i+1:m,i)
call stdlib${ii}$_slarfg( m-i+1, a( i, i ), a( min( i+1, m ), i ), 1_${ik}$,tauq( i ) )
d( i ) = a( i, i )
a( i, i ) = one
! apply h(i) to a(i:m,i+1:n) from the left
if( i<n )call stdlib${ii}$_slarf( 'LEFT', m-i+1, n-i, a( i, i ), 1_${ik}$, tauq( i ),a( i, i+&
1_${ik}$ ), lda, work )
a( i, i ) = d( i )
if( i<n ) then
! generate elementary reflector g(i) to annihilate
! a(i,i+2:n)
call stdlib${ii}$_slarfg( n-i, a( i, i+1 ), a( i, min( i+2, n ) ),lda, taup( i ) )
e( i ) = a( i, i+1 )
a( i, i+1 ) = one
! apply g(i) to a(i+1:m,i+1:n) from the right
call stdlib${ii}$_slarf( 'RIGHT', m-i, n-i, a( i, i+1 ), lda,taup( i ), a( i+1, i+1 &
), lda, work )
a( i, i+1 ) = e( i )
else
taup( i ) = zero
end if
end do
else
! reduce to lower bidiagonal form
do i = 1, m
! generate elementary reflector g(i) to annihilate a(i,i+1:n)
call stdlib${ii}$_slarfg( n-i+1, a( i, i ), a( i, min( i+1, n ) ), lda,taup( i ) )
d( i ) = a( i, i )
a( i, i ) = one
! apply g(i) to a(i+1:m,i:n) from the right
if( i<m )call stdlib${ii}$_slarf( 'RIGHT', m-i, n-i+1, a( i, i ), lda,taup( i ), a( i+&
1_${ik}$, i ), lda, work )
a( i, i ) = d( i )
if( i<m ) then
! generate elementary reflector h(i) to annihilate
! a(i+2:m,i)
call stdlib${ii}$_slarfg( m-i, a( i+1, i ), a( min( i+2, m ), i ), 1_${ik}$,tauq( i ) )
e( i ) = a( i+1, i )
a( i+1, i ) = one
! apply h(i) to a(i+1:m,i+1:n) from the left
call stdlib${ii}$_slarf( 'LEFT', m-i, n-i, a( i+1, i ), 1_${ik}$, tauq( i ),a( i+1, i+1 ), &
lda, work )
a( i+1, i ) = e( i )
else
tauq( i ) = zero
end if
end do
end if
return
end subroutine stdlib${ii}$_sgebd2
pure module subroutine stdlib${ii}$_dgebd2( m, n, a, lda, d, e, tauq, taup, work, info )
!! DGEBD2 reduces a real general m by n matrix A to upper or lower
!! bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
!! If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, m, n
! Array Arguments
real(dp), intent(inout) :: a(lda,*)
real(dp), intent(out) :: d(*), e(*), taup(*), tauq(*), work(*)
! =====================================================================
! Local Scalars
integer(${ik}$) :: i
! Intrinsic Functions
! Executable Statements
! test the input parameters
info = 0_${ik}$
if( m<0_${ik}$ ) then
info = -1_${ik}$
else if( n<0_${ik}$ ) then
info = -2_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -4_${ik}$
end if
if( info<0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DGEBD2', -info )
return
end if
if( m>=n ) then
! reduce to upper bidiagonal form
do i = 1, n
! generate elementary reflector h(i) to annihilate a(i+1:m,i)
call stdlib${ii}$_dlarfg( m-i+1, a( i, i ), a( min( i+1, m ), i ), 1_${ik}$,tauq( i ) )
d( i ) = a( i, i )
a( i, i ) = one
! apply h(i) to a(i:m,i+1:n) from the left
if( i<n )call stdlib${ii}$_dlarf( 'LEFT', m-i+1, n-i, a( i, i ), 1_${ik}$, tauq( i ),a( i, i+&
1_${ik}$ ), lda, work )
a( i, i ) = d( i )
if( i<n ) then
! generate elementary reflector g(i) to annihilate
! a(i,i+2:n)
call stdlib${ii}$_dlarfg( n-i, a( i, i+1 ), a( i, min( i+2, n ) ),lda, taup( i ) )
e( i ) = a( i, i+1 )
a( i, i+1 ) = one
! apply g(i) to a(i+1:m,i+1:n) from the right
call stdlib${ii}$_dlarf( 'RIGHT', m-i, n-i, a( i, i+1 ), lda,taup( i ), a( i+1, i+1 &
), lda, work )
a( i, i+1 ) = e( i )
else
taup( i ) = zero
end if
end do
else
! reduce to lower bidiagonal form
do i = 1, m
! generate elementary reflector g(i) to annihilate a(i,i+1:n)
call stdlib${ii}$_dlarfg( n-i+1, a( i, i ), a( i, min( i+1, n ) ), lda,taup( i ) )
d( i ) = a( i, i )
a( i, i ) = one
! apply g(i) to a(i+1:m,i:n) from the right
if( i<m )call stdlib${ii}$_dlarf( 'RIGHT', m-i, n-i+1, a( i, i ), lda,taup( i ), a( i+&
1_${ik}$, i ), lda, work )
a( i, i ) = d( i )
if( i<m ) then
! generate elementary reflector h(i) to annihilate
! a(i+2:m,i)
call stdlib${ii}$_dlarfg( m-i, a( i+1, i ), a( min( i+2, m ), i ), 1_${ik}$,tauq( i ) )
e( i ) = a( i+1, i )
a( i+1, i ) = one
! apply h(i) to a(i+1:m,i+1:n) from the left
call stdlib${ii}$_dlarf( 'LEFT', m-i, n-i, a( i+1, i ), 1_${ik}$, tauq( i ),a( i+1, i+1 ), &
lda, work )
a( i+1, i ) = e( i )
else
tauq( i ) = zero
end if
end do
end if
return
end subroutine stdlib${ii}$_dgebd2
#:for rk,rt,ri in REAL_KINDS_TYPES
#:if not rk in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ri}$gebd2( m, n, a, lda, d, e, tauq, taup, work, info )
!! DGEBD2: reduces a real general m by n matrix A to upper or lower
!! bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
!! If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${rk}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, m, n
! Array Arguments
real(${rk}$), intent(inout) :: a(lda,*)
real(${rk}$), intent(out) :: d(*), e(*), taup(*), tauq(*), work(*)
! =====================================================================
! Local Scalars
integer(${ik}$) :: i
! Intrinsic Functions
! Executable Statements
! test the input parameters
info = 0_${ik}$
if( m<0_${ik}$ ) then
info = -1_${ik}$
else if( n<0_${ik}$ ) then
info = -2_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -4_${ik}$
end if
if( info<0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DGEBD2', -info )
return
end if
if( m>=n ) then
! reduce to upper bidiagonal form
do i = 1, n
! generate elementary reflector h(i) to annihilate a(i+1:m,i)
call stdlib${ii}$_${ri}$larfg( m-i+1, a( i, i ), a( min( i+1, m ), i ), 1_${ik}$,tauq( i ) )
d( i ) = a( i, i )
a( i, i ) = one
! apply h(i) to a(i:m,i+1:n) from the left
if( i<n )call stdlib${ii}$_${ri}$larf( 'LEFT', m-i+1, n-i, a( i, i ), 1_${ik}$, tauq( i ),a( i, i+&
1_${ik}$ ), lda, work )
a( i, i ) = d( i )
if( i<n ) then
! generate elementary reflector g(i) to annihilate
! a(i,i+2:n)
call stdlib${ii}$_${ri}$larfg( n-i, a( i, i+1 ), a( i, min( i+2, n ) ),lda, taup( i ) )
e( i ) = a( i, i+1 )
a( i, i+1 ) = one
! apply g(i) to a(i+1:m,i+1:n) from the right
call stdlib${ii}$_${ri}$larf( 'RIGHT', m-i, n-i, a( i, i+1 ), lda,taup( i ), a( i+1, i+1 &
), lda, work )
a( i, i+1 ) = e( i )
else
taup( i ) = zero
end if
end do
else
! reduce to lower bidiagonal form
do i = 1, m
! generate elementary reflector g(i) to annihilate a(i,i+1:n)
call stdlib${ii}$_${ri}$larfg( n-i+1, a( i, i ), a( i, min( i+1, n ) ), lda,taup( i ) )
d( i ) = a( i, i )
a( i, i ) = one
! apply g(i) to a(i+1:m,i:n) from the right
if( i<m )call stdlib${ii}$_${ri}$larf( 'RIGHT', m-i, n-i+1, a( i, i ), lda,taup( i ), a( i+&
1_${ik}$, i ), lda, work )
a( i, i ) = d( i )
if( i<m ) then
! generate elementary reflector h(i) to annihilate
! a(i+2:m,i)
call stdlib${ii}$_${ri}$larfg( m-i, a( i+1, i ), a( min( i+2, m ), i ), 1_${ik}$,tauq( i ) )
e( i ) = a( i+1, i )
a( i+1, i ) = one
! apply h(i) to a(i+1:m,i+1:n) from the left
call stdlib${ii}$_${ri}$larf( 'LEFT', m-i, n-i, a( i+1, i ), 1_${ik}$, tauq( i ),a( i+1, i+1 ), &
lda, work )
a( i+1, i ) = e( i )
else
tauq( i ) = zero
end if
end do
end if
return
end subroutine stdlib${ii}$_${ri}$gebd2
#:endif
#:endfor
pure module subroutine stdlib${ii}$_cgebd2( m, n, a, lda, d, e, tauq, taup, work, info )
!! CGEBD2 reduces a complex general m by n matrix A to upper or lower
!! real bidiagonal form B by a unitary transformation: Q**H * A * P = B.
!! If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, m, n
! Array Arguments
real(sp), intent(out) :: d(*), e(*)
complex(sp), intent(inout) :: a(lda,*)
complex(sp), intent(out) :: taup(*), tauq(*), work(*)
! =====================================================================
! Local Scalars
integer(${ik}$) :: i
complex(sp) :: alpha
! Intrinsic Functions
! Executable Statements
! test the input parameters
info = 0_${ik}$
if( m<0_${ik}$ ) then
info = -1_${ik}$
else if( n<0_${ik}$ ) then
info = -2_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -4_${ik}$
end if
if( info<0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'CGEBD2', -info )
return
end if
if( m>=n ) then
! reduce to upper bidiagonal form
do i = 1, n
! generate elementary reflector h(i) to annihilate a(i+1:m,i)
alpha = a( i, i )
call stdlib${ii}$_clarfg( m-i+1, alpha, a( min( i+1, m ), i ), 1_${ik}$,tauq( i ) )
d( i ) = real( alpha,KIND=sp)
a( i, i ) = cone
! apply h(i)**h to a(i:m,i+1:n) from the left
if( i<n )call stdlib${ii}$_clarf( 'LEFT', m-i+1, n-i, a( i, i ), 1_${ik}$,conjg( tauq( i ) ), &
a( i, i+1 ), lda, work )
a( i, i ) = d( i )
if( i<n ) then
! generate elementary reflector g(i) to annihilate
! a(i,i+2:n)
call stdlib${ii}$_clacgv( n-i, a( i, i+1 ), lda )
alpha = a( i, i+1 )
call stdlib${ii}$_clarfg( n-i, alpha, a( i, min( i+2, n ) ),lda, taup( i ) )
e( i ) = real( alpha,KIND=sp)
a( i, i+1 ) = cone
! apply g(i) to a(i+1:m,i+1:n) from the right
call stdlib${ii}$_clarf( 'RIGHT', m-i, n-i, a( i, i+1 ), lda,taup( i ), a( i+1, i+1 &
), lda, work )
call stdlib${ii}$_clacgv( n-i, a( i, i+1 ), lda )
a( i, i+1 ) = e( i )
else
taup( i ) = czero
end if
end do
else
! reduce to lower bidiagonal form
do i = 1, m
! generate elementary reflector g(i) to annihilate a(i,i+1:n)
call stdlib${ii}$_clacgv( n-i+1, a( i, i ), lda )
alpha = a( i, i )
call stdlib${ii}$_clarfg( n-i+1, alpha, a( i, min( i+1, n ) ), lda,taup( i ) )
d( i ) = real( alpha,KIND=sp)
a( i, i ) = cone
! apply g(i) to a(i+1:m,i:n) from the right
if( i<m )call stdlib${ii}$_clarf( 'RIGHT', m-i, n-i+1, a( i, i ), lda,taup( i ), a( i+&
1_${ik}$, i ), lda, work )
call stdlib${ii}$_clacgv( n-i+1, a( i, i ), lda )
a( i, i ) = d( i )
if( i<m ) then
! generate elementary reflector h(i) to annihilate
! a(i+2:m,i)
alpha = a( i+1, i )
call stdlib${ii}$_clarfg( m-i, alpha, a( min( i+2, m ), i ), 1_${ik}$,tauq( i ) )
e( i ) = real( alpha,KIND=sp)
a( i+1, i ) = cone
! apply h(i)**h to a(i+1:m,i+1:n) from the left
call stdlib${ii}$_clarf( 'LEFT', m-i, n-i, a( i+1, i ), 1_${ik}$,conjg( tauq( i ) ), a( i+&
1_${ik}$, i+1 ), lda,work )
a( i+1, i ) = e( i )
else
tauq( i ) = czero
end if
end do
end if
return
end subroutine stdlib${ii}$_cgebd2
pure module subroutine stdlib${ii}$_zgebd2( m, n, a, lda, d, e, tauq, taup, work, info )
!! ZGEBD2 reduces a complex general m by n matrix A to upper or lower
!! real bidiagonal form B by a unitary transformation: Q**H * A * P = B.
!! If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, m, n
! Array Arguments
real(dp), intent(out) :: d(*), e(*)
complex(dp), intent(inout) :: a(lda,*)
complex(dp), intent(out) :: taup(*), tauq(*), work(*)
! =====================================================================
! Local Scalars
integer(${ik}$) :: i
complex(dp) :: alpha
! Intrinsic Functions
! Executable Statements
! test the input parameters
info = 0_${ik}$
if( m<0_${ik}$ ) then
info = -1_${ik}$
else if( n<0_${ik}$ ) then
info = -2_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -4_${ik}$
end if
if( info<0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZGEBD2', -info )
return
end if
if( m>=n ) then
! reduce to upper bidiagonal form
do i = 1, n
! generate elementary reflector h(i) to annihilate a(i+1:m,i)
alpha = a( i, i )
call stdlib${ii}$_zlarfg( m-i+1, alpha, a( min( i+1, m ), i ), 1_${ik}$,tauq( i ) )
d( i ) = real( alpha,KIND=dp)
a( i, i ) = cone
! apply h(i)**h to a(i:m,i+1:n) from the left
if( i<n )call stdlib${ii}$_zlarf( 'LEFT', m-i+1, n-i, a( i, i ), 1_${ik}$,conjg( tauq( i ) ), &
a( i, i+1 ), lda, work )
a( i, i ) = d( i )
if( i<n ) then
! generate elementary reflector g(i) to annihilate
! a(i,i+2:n)
call stdlib${ii}$_zlacgv( n-i, a( i, i+1 ), lda )
alpha = a( i, i+1 )
call stdlib${ii}$_zlarfg( n-i, alpha, a( i, min( i+2, n ) ), lda,taup( i ) )
e( i ) = real( alpha,KIND=dp)
a( i, i+1 ) = cone
! apply g(i) to a(i+1:m,i+1:n) from the right
call stdlib${ii}$_zlarf( 'RIGHT', m-i, n-i, a( i, i+1 ), lda,taup( i ), a( i+1, i+1 &
), lda, work )
call stdlib${ii}$_zlacgv( n-i, a( i, i+1 ), lda )
a( i, i+1 ) = e( i )
else
taup( i ) = czero
end if
end do
else
! reduce to lower bidiagonal form
do i = 1, m
! generate elementary reflector g(i) to annihilate a(i,i+1:n)
call stdlib${ii}$_zlacgv( n-i+1, a( i, i ), lda )
alpha = a( i, i )
call stdlib${ii}$_zlarfg( n-i+1, alpha, a( i, min( i+1, n ) ), lda,taup( i ) )
d( i ) = real( alpha,KIND=dp)
a( i, i ) = cone
! apply g(i) to a(i+1:m,i:n) from the right
if( i<m )call stdlib${ii}$_zlarf( 'RIGHT', m-i, n-i+1, a( i, i ), lda,taup( i ), a( i+&
1_${ik}$, i ), lda, work )
call stdlib${ii}$_zlacgv( n-i+1, a( i, i ), lda )
a( i, i ) = d( i )
if( i<m ) then
! generate elementary reflector h(i) to annihilate
! a(i+2:m,i)
alpha = a( i+1, i )
call stdlib${ii}$_zlarfg( m-i, alpha, a( min( i+2, m ), i ), 1_${ik}$,tauq( i ) )
e( i ) = real( alpha,KIND=dp)
a( i+1, i ) = cone
! apply h(i)**h to a(i+1:m,i+1:n) from the left
call stdlib${ii}$_zlarf( 'LEFT', m-i, n-i, a( i+1, i ), 1_${ik}$,conjg( tauq( i ) ), a( i+&
1_${ik}$, i+1 ), lda,work )
a( i+1, i ) = e( i )
else
tauq( i ) = czero
end if
end do
end if
return
end subroutine stdlib${ii}$_zgebd2
#:for ck,ct,ci in CMPLX_KINDS_TYPES
#:if not ck in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ci}$gebd2( m, n, a, lda, d, e, tauq, taup, work, info )
!! ZGEBD2: reduces a complex general m by n matrix A to upper or lower
!! real bidiagonal form B by a unitary transformation: Q**H * A * P = B.
!! If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${ck}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, m, n
! Array Arguments
real(${ck}$), intent(out) :: d(*), e(*)
complex(${ck}$), intent(inout) :: a(lda,*)
complex(${ck}$), intent(out) :: taup(*), tauq(*), work(*)
! =====================================================================
! Local Scalars
integer(${ik}$) :: i
complex(${ck}$) :: alpha
! Intrinsic Functions
! Executable Statements
! test the input parameters
info = 0_${ik}$
if( m<0_${ik}$ ) then
info = -1_${ik}$
else if( n<0_${ik}$ ) then
info = -2_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -4_${ik}$
end if
if( info<0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZGEBD2', -info )
return
end if
if( m>=n ) then
! reduce to upper bidiagonal form
do i = 1, n
! generate elementary reflector h(i) to annihilate a(i+1:m,i)
alpha = a( i, i )
call stdlib${ii}$_${ci}$larfg( m-i+1, alpha, a( min( i+1, m ), i ), 1_${ik}$,tauq( i ) )
d( i ) = real( alpha,KIND=${ck}$)
a( i, i ) = cone
! apply h(i)**h to a(i:m,i+1:n) from the left
if( i<n )call stdlib${ii}$_${ci}$larf( 'LEFT', m-i+1, n-i, a( i, i ), 1_${ik}$,conjg( tauq( i ) ), &
a( i, i+1 ), lda, work )
a( i, i ) = d( i )
if( i<n ) then
! generate elementary reflector g(i) to annihilate
! a(i,i+2:n)
call stdlib${ii}$_${ci}$lacgv( n-i, a( i, i+1 ), lda )
alpha = a( i, i+1 )
call stdlib${ii}$_${ci}$larfg( n-i, alpha, a( i, min( i+2, n ) ), lda,taup( i ) )
e( i ) = real( alpha,KIND=${ck}$)
a( i, i+1 ) = cone
! apply g(i) to a(i+1:m,i+1:n) from the right
call stdlib${ii}$_${ci}$larf( 'RIGHT', m-i, n-i, a( i, i+1 ), lda,taup( i ), a( i+1, i+1 &
), lda, work )
call stdlib${ii}$_${ci}$lacgv( n-i, a( i, i+1 ), lda )
a( i, i+1 ) = e( i )
else
taup( i ) = czero
end if
end do
else
! reduce to lower bidiagonal form
do i = 1, m
! generate elementary reflector g(i) to annihilate a(i,i+1:n)
call stdlib${ii}$_${ci}$lacgv( n-i+1, a( i, i ), lda )
alpha = a( i, i )
call stdlib${ii}$_${ci}$larfg( n-i+1, alpha, a( i, min( i+1, n ) ), lda,taup( i ) )
d( i ) = real( alpha,KIND=${ck}$)
a( i, i ) = cone
! apply g(i) to a(i+1:m,i:n) from the right
if( i<m )call stdlib${ii}$_${ci}$larf( 'RIGHT', m-i, n-i+1, a( i, i ), lda,taup( i ), a( i+&
1_${ik}$, i ), lda, work )
call stdlib${ii}$_${ci}$lacgv( n-i+1, a( i, i ), lda )
a( i, i ) = d( i )
if( i<m ) then
! generate elementary reflector h(i) to annihilate
! a(i+2:m,i)
alpha = a( i+1, i )
call stdlib${ii}$_${ci}$larfg( m-i, alpha, a( min( i+2, m ), i ), 1_${ik}$,tauq( i ) )
e( i ) = real( alpha,KIND=${ck}$)
a( i+1, i ) = cone
! apply h(i)**h to a(i+1:m,i+1:n) from the left
call stdlib${ii}$_${ci}$larf( 'LEFT', m-i, n-i, a( i+1, i ), 1_${ik}$,conjg( tauq( i ) ), a( i+&
1_${ik}$, i+1 ), lda,work )
a( i+1, i ) = e( i )
else
tauq( i ) = czero
end if
end do
end if
return
end subroutine stdlib${ii}$_${ci}$gebd2
#:endif
#:endfor
pure module subroutine stdlib${ii}$_sgbbrd( vect, m, n, ncc, kl, ku, ab, ldab, d, e, q,ldq, pt, ldpt, c, &
!! SGBBRD reduces a real general m-by-n band matrix A to upper
!! bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
!! The routine computes B, and optionally forms Q or P**T, or computes
!! Q**T*C for a given matrix C.
ldc, work, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: kl, ku, ldab, ldc, ldpt, ldq, m, n, ncc
! Array Arguments
real(sp), intent(inout) :: ab(ldab,*), c(ldc,*)
real(sp), intent(out) :: d(*), e(*), pt(ldpt,*), q(ldq,*), work(*)
! =====================================================================
! Local Scalars
logical(lk) :: wantb, wantc, wantpt, wantq
integer(${ik}$) :: i, inca, j, j1, j2, kb, kb1, kk, klm, klu1, kun, l, minmn, ml, ml0, mn,&
mu, mu0, nr, nrt
real(sp) :: ra, rb, rc, rs
! Intrinsic Functions
! Executable Statements
! test the input parameters
wantb = stdlib_lsame( vect, 'B' )
wantq = stdlib_lsame( vect, 'Q' ) .or. wantb
wantpt = stdlib_lsame( vect, 'P' ) .or. wantb
wantc = ncc>0_${ik}$
klu1 = kl + ku + 1_${ik}$
info = 0_${ik}$
if( .not.wantq .and. .not.wantpt .and. .not.stdlib_lsame( vect, 'N' ) )then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( n<0_${ik}$ ) then
info = -3_${ik}$
else if( ncc<0_${ik}$ ) then
info = -4_${ik}$
else if( kl<0_${ik}$ ) then
info = -5_${ik}$
else if( ku<0_${ik}$ ) then
info = -6_${ik}$
else if( ldab<klu1 ) then
info = -8_${ik}$
else if( ldq<1_${ik}$ .or. wantq .and. ldq<max( 1_${ik}$, m ) ) then
info = -12_${ik}$
else if( ldpt<1_${ik}$ .or. wantpt .and. ldpt<max( 1_${ik}$, n ) ) then
info = -14_${ik}$
else if( ldc<1_${ik}$ .or. wantc .and. ldc<max( 1_${ik}$, m ) ) then
info = -16_${ik}$
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'SGBBRD', -info )
return
end if
! initialize q and p**t to the unit matrix, if needed
if( wantq )call stdlib${ii}$_slaset( 'FULL', m, m, zero, one, q, ldq )
if( wantpt )call stdlib${ii}$_slaset( 'FULL', n, n, zero, one, pt, ldpt )
! quick return if possible.
if( m==0 .or. n==0 )return
minmn = min( m, n )
if( kl+ku>1_${ik}$ ) then
! reduce to upper bidiagonal form if ku > 0; if ku = 0, reduce
! first to lower bidiagonal form and then transform to upper
! bidiagonal
if( ku>0_${ik}$ ) then
ml0 = 1_${ik}$
mu0 = 2_${ik}$
else
ml0 = 2_${ik}$
mu0 = 1_${ik}$
end if
! wherever possible, plane rotations are generated and applied in
! vector operations of length nr over the index set j1:j2:klu1.
! the sines of the plane rotations are stored in work(1:max(m,n))
! and the cosines in work(max(m,n)+1:2*max(m,n)).
mn = max( m, n )
klm = min( m-1, kl )
kun = min( n-1, ku )
kb = klm + kun
kb1 = kb + 1_${ik}$
inca = kb1*ldab
nr = 0_${ik}$
j1 = klm + 2_${ik}$
j2 = 1_${ik}$ - kun
loop_90: do i = 1, minmn
! reduce i-th column and i-th row of matrix to bidiagonal form
ml = klm + 1_${ik}$
mu = kun + 1_${ik}$
loop_80: do kk = 1, kb
j1 = j1 + kb
j2 = j2 + kb
! generate plane rotations to annihilate nonzero elements
! which have been created below the band
if( nr>0_${ik}$ )call stdlib${ii}$_slargv( nr, ab( klu1, j1-klm-1 ), inca,work( j1 ), kb1, &
work( mn+j1 ), kb1 )
! apply plane rotations from the left
do l = 1, kb
if( j2-klm+l-1>n ) then
nrt = nr - 1_${ik}$
else
nrt = nr
end if
if( nrt>0_${ik}$ )call stdlib${ii}$_slartv( nrt, ab( klu1-l, j1-klm+l-1 ), inca,ab( &
klu1-l+1, j1-klm+l-1 ), inca,work( mn+j1 ), work( j1 ), kb1 )
end do
if( ml>ml0 ) then
if( ml<=m-i+1 ) then
! generate plane rotation to annihilate a(i+ml-1,i)
! within the band, and apply rotation from the left
call stdlib${ii}$_slartg( ab( ku+ml-1, i ), ab( ku+ml, i ),work( mn+i+ml-1 ), &
work( i+ml-1 ),ra )
ab( ku+ml-1, i ) = ra
if( i<n )call stdlib${ii}$_srot( min( ku+ml-2, n-i ),ab( ku+ml-2, i+1 ), ldab-&
1_${ik}$,ab( ku+ml-1, i+1 ), ldab-1,work( mn+i+ml-1 ), work( i+ml-1 ) )
end if
nr = nr + 1_${ik}$
j1 = j1 - kb1
end if
if( wantq ) then
! accumulate product of plane rotations in q
do j = j1, j2, kb1
call stdlib${ii}$_srot( m, q( 1_${ik}$, j-1 ), 1_${ik}$, q( 1_${ik}$, j ), 1_${ik}$,work( mn+j ), work( j &
) )
end do
end if
if( wantc ) then
! apply plane rotations to c
do j = j1, j2, kb1
call stdlib${ii}$_srot( ncc, c( j-1, 1_${ik}$ ), ldc, c( j, 1_${ik}$ ), ldc,work( mn+j ), &
work( j ) )
end do
end if
if( j2+kun>n ) then
! adjust j2 to keep within the bounds of the matrix
nr = nr - 1_${ik}$
j2 = j2 - kb1
end if
do j = j1, j2, kb1
! create nonzero element a(j-1,j+ku) above the band
! and store it in work(n+1:2*n)
work( j+kun ) = work( j )*ab( 1_${ik}$, j+kun )
ab( 1_${ik}$, j+kun ) = work( mn+j )*ab( 1_${ik}$, j+kun )
end do
! generate plane rotations to annihilate nonzero elements
! which have been generated above the band
if( nr>0_${ik}$ )call stdlib${ii}$_slargv( nr, ab( 1_${ik}$, j1+kun-1 ), inca,work( j1+kun ), kb1,&
work( mn+j1+kun ),kb1 )
! apply plane rotations from the right
do l = 1, kb
if( j2+l-1>m ) then
nrt = nr - 1_${ik}$
else
nrt = nr
end if
if( nrt>0_${ik}$ )call stdlib${ii}$_slartv( nrt, ab( l+1, j1+kun-1 ), inca,ab( l, j1+&
kun ), inca,work( mn+j1+kun ), work( j1+kun ),kb1 )
end do
if( ml==ml0 .and. mu>mu0 ) then
if( mu<=n-i+1 ) then
! generate plane rotation to annihilate a(i,i+mu-1)
! within the band, and apply rotation from the right
call stdlib${ii}$_slartg( ab( ku-mu+3, i+mu-2 ),ab( ku-mu+2, i+mu-1 ),work( &
mn+i+mu-1 ), work( i+mu-1 ),ra )
ab( ku-mu+3, i+mu-2 ) = ra
call stdlib${ii}$_srot( min( kl+mu-2, m-i ),ab( ku-mu+4, i+mu-2 ), 1_${ik}$,ab( ku-&
mu+3, i+mu-1 ), 1_${ik}$,work( mn+i+mu-1 ), work( i+mu-1 ) )
end if
nr = nr + 1_${ik}$
j1 = j1 - kb1
end if
if( wantpt ) then
! accumulate product of plane rotations in p**t
do j = j1, j2, kb1
call stdlib${ii}$_srot( n, pt( j+kun-1, 1_${ik}$ ), ldpt,pt( j+kun, 1_${ik}$ ), ldpt, work( &
mn+j+kun ),work( j+kun ) )
end do
end if
if( j2+kb>m ) then
! adjust j2 to keep within the bounds of the matrix
nr = nr - 1_${ik}$
j2 = j2 - kb1
end if
do j = j1, j2, kb1
! create nonzero element a(j+kl+ku,j+ku-1) below the
! band and store it in work(1:n)
work( j+kb ) = work( j+kun )*ab( klu1, j+kun )
ab( klu1, j+kun ) = work( mn+j+kun )*ab( klu1, j+kun )
end do
if( ml>ml0 ) then
ml = ml - 1_${ik}$
else
mu = mu - 1_${ik}$
end if
end do loop_80
end do loop_90
end if
if( ku==0_${ik}$ .and. kl>0_${ik}$ ) then
! a has been reduced to lower bidiagonal form
! transform lower bidiagonal form to upper bidiagonal by applying
! plane rotations from the left, storing diagonal elements in d
! and off-diagonal elements in e
do i = 1, min( m-1, n )
call stdlib${ii}$_slartg( ab( 1_${ik}$, i ), ab( 2_${ik}$, i ), rc, rs, ra )
d( i ) = ra
if( i<n ) then
e( i ) = rs*ab( 1_${ik}$, i+1 )
ab( 1_${ik}$, i+1 ) = rc*ab( 1_${ik}$, i+1 )
end if
if( wantq )call stdlib${ii}$_srot( m, q( 1_${ik}$, i ), 1_${ik}$, q( 1_${ik}$, i+1 ), 1_${ik}$, rc, rs )
if( wantc )call stdlib${ii}$_srot( ncc, c( i, 1_${ik}$ ), ldc, c( i+1, 1_${ik}$ ), ldc, rc,rs )
end do
if( m<=n )d( m ) = ab( 1_${ik}$, m )
else if( ku>0_${ik}$ ) then
! a has been reduced to upper bidiagonal form
if( m<n ) then
! annihilate a(m,m+1) by applying plane rotations from the
! right, storing diagonal elements in d and off-diagonal
! elements in e
rb = ab( ku, m+1 )
do i = m, 1, -1
call stdlib${ii}$_slartg( ab( ku+1, i ), rb, rc, rs, ra )
d( i ) = ra
if( i>1_${ik}$ ) then
rb = -rs*ab( ku, i )
e( i-1 ) = rc*ab( ku, i )
end if
if( wantpt )call stdlib${ii}$_srot( n, pt( i, 1_${ik}$ ), ldpt, pt( m+1, 1_${ik}$ ), ldpt,rc, rs )
end do
else
! copy off-diagonal elements to e and diagonal elements to d
do i = 1, minmn - 1
e( i ) = ab( ku, i+1 )
end do
do i = 1, minmn
d( i ) = ab( ku+1, i )
end do
end if
else
! a is diagonal. set elements of e to zero and copy diagonal
! elements to d.
do i = 1, minmn - 1
e( i ) = zero
end do
do i = 1, minmn
d( i ) = ab( 1_${ik}$, i )
end do
end if
return
end subroutine stdlib${ii}$_sgbbrd
pure module subroutine stdlib${ii}$_dgbbrd( vect, m, n, ncc, kl, ku, ab, ldab, d, e, q,ldq, pt, ldpt, c, &
!! DGBBRD reduces a real general m-by-n band matrix A to upper
!! bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
!! The routine computes B, and optionally forms Q or P**T, or computes
!! Q**T*C for a given matrix C.
ldc, work, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: kl, ku, ldab, ldc, ldpt, ldq, m, n, ncc
! Array Arguments
real(dp), intent(inout) :: ab(ldab,*), c(ldc,*)
real(dp), intent(out) :: d(*), e(*), pt(ldpt,*), q(ldq,*), work(*)
! =====================================================================
! Local Scalars
logical(lk) :: wantb, wantc, wantpt, wantq
integer(${ik}$) :: i, inca, j, j1, j2, kb, kb1, kk, klm, klu1, kun, l, minmn, ml, ml0, mn,&
mu, mu0, nr, nrt
real(dp) :: ra, rb, rc, rs
! Intrinsic Functions
! Executable Statements
! test the input parameters
wantb = stdlib_lsame( vect, 'B' )
wantq = stdlib_lsame( vect, 'Q' ) .or. wantb
wantpt = stdlib_lsame( vect, 'P' ) .or. wantb
wantc = ncc>0_${ik}$
klu1 = kl + ku + 1_${ik}$
info = 0_${ik}$
if( .not.wantq .and. .not.wantpt .and. .not.stdlib_lsame( vect, 'N' ) )then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( n<0_${ik}$ ) then
info = -3_${ik}$
else if( ncc<0_${ik}$ ) then
info = -4_${ik}$
else if( kl<0_${ik}$ ) then
info = -5_${ik}$
else if( ku<0_${ik}$ ) then
info = -6_${ik}$
else if( ldab<klu1 ) then
info = -8_${ik}$
else if( ldq<1_${ik}$ .or. wantq .and. ldq<max( 1_${ik}$, m ) ) then
info = -12_${ik}$
else if( ldpt<1_${ik}$ .or. wantpt .and. ldpt<max( 1_${ik}$, n ) ) then
info = -14_${ik}$
else if( ldc<1_${ik}$ .or. wantc .and. ldc<max( 1_${ik}$, m ) ) then
info = -16_${ik}$
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DGBBRD', -info )
return
end if
! initialize q and p**t to the unit matrix, if needed
if( wantq )call stdlib${ii}$_dlaset( 'FULL', m, m, zero, one, q, ldq )
if( wantpt )call stdlib${ii}$_dlaset( 'FULL', n, n, zero, one, pt, ldpt )
! quick return if possible.
if( m==0 .or. n==0 )return
minmn = min( m, n )
if( kl+ku>1_${ik}$ ) then
! reduce to upper bidiagonal form if ku > 0; if ku = 0, reduce
! first to lower bidiagonal form and then transform to upper
! bidiagonal
if( ku>0_${ik}$ ) then
ml0 = 1_${ik}$
mu0 = 2_${ik}$
else
ml0 = 2_${ik}$
mu0 = 1_${ik}$
end if
! wherever possible, plane rotations are generated and applied in
! vector operations of length nr over the index set j1:j2:klu1.
! the sines of the plane rotations are stored in work(1:max(m,n))
! and the cosines in work(max(m,n)+1:2*max(m,n)).
mn = max( m, n )
klm = min( m-1, kl )
kun = min( n-1, ku )
kb = klm + kun
kb1 = kb + 1_${ik}$
inca = kb1*ldab
nr = 0_${ik}$
j1 = klm + 2_${ik}$
j2 = 1_${ik}$ - kun
loop_90: do i = 1, minmn
! reduce i-th column and i-th row of matrix to bidiagonal form
ml = klm + 1_${ik}$
mu = kun + 1_${ik}$
loop_80: do kk = 1, kb
j1 = j1 + kb
j2 = j2 + kb
! generate plane rotations to annihilate nonzero elements
! which have been created below the band
if( nr>0_${ik}$ )call stdlib${ii}$_dlargv( nr, ab( klu1, j1-klm-1 ), inca,work( j1 ), kb1, &
work( mn+j1 ), kb1 )
! apply plane rotations from the left
do l = 1, kb
if( j2-klm+l-1>n ) then
nrt = nr - 1_${ik}$
else
nrt = nr
end if
if( nrt>0_${ik}$ )call stdlib${ii}$_dlartv( nrt, ab( klu1-l, j1-klm+l-1 ), inca,ab( &
klu1-l+1, j1-klm+l-1 ), inca,work( mn+j1 ), work( j1 ), kb1 )
end do
if( ml>ml0 ) then
if( ml<=m-i+1 ) then
! generate plane rotation to annihilate a(i+ml-1,i)
! within the band, and apply rotation from the left
call stdlib${ii}$_dlartg( ab( ku+ml-1, i ), ab( ku+ml, i ),work( mn+i+ml-1 ), &
work( i+ml-1 ),ra )
ab( ku+ml-1, i ) = ra
if( i<n )call stdlib${ii}$_drot( min( ku+ml-2, n-i ),ab( ku+ml-2, i+1 ), ldab-&
1_${ik}$,ab( ku+ml-1, i+1 ), ldab-1,work( mn+i+ml-1 ), work( i+ml-1 ) )
end if
nr = nr + 1_${ik}$
j1 = j1 - kb1
end if
if( wantq ) then
! accumulate product of plane rotations in q
do j = j1, j2, kb1
call stdlib${ii}$_drot( m, q( 1_${ik}$, j-1 ), 1_${ik}$, q( 1_${ik}$, j ), 1_${ik}$,work( mn+j ), work( j &
) )
end do
end if
if( wantc ) then
! apply plane rotations to c
do j = j1, j2, kb1
call stdlib${ii}$_drot( ncc, c( j-1, 1_${ik}$ ), ldc, c( j, 1_${ik}$ ), ldc,work( mn+j ), &
work( j ) )
end do
end if
if( j2+kun>n ) then
! adjust j2 to keep within the bounds of the matrix
nr = nr - 1_${ik}$
j2 = j2 - kb1
end if
do j = j1, j2, kb1
! create nonzero element a(j-1,j+ku) above the band
! and store it in work(n+1:2*n)
work( j+kun ) = work( j )*ab( 1_${ik}$, j+kun )
ab( 1_${ik}$, j+kun ) = work( mn+j )*ab( 1_${ik}$, j+kun )
end do
! generate plane rotations to annihilate nonzero elements
! which have been generated above the band
if( nr>0_${ik}$ )call stdlib${ii}$_dlargv( nr, ab( 1_${ik}$, j1+kun-1 ), inca,work( j1+kun ), kb1,&
work( mn+j1+kun ),kb1 )
! apply plane rotations from the right
do l = 1, kb
if( j2+l-1>m ) then
nrt = nr - 1_${ik}$
else
nrt = nr
end if
if( nrt>0_${ik}$ )call stdlib${ii}$_dlartv( nrt, ab( l+1, j1+kun-1 ), inca,ab( l, j1+&
kun ), inca,work( mn+j1+kun ), work( j1+kun ),kb1 )
end do
if( ml==ml0 .and. mu>mu0 ) then
if( mu<=n-i+1 ) then
! generate plane rotation to annihilate a(i,i+mu-1)
! within the band, and apply rotation from the right
call stdlib${ii}$_dlartg( ab( ku-mu+3, i+mu-2 ),ab( ku-mu+2, i+mu-1 ),work( &
mn+i+mu-1 ), work( i+mu-1 ),ra )
ab( ku-mu+3, i+mu-2 ) = ra
call stdlib${ii}$_drot( min( kl+mu-2, m-i ),ab( ku-mu+4, i+mu-2 ), 1_${ik}$,ab( ku-&
mu+3, i+mu-1 ), 1_${ik}$,work( mn+i+mu-1 ), work( i+mu-1 ) )
end if
nr = nr + 1_${ik}$
j1 = j1 - kb1
end if
if( wantpt ) then
! accumulate product of plane rotations in p**t
do j = j1, j2, kb1
call stdlib${ii}$_drot( n, pt( j+kun-1, 1_${ik}$ ), ldpt,pt( j+kun, 1_${ik}$ ), ldpt, work( &
mn+j+kun ),work( j+kun ) )
end do
end if
if( j2+kb>m ) then
! adjust j2 to keep within the bounds of the matrix
nr = nr - 1_${ik}$
j2 = j2 - kb1
end if
do j = j1, j2, kb1
! create nonzero element a(j+kl+ku,j+ku-1) below the
! band and store it in work(1:n)
work( j+kb ) = work( j+kun )*ab( klu1, j+kun )
ab( klu1, j+kun ) = work( mn+j+kun )*ab( klu1, j+kun )
end do
if( ml>ml0 ) then
ml = ml - 1_${ik}$
else
mu = mu - 1_${ik}$
end if
end do loop_80
end do loop_90
end if
if( ku==0_${ik}$ .and. kl>0_${ik}$ ) then
! a has been reduced to lower bidiagonal form
! transform lower bidiagonal form to upper bidiagonal by applying
! plane rotations from the left, storing diagonal elements in d
! and off-diagonal elements in e
do i = 1, min( m-1, n )
call stdlib${ii}$_dlartg( ab( 1_${ik}$, i ), ab( 2_${ik}$, i ), rc, rs, ra )
d( i ) = ra
if( i<n ) then
e( i ) = rs*ab( 1_${ik}$, i+1 )
ab( 1_${ik}$, i+1 ) = rc*ab( 1_${ik}$, i+1 )
end if
if( wantq )call stdlib${ii}$_drot( m, q( 1_${ik}$, i ), 1_${ik}$, q( 1_${ik}$, i+1 ), 1_${ik}$, rc, rs )
if( wantc )call stdlib${ii}$_drot( ncc, c( i, 1_${ik}$ ), ldc, c( i+1, 1_${ik}$ ), ldc, rc,rs )
end do
if( m<=n )d( m ) = ab( 1_${ik}$, m )
else if( ku>0_${ik}$ ) then
! a has been reduced to upper bidiagonal form
if( m<n ) then
! annihilate a(m,m+1) by applying plane rotations from the
! right, storing diagonal elements in d and off-diagonal
! elements in e
rb = ab( ku, m+1 )
do i = m, 1, -1
call stdlib${ii}$_dlartg( ab( ku+1, i ), rb, rc, rs, ra )
d( i ) = ra
if( i>1_${ik}$ ) then
rb = -rs*ab( ku, i )
e( i-1 ) = rc*ab( ku, i )
end if
if( wantpt )call stdlib${ii}$_drot( n, pt( i, 1_${ik}$ ), ldpt, pt( m+1, 1_${ik}$ ), ldpt,rc, rs )
end do
else
! copy off-diagonal elements to e and diagonal elements to d
do i = 1, minmn - 1
e( i ) = ab( ku, i+1 )
end do
do i = 1, minmn
d( i ) = ab( ku+1, i )
end do
end if
else
! a is diagonal. set elements of e to zero and copy diagonal
! elements to d.
do i = 1, minmn - 1
e( i ) = zero
end do
do i = 1, minmn
d( i ) = ab( 1_${ik}$, i )
end do
end if
return
end subroutine stdlib${ii}$_dgbbrd
#:for rk,rt,ri in REAL_KINDS_TYPES
#:if not rk in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ri}$gbbrd( vect, m, n, ncc, kl, ku, ab, ldab, d, e, q,ldq, pt, ldpt, c, &
!! DGBBRD: reduces a real general m-by-n band matrix A to upper
!! bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
!! The routine computes B, and optionally forms Q or P**T, or computes
!! Q**T*C for a given matrix C.
ldc, work, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${rk}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: kl, ku, ldab, ldc, ldpt, ldq, m, n, ncc
! Array Arguments
real(${rk}$), intent(inout) :: ab(ldab,*), c(ldc,*)
real(${rk}$), intent(out) :: d(*), e(*), pt(ldpt,*), q(ldq,*), work(*)
! =====================================================================
! Local Scalars
logical(lk) :: wantb, wantc, wantpt, wantq
integer(${ik}$) :: i, inca, j, j1, j2, kb, kb1, kk, klm, klu1, kun, l, minmn, ml, ml0, mn,&
mu, mu0, nr, nrt
real(${rk}$) :: ra, rb, rc, rs
! Intrinsic Functions
! Executable Statements
! test the input parameters
wantb = stdlib_lsame( vect, 'B' )
wantq = stdlib_lsame( vect, 'Q' ) .or. wantb
wantpt = stdlib_lsame( vect, 'P' ) .or. wantb
wantc = ncc>0_${ik}$
klu1 = kl + ku + 1_${ik}$
info = 0_${ik}$
if( .not.wantq .and. .not.wantpt .and. .not.stdlib_lsame( vect, 'N' ) )then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( n<0_${ik}$ ) then
info = -3_${ik}$
else if( ncc<0_${ik}$ ) then
info = -4_${ik}$
else if( kl<0_${ik}$ ) then
info = -5_${ik}$
else if( ku<0_${ik}$ ) then
info = -6_${ik}$
else if( ldab<klu1 ) then
info = -8_${ik}$
else if( ldq<1_${ik}$ .or. wantq .and. ldq<max( 1_${ik}$, m ) ) then
info = -12_${ik}$
else if( ldpt<1_${ik}$ .or. wantpt .and. ldpt<max( 1_${ik}$, n ) ) then
info = -14_${ik}$
else if( ldc<1_${ik}$ .or. wantc .and. ldc<max( 1_${ik}$, m ) ) then
info = -16_${ik}$
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DGBBRD', -info )
return
end if
! initialize q and p**t to the unit matrix, if needed
if( wantq )call stdlib${ii}$_${ri}$laset( 'FULL', m, m, zero, one, q, ldq )
if( wantpt )call stdlib${ii}$_${ri}$laset( 'FULL', n, n, zero, one, pt, ldpt )
! quick return if possible.
if( m==0 .or. n==0 )return
minmn = min( m, n )
if( kl+ku>1_${ik}$ ) then
! reduce to upper bidiagonal form if ku > 0; if ku = 0, reduce
! first to lower bidiagonal form and then transform to upper
! bidiagonal
if( ku>0_${ik}$ ) then
ml0 = 1_${ik}$
mu0 = 2_${ik}$
else
ml0 = 2_${ik}$
mu0 = 1_${ik}$
end if
! wherever possible, plane rotations are generated and applied in
! vector operations of length nr over the index set j1:j2:klu1.
! the sines of the plane rotations are stored in work(1:max(m,n))
! and the cosines in work(max(m,n)+1:2*max(m,n)).
mn = max( m, n )
klm = min( m-1, kl )
kun = min( n-1, ku )
kb = klm + kun
kb1 = kb + 1_${ik}$
inca = kb1*ldab
nr = 0_${ik}$
j1 = klm + 2_${ik}$
j2 = 1_${ik}$ - kun
loop_90: do i = 1, minmn
! reduce i-th column and i-th row of matrix to bidiagonal form
ml = klm + 1_${ik}$
mu = kun + 1_${ik}$
loop_80: do kk = 1, kb
j1 = j1 + kb
j2 = j2 + kb
! generate plane rotations to annihilate nonzero elements
! which have been created below the band
if( nr>0_${ik}$ )call stdlib${ii}$_${ri}$largv( nr, ab( klu1, j1-klm-1 ), inca,work( j1 ), kb1, &
work( mn+j1 ), kb1 )
! apply plane rotations from the left
do l = 1, kb
if( j2-klm+l-1>n ) then
nrt = nr - 1_${ik}$
else
nrt = nr
end if
if( nrt>0_${ik}$ )call stdlib${ii}$_${ri}$lartv( nrt, ab( klu1-l, j1-klm+l-1 ), inca,ab( &
klu1-l+1, j1-klm+l-1 ), inca,work( mn+j1 ), work( j1 ), kb1 )
end do
if( ml>ml0 ) then
if( ml<=m-i+1 ) then
! generate plane rotation to annihilate a(i+ml-1,i)
! within the band, and apply rotation from the left
call stdlib${ii}$_${ri}$lartg( ab( ku+ml-1, i ), ab( ku+ml, i ),work( mn+i+ml-1 ), &
work( i+ml-1 ),ra )
ab( ku+ml-1, i ) = ra
if( i<n )call stdlib${ii}$_${ri}$rot( min( ku+ml-2, n-i ),ab( ku+ml-2, i+1 ), ldab-&
1_${ik}$,ab( ku+ml-1, i+1 ), ldab-1,work( mn+i+ml-1 ), work( i+ml-1 ) )
end if
nr = nr + 1_${ik}$
j1 = j1 - kb1
end if
if( wantq ) then
! accumulate product of plane rotations in q
do j = j1, j2, kb1
call stdlib${ii}$_${ri}$rot( m, q( 1_${ik}$, j-1 ), 1_${ik}$, q( 1_${ik}$, j ), 1_${ik}$,work( mn+j ), work( j &
) )
end do
end if
if( wantc ) then
! apply plane rotations to c
do j = j1, j2, kb1
call stdlib${ii}$_${ri}$rot( ncc, c( j-1, 1_${ik}$ ), ldc, c( j, 1_${ik}$ ), ldc,work( mn+j ), &
work( j ) )
end do
end if
if( j2+kun>n ) then
! adjust j2 to keep within the bounds of the matrix
nr = nr - 1_${ik}$
j2 = j2 - kb1
end if
do j = j1, j2, kb1
! create nonzero element a(j-1,j+ku) above the band
! and store it in work(n+1:2*n)
work( j+kun ) = work( j )*ab( 1_${ik}$, j+kun )
ab( 1_${ik}$, j+kun ) = work( mn+j )*ab( 1_${ik}$, j+kun )
end do
! generate plane rotations to annihilate nonzero elements
! which have been generated above the band
if( nr>0_${ik}$ )call stdlib${ii}$_${ri}$largv( nr, ab( 1_${ik}$, j1+kun-1 ), inca,work( j1+kun ), kb1,&
work( mn+j1+kun ),kb1 )
! apply plane rotations from the right
do l = 1, kb
if( j2+l-1>m ) then
nrt = nr - 1_${ik}$
else
nrt = nr
end if
if( nrt>0_${ik}$ )call stdlib${ii}$_${ri}$lartv( nrt, ab( l+1, j1+kun-1 ), inca,ab( l, j1+&
kun ), inca,work( mn+j1+kun ), work( j1+kun ),kb1 )
end do
if( ml==ml0 .and. mu>mu0 ) then
if( mu<=n-i+1 ) then
! generate plane rotation to annihilate a(i,i+mu-1)
! within the band, and apply rotation from the right
call stdlib${ii}$_${ri}$lartg( ab( ku-mu+3, i+mu-2 ),ab( ku-mu+2, i+mu-1 ),work( &
mn+i+mu-1 ), work( i+mu-1 ),ra )
ab( ku-mu+3, i+mu-2 ) = ra
call stdlib${ii}$_${ri}$rot( min( kl+mu-2, m-i ),ab( ku-mu+4, i+mu-2 ), 1_${ik}$,ab( ku-&
mu+3, i+mu-1 ), 1_${ik}$,work( mn+i+mu-1 ), work( i+mu-1 ) )
end if
nr = nr + 1_${ik}$
j1 = j1 - kb1
end if
if( wantpt ) then
! accumulate product of plane rotations in p**t
do j = j1, j2, kb1
call stdlib${ii}$_${ri}$rot( n, pt( j+kun-1, 1_${ik}$ ), ldpt,pt( j+kun, 1_${ik}$ ), ldpt, work( &
mn+j+kun ),work( j+kun ) )
end do
end if
if( j2+kb>m ) then
! adjust j2 to keep within the bounds of the matrix
nr = nr - 1_${ik}$
j2 = j2 - kb1
end if
do j = j1, j2, kb1
! create nonzero element a(j+kl+ku,j+ku-1) below the
! band and store it in work(1:n)
work( j+kb ) = work( j+kun )*ab( klu1, j+kun )
ab( klu1, j+kun ) = work( mn+j+kun )*ab( klu1, j+kun )
end do
if( ml>ml0 ) then
ml = ml - 1_${ik}$
else
mu = mu - 1_${ik}$
end if
end do loop_80
end do loop_90
end if
if( ku==0_${ik}$ .and. kl>0_${ik}$ ) then
! a has been reduced to lower bidiagonal form
! transform lower bidiagonal form to upper bidiagonal by applying
! plane rotations from the left, storing diagonal elements in d
! and off-diagonal elements in e
do i = 1, min( m-1, n )
call stdlib${ii}$_${ri}$lartg( ab( 1_${ik}$, i ), ab( 2_${ik}$, i ), rc, rs, ra )
d( i ) = ra
if( i<n ) then
e( i ) = rs*ab( 1_${ik}$, i+1 )
ab( 1_${ik}$, i+1 ) = rc*ab( 1_${ik}$, i+1 )
end if
if( wantq )call stdlib${ii}$_${ri}$rot( m, q( 1_${ik}$, i ), 1_${ik}$, q( 1_${ik}$, i+1 ), 1_${ik}$, rc, rs )
if( wantc )call stdlib${ii}$_${ri}$rot( ncc, c( i, 1_${ik}$ ), ldc, c( i+1, 1_${ik}$ ), ldc, rc,rs )
end do
if( m<=n )d( m ) = ab( 1_${ik}$, m )
else if( ku>0_${ik}$ ) then
! a has been reduced to upper bidiagonal form
if( m<n ) then
! annihilate a(m,m+1) by applying plane rotations from the
! right, storing diagonal elements in d and off-diagonal
! elements in e
rb = ab( ku, m+1 )
do i = m, 1, -1
call stdlib${ii}$_${ri}$lartg( ab( ku+1, i ), rb, rc, rs, ra )
d( i ) = ra
if( i>1_${ik}$ ) then
rb = -rs*ab( ku, i )
e( i-1 ) = rc*ab( ku, i )
end if
if( wantpt )call stdlib${ii}$_${ri}$rot( n, pt( i, 1_${ik}$ ), ldpt, pt( m+1, 1_${ik}$ ), ldpt,rc, rs )
end do
else
! copy off-diagonal elements to e and diagonal elements to d
do i = 1, minmn - 1
e( i ) = ab( ku, i+1 )
end do
do i = 1, minmn
d( i ) = ab( ku+1, i )
end do
end if
else
! a is diagonal. set elements of e to zero and copy diagonal
! elements to d.
do i = 1, minmn - 1
e( i ) = zero
end do
do i = 1, minmn
d( i ) = ab( 1_${ik}$, i )
end do
end if
return
end subroutine stdlib${ii}$_${ri}$gbbrd
#:endif
#:endfor
pure module subroutine stdlib${ii}$_cgbbrd( vect, m, n, ncc, kl, ku, ab, ldab, d, e, q,ldq, pt, ldpt, c, &
!! CGBBRD reduces a complex general m-by-n band matrix A to real upper
!! bidiagonal form B by a unitary transformation: Q**H * A * P = B.
!! The routine computes B, and optionally forms Q or P**H, or computes
!! Q**H*C for a given matrix C.
ldc, work, rwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: kl, ku, ldab, ldc, ldpt, ldq, m, n, ncc
! Array Arguments
real(sp), intent(out) :: d(*), e(*), rwork(*)
complex(sp), intent(inout) :: ab(ldab,*), c(ldc,*)
complex(sp), intent(out) :: pt(ldpt,*), q(ldq,*), work(*)
! =====================================================================
! Local Scalars
logical(lk) :: wantb, wantc, wantpt, wantq
integer(${ik}$) :: i, inca, j, j1, j2, kb, kb1, kk, klm, klu1, kun, l, minmn, ml, ml0, mu,&
mu0, nr, nrt
real(sp) :: abst, rc
complex(sp) :: ra, rb, rs, t
! Intrinsic Functions
! Executable Statements
! test the input parameters
wantb = stdlib_lsame( vect, 'B' )
wantq = stdlib_lsame( vect, 'Q' ) .or. wantb
wantpt = stdlib_lsame( vect, 'P' ) .or. wantb
wantc = ncc>0_${ik}$
klu1 = kl + ku + 1_${ik}$
info = 0_${ik}$
if( .not.wantq .and. .not.wantpt .and. .not.stdlib_lsame( vect, 'N' ) )then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( n<0_${ik}$ ) then
info = -3_${ik}$
else if( ncc<0_${ik}$ ) then
info = -4_${ik}$
else if( kl<0_${ik}$ ) then
info = -5_${ik}$
else if( ku<0_${ik}$ ) then
info = -6_${ik}$
else if( ldab<klu1 ) then
info = -8_${ik}$
else if( ldq<1_${ik}$ .or. wantq .and. ldq<max( 1_${ik}$, m ) ) then
info = -12_${ik}$
else if( ldpt<1_${ik}$ .or. wantpt .and. ldpt<max( 1_${ik}$, n ) ) then
info = -14_${ik}$
else if( ldc<1_${ik}$ .or. wantc .and. ldc<max( 1_${ik}$, m ) ) then
info = -16_${ik}$
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'CGBBRD', -info )
return
end if
! initialize q and p**h to the unit matrix, if needed
if( wantq )call stdlib${ii}$_claset( 'FULL', m, m, czero, cone, q, ldq )
if( wantpt )call stdlib${ii}$_claset( 'FULL', n, n, czero, cone, pt, ldpt )
! quick return if possible.
if( m==0 .or. n==0 )return
minmn = min( m, n )
if( kl+ku>1_${ik}$ ) then
! reduce to upper bidiagonal form if ku > 0; if ku = 0, reduce
! first to lower bidiagonal form and then transform to upper
! bidiagonal
if( ku>0_${ik}$ ) then
ml0 = 1_${ik}$
mu0 = 2_${ik}$
else
ml0 = 2_${ik}$
mu0 = 1_${ik}$
end if
! wherever possible, plane rotations are generated and applied in
! vector operations of length nr over the index set j1:j2:klu1.
! the complex sines of the plane rotations are stored in work,
! and the real cosines in rwork.
klm = min( m-1, kl )
kun = min( n-1, ku )
kb = klm + kun
kb1 = kb + 1_${ik}$
inca = kb1*ldab
nr = 0_${ik}$
j1 = klm + 2_${ik}$
j2 = 1_${ik}$ - kun
loop_90: do i = 1, minmn
! reduce i-th column and i-th row of matrix to bidiagonal form
ml = klm + 1_${ik}$
mu = kun + 1_${ik}$
loop_80: do kk = 1, kb
j1 = j1 + kb
j2 = j2 + kb
! generate plane rotations to annihilate nonzero elements
! which have been created below the band
if( nr>0_${ik}$ )call stdlib${ii}$_clargv( nr, ab( klu1, j1-klm-1 ), inca,work( j1 ), kb1, &
rwork( j1 ), kb1 )
! apply plane rotations from the left
do l = 1, kb
if( j2-klm+l-1>n ) then
nrt = nr - 1_${ik}$
else
nrt = nr
end if
if( nrt>0_${ik}$ )call stdlib${ii}$_clartv( nrt, ab( klu1-l, j1-klm+l-1 ), inca,ab( &
klu1-l+1, j1-klm+l-1 ), inca,rwork( j1 ), work( j1 ), kb1 )
end do
if( ml>ml0 ) then
if( ml<=m-i+1 ) then
! generate plane rotation to annihilate a(i+ml-1,i)
! within the band, and apply rotation from the left
call stdlib${ii}$_clartg( ab( ku+ml-1, i ), ab( ku+ml, i ),rwork( i+ml-1 ), &
work( i+ml-1 ), ra )
ab( ku+ml-1, i ) = ra
if( i<n )call stdlib${ii}$_crot( min( ku+ml-2, n-i ),ab( ku+ml-2, i+1 ), ldab-&
1_${ik}$,ab( ku+ml-1, i+1 ), ldab-1,rwork( i+ml-1 ), work( i+ml-1 ) )
end if
nr = nr + 1_${ik}$
j1 = j1 - kb1
end if
if( wantq ) then
! accumulate product of plane rotations in q
do j = j1, j2, kb1
call stdlib${ii}$_crot( m, q( 1_${ik}$, j-1 ), 1_${ik}$, q( 1_${ik}$, j ), 1_${ik}$,rwork( j ), conjg( &
work( j ) ) )
end do
end if
if( wantc ) then
! apply plane rotations to c
do j = j1, j2, kb1
call stdlib${ii}$_crot( ncc, c( j-1, 1_${ik}$ ), ldc, c( j, 1_${ik}$ ), ldc,rwork( j ), &
work( j ) )
end do
end if
if( j2+kun>n ) then
! adjust j2 to keep within the bounds of the matrix
nr = nr - 1_${ik}$
j2 = j2 - kb1
end if
do j = j1, j2, kb1
! create nonzero element a(j-1,j+ku) above the band
! and store it in work(n+1:2*n)
work( j+kun ) = work( j )*ab( 1_${ik}$, j+kun )
ab( 1_${ik}$, j+kun ) = rwork( j )*ab( 1_${ik}$, j+kun )
end do
! generate plane rotations to annihilate nonzero elements
! which have been generated above the band
if( nr>0_${ik}$ )call stdlib${ii}$_clargv( nr, ab( 1_${ik}$, j1+kun-1 ), inca,work( j1+kun ), kb1,&
rwork( j1+kun ),kb1 )
! apply plane rotations from the right
do l = 1, kb
if( j2+l-1>m ) then
nrt = nr - 1_${ik}$
else
nrt = nr
end if
if( nrt>0_${ik}$ )call stdlib${ii}$_clartv( nrt, ab( l+1, j1+kun-1 ), inca,ab( l, j1+&
kun ), inca,rwork( j1+kun ), work( j1+kun ), kb1 )
end do
if( ml==ml0 .and. mu>mu0 ) then
if( mu<=n-i+1 ) then
! generate plane rotation to annihilate a(i,i+mu-1)
! within the band, and apply rotation from the right
call stdlib${ii}$_clartg( ab( ku-mu+3, i+mu-2 ),ab( ku-mu+2, i+mu-1 ),rwork( &
i+mu-1 ), work( i+mu-1 ), ra )
ab( ku-mu+3, i+mu-2 ) = ra
call stdlib${ii}$_crot( min( kl+mu-2, m-i ),ab( ku-mu+4, i+mu-2 ), 1_${ik}$,ab( ku-&
mu+3, i+mu-1 ), 1_${ik}$,rwork( i+mu-1 ), work( i+mu-1 ) )
end if
nr = nr + 1_${ik}$
j1 = j1 - kb1
end if
if( wantpt ) then
! accumulate product of plane rotations in p**h
do j = j1, j2, kb1
call stdlib${ii}$_crot( n, pt( j+kun-1, 1_${ik}$ ), ldpt,pt( j+kun, 1_${ik}$ ), ldpt, rwork(&
j+kun ),conjg( work( j+kun ) ) )
end do
end if
if( j2+kb>m ) then
! adjust j2 to keep within the bounds of the matrix
nr = nr - 1_${ik}$
j2 = j2 - kb1
end if
do j = j1, j2, kb1
! create nonzero element a(j+kl+ku,j+ku-1) below the
! band and store it in work(1:n)
work( j+kb ) = work( j+kun )*ab( klu1, j+kun )
ab( klu1, j+kun ) = rwork( j+kun )*ab( klu1, j+kun )
end do
if( ml>ml0 ) then
ml = ml - 1_${ik}$
else
mu = mu - 1_${ik}$
end if
end do loop_80
end do loop_90
end if
if( ku==0_${ik}$ .and. kl>0_${ik}$ ) then
! a has been reduced to complex lower bidiagonal form
! transform lower bidiagonal form to upper bidiagonal by applying
! plane rotations from the left, overwriting superdiagonal
! elements on subdiagonal elements
do i = 1, min( m-1, n )
call stdlib${ii}$_clartg( ab( 1_${ik}$, i ), ab( 2_${ik}$, i ), rc, rs, ra )
ab( 1_${ik}$, i ) = ra
if( i<n ) then
ab( 2_${ik}$, i ) = rs*ab( 1_${ik}$, i+1 )
ab( 1_${ik}$, i+1 ) = rc*ab( 1_${ik}$, i+1 )
end if
if( wantq )call stdlib${ii}$_crot( m, q( 1_${ik}$, i ), 1_${ik}$, q( 1_${ik}$, i+1 ), 1_${ik}$, rc,conjg( rs ) )
if( wantc )call stdlib${ii}$_crot( ncc, c( i, 1_${ik}$ ), ldc, c( i+1, 1_${ik}$ ), ldc, rc,rs )
end do
else
! a has been reduced to complex upper bidiagonal form or is
! diagonal
if( ku>0_${ik}$ .and. m<n ) then
! annihilate a(m,m+1) by applying plane rotations from the
! right
rb = ab( ku, m+1 )
do i = m, 1, -1
call stdlib${ii}$_clartg( ab( ku+1, i ), rb, rc, rs, ra )
ab( ku+1, i ) = ra
if( i>1_${ik}$ ) then
rb = -conjg( rs )*ab( ku, i )
ab( ku, i ) = rc*ab( ku, i )
end if
if( wantpt )call stdlib${ii}$_crot( n, pt( i, 1_${ik}$ ), ldpt, pt( m+1, 1_${ik}$ ), ldpt,rc, &
conjg( rs ) )
end do
end if
end if
! make diagonal and superdiagonal elements real, storing them in d
! and e
t = ab( ku+1, 1_${ik}$ )
loop_120: do i = 1, minmn
abst = abs( t )
d( i ) = abst
if( abst/=zero ) then
t = t / abst
else
t = cone
end if
if( wantq )call stdlib${ii}$_cscal( m, t, q( 1_${ik}$, i ), 1_${ik}$ )
if( wantc )call stdlib${ii}$_cscal( ncc, conjg( t ), c( i, 1_${ik}$ ), ldc )
if( i<minmn ) then
if( ku==0_${ik}$ .and. kl==0_${ik}$ ) then
e( i ) = zero
t = ab( 1_${ik}$, i+1 )
else
if( ku==0_${ik}$ ) then
t = ab( 2_${ik}$, i )*conjg( t )
else
t = ab( ku, i+1 )*conjg( t )
end if
abst = abs( t )
e( i ) = abst
if( abst/=zero ) then
t = t / abst
else
t = cone
end if
if( wantpt )call stdlib${ii}$_cscal( n, t, pt( i+1, 1_${ik}$ ), ldpt )
t = ab( ku+1, i+1 )*conjg( t )
end if
end if
end do loop_120
return
end subroutine stdlib${ii}$_cgbbrd
pure module subroutine stdlib${ii}$_zgbbrd( vect, m, n, ncc, kl, ku, ab, ldab, d, e, q,ldq, pt, ldpt, c, &
!! ZGBBRD reduces a complex general m-by-n band matrix A to real upper
!! bidiagonal form B by a unitary transformation: Q**H * A * P = B.
!! The routine computes B, and optionally forms Q or P**H, or computes
!! Q**H*C for a given matrix C.
ldc, work, rwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: kl, ku, ldab, ldc, ldpt, ldq, m, n, ncc
! Array Arguments
real(dp), intent(out) :: d(*), e(*), rwork(*)
complex(dp), intent(inout) :: ab(ldab,*), c(ldc,*)
complex(dp), intent(out) :: pt(ldpt,*), q(ldq,*), work(*)
! =====================================================================
! Local Scalars
logical(lk) :: wantb, wantc, wantpt, wantq
integer(${ik}$) :: i, inca, j, j1, j2, kb, kb1, kk, klm, klu1, kun, l, minmn, ml, ml0, mu,&
mu0, nr, nrt
real(dp) :: abst, rc
complex(dp) :: ra, rb, rs, t
! Intrinsic Functions
! Executable Statements
! test the input parameters
wantb = stdlib_lsame( vect, 'B' )
wantq = stdlib_lsame( vect, 'Q' ) .or. wantb
wantpt = stdlib_lsame( vect, 'P' ) .or. wantb
wantc = ncc>0_${ik}$
klu1 = kl + ku + 1_${ik}$
info = 0_${ik}$
if( .not.wantq .and. .not.wantpt .and. .not.stdlib_lsame( vect, 'N' ) )then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( n<0_${ik}$ ) then
info = -3_${ik}$
else if( ncc<0_${ik}$ ) then
info = -4_${ik}$
else if( kl<0_${ik}$ ) then
info = -5_${ik}$
else if( ku<0_${ik}$ ) then
info = -6_${ik}$
else if( ldab<klu1 ) then
info = -8_${ik}$
else if( ldq<1_${ik}$ .or. wantq .and. ldq<max( 1_${ik}$, m ) ) then
info = -12_${ik}$
else if( ldpt<1_${ik}$ .or. wantpt .and. ldpt<max( 1_${ik}$, n ) ) then
info = -14_${ik}$
else if( ldc<1_${ik}$ .or. wantc .and. ldc<max( 1_${ik}$, m ) ) then
info = -16_${ik}$
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZGBBRD', -info )
return
end if
! initialize q and p**h to the unit matrix, if needed
if( wantq )call stdlib${ii}$_zlaset( 'FULL', m, m, czero, cone, q, ldq )
if( wantpt )call stdlib${ii}$_zlaset( 'FULL', n, n, czero, cone, pt, ldpt )
! quick return if possible.
if( m==0 .or. n==0 )return
minmn = min( m, n )
if( kl+ku>1_${ik}$ ) then
! reduce to upper bidiagonal form if ku > 0; if ku = 0, reduce
! first to lower bidiagonal form and then transform to upper
! bidiagonal
if( ku>0_${ik}$ ) then
ml0 = 1_${ik}$
mu0 = 2_${ik}$
else
ml0 = 2_${ik}$
mu0 = 1_${ik}$
end if
! wherever possible, plane rotations are generated and applied in
! vector operations of length nr over the index set j1:j2:klu1.
! the complex sines of the plane rotations are stored in work,
! and the real cosines in rwork.
klm = min( m-1, kl )
kun = min( n-1, ku )
kb = klm + kun
kb1 = kb + 1_${ik}$
inca = kb1*ldab
nr = 0_${ik}$
j1 = klm + 2_${ik}$
j2 = 1_${ik}$ - kun
loop_90: do i = 1, minmn
! reduce i-th column and i-th row of matrix to bidiagonal form
ml = klm + 1_${ik}$
mu = kun + 1_${ik}$
loop_80: do kk = 1, kb
j1 = j1 + kb
j2 = j2 + kb
! generate plane rotations to annihilate nonzero elements
! which have been created below the band
if( nr>0_${ik}$ )call stdlib${ii}$_zlargv( nr, ab( klu1, j1-klm-1 ), inca,work( j1 ), kb1, &
rwork( j1 ), kb1 )
! apply plane rotations from the left
do l = 1, kb
if( j2-klm+l-1>n ) then
nrt = nr - 1_${ik}$
else
nrt = nr
end if
if( nrt>0_${ik}$ )call stdlib${ii}$_zlartv( nrt, ab( klu1-l, j1-klm+l-1 ), inca,ab( &
klu1-l+1, j1-klm+l-1 ), inca,rwork( j1 ), work( j1 ), kb1 )
end do
if( ml>ml0 ) then
if( ml<=m-i+1 ) then
! generate plane rotation to annihilate a(i+ml-1,i)
! within the band, and apply rotation from the left
call stdlib${ii}$_zlartg( ab( ku+ml-1, i ), ab( ku+ml, i ),rwork( i+ml-1 ), &
work( i+ml-1 ), ra )
ab( ku+ml-1, i ) = ra
if( i<n )call stdlib${ii}$_zrot( min( ku+ml-2, n-i ),ab( ku+ml-2, i+1 ), ldab-&
1_${ik}$,ab( ku+ml-1, i+1 ), ldab-1,rwork( i+ml-1 ), work( i+ml-1 ) )
end if
nr = nr + 1_${ik}$
j1 = j1 - kb1
end if
if( wantq ) then
! accumulate product of plane rotations in q
do j = j1, j2, kb1
call stdlib${ii}$_zrot( m, q( 1_${ik}$, j-1 ), 1_${ik}$, q( 1_${ik}$, j ), 1_${ik}$,rwork( j ), conjg( &
work( j ) ) )
end do
end if
if( wantc ) then
! apply plane rotations to c
do j = j1, j2, kb1
call stdlib${ii}$_zrot( ncc, c( j-1, 1_${ik}$ ), ldc, c( j, 1_${ik}$ ), ldc,rwork( j ), &
work( j ) )
end do
end if
if( j2+kun>n ) then
! adjust j2 to keep within the bounds of the matrix
nr = nr - 1_${ik}$
j2 = j2 - kb1
end if
do j = j1, j2, kb1
! create nonzero element a(j-1,j+ku) above the band
! and store it in work(n+1:2*n)
work( j+kun ) = work( j )*ab( 1_${ik}$, j+kun )
ab( 1_${ik}$, j+kun ) = rwork( j )*ab( 1_${ik}$, j+kun )
end do
! generate plane rotations to annihilate nonzero elements
! which have been generated above the band
if( nr>0_${ik}$ )call stdlib${ii}$_zlargv( nr, ab( 1_${ik}$, j1+kun-1 ), inca,work( j1+kun ), kb1,&
rwork( j1+kun ),kb1 )
! apply plane rotations from the right
do l = 1, kb
if( j2+l-1>m ) then
nrt = nr - 1_${ik}$
else
nrt = nr
end if
if( nrt>0_${ik}$ )call stdlib${ii}$_zlartv( nrt, ab( l+1, j1+kun-1 ), inca,ab( l, j1+&
kun ), inca,rwork( j1+kun ), work( j1+kun ), kb1 )
end do
if( ml==ml0 .and. mu>mu0 ) then
if( mu<=n-i+1 ) then
! generate plane rotation to annihilate a(i,i+mu-1)
! within the band, and apply rotation from the right
call stdlib${ii}$_zlartg( ab( ku-mu+3, i+mu-2 ),ab( ku-mu+2, i+mu-1 ),rwork( &
i+mu-1 ), work( i+mu-1 ), ra )
ab( ku-mu+3, i+mu-2 ) = ra
call stdlib${ii}$_zrot( min( kl+mu-2, m-i ),ab( ku-mu+4, i+mu-2 ), 1_${ik}$,ab( ku-&
mu+3, i+mu-1 ), 1_${ik}$,rwork( i+mu-1 ), work( i+mu-1 ) )
end if
nr = nr + 1_${ik}$
j1 = j1 - kb1
end if
if( wantpt ) then
! accumulate product of plane rotations in p**h
do j = j1, j2, kb1
call stdlib${ii}$_zrot( n, pt( j+kun-1, 1_${ik}$ ), ldpt,pt( j+kun, 1_${ik}$ ), ldpt, rwork(&
j+kun ),conjg( work( j+kun ) ) )
end do
end if
if( j2+kb>m ) then
! adjust j2 to keep within the bounds of the matrix
nr = nr - 1_${ik}$
j2 = j2 - kb1
end if
do j = j1, j2, kb1
! create nonzero element a(j+kl+ku,j+ku-1) below the
! band and store it in work(1:n)
work( j+kb ) = work( j+kun )*ab( klu1, j+kun )
ab( klu1, j+kun ) = rwork( j+kun )*ab( klu1, j+kun )
end do
if( ml>ml0 ) then
ml = ml - 1_${ik}$
else
mu = mu - 1_${ik}$
end if
end do loop_80
end do loop_90
end if
if( ku==0_${ik}$ .and. kl>0_${ik}$ ) then
! a has been reduced to complex lower bidiagonal form
! transform lower bidiagonal form to upper bidiagonal by applying
! plane rotations from the left, overwriting superdiagonal
! elements on subdiagonal elements
do i = 1, min( m-1, n )
call stdlib${ii}$_zlartg( ab( 1_${ik}$, i ), ab( 2_${ik}$, i ), rc, rs, ra )
ab( 1_${ik}$, i ) = ra
if( i<n ) then
ab( 2_${ik}$, i ) = rs*ab( 1_${ik}$, i+1 )
ab( 1_${ik}$, i+1 ) = rc*ab( 1_${ik}$, i+1 )
end if
if( wantq )call stdlib${ii}$_zrot( m, q( 1_${ik}$, i ), 1_${ik}$, q( 1_${ik}$, i+1 ), 1_${ik}$, rc,conjg( rs ) )
if( wantc )call stdlib${ii}$_zrot( ncc, c( i, 1_${ik}$ ), ldc, c( i+1, 1_${ik}$ ), ldc, rc,rs )
end do
else
! a has been reduced to complex upper bidiagonal form or is
! diagonal
if( ku>0_${ik}$ .and. m<n ) then
! annihilate a(m,m+1) by applying plane rotations from the
! right
rb = ab( ku, m+1 )
do i = m, 1, -1
call stdlib${ii}$_zlartg( ab( ku+1, i ), rb, rc, rs, ra )
ab( ku+1, i ) = ra
if( i>1_${ik}$ ) then
rb = -conjg( rs )*ab( ku, i )
ab( ku, i ) = rc*ab( ku, i )
end if
if( wantpt )call stdlib${ii}$_zrot( n, pt( i, 1_${ik}$ ), ldpt, pt( m+1, 1_${ik}$ ), ldpt,rc, &
conjg( rs ) )
end do
end if
end if
! make diagonal and superdiagonal elements real, storing them in d
! and e
t = ab( ku+1, 1_${ik}$ )
loop_120: do i = 1, minmn
abst = abs( t )
d( i ) = abst
if( abst/=zero ) then
t = t / abst
else
t = cone
end if
if( wantq )call stdlib${ii}$_zscal( m, t, q( 1_${ik}$, i ), 1_${ik}$ )
if( wantc )call stdlib${ii}$_zscal( ncc, conjg( t ), c( i, 1_${ik}$ ), ldc )
if( i<minmn ) then
if( ku==0_${ik}$ .and. kl==0_${ik}$ ) then
e( i ) = zero
t = ab( 1_${ik}$, i+1 )
else
if( ku==0_${ik}$ ) then
t = ab( 2_${ik}$, i )*conjg( t )
else
t = ab( ku, i+1 )*conjg( t )
end if
abst = abs( t )
e( i ) = abst
if( abst/=zero ) then
t = t / abst
else
t = cone
end if
if( wantpt )call stdlib${ii}$_zscal( n, t, pt( i+1, 1_${ik}$ ), ldpt )
t = ab( ku+1, i+1 )*conjg( t )
end if
end if
end do loop_120
return
end subroutine stdlib${ii}$_zgbbrd
#:for ck,ct,ci in CMPLX_KINDS_TYPES
#:if not ck in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ci}$gbbrd( vect, m, n, ncc, kl, ku, ab, ldab, d, e, q,ldq, pt, ldpt, c, &
!! ZGBBRD: reduces a complex general m-by-n band matrix A to real upper
!! bidiagonal form B by a unitary transformation: Q**H * A * P = B.
!! The routine computes B, and optionally forms Q or P**H, or computes
!! Q**H*C for a given matrix C.
ldc, work, rwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${ck}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: kl, ku, ldab, ldc, ldpt, ldq, m, n, ncc
! Array Arguments
real(${ck}$), intent(out) :: d(*), e(*), rwork(*)
complex(${ck}$), intent(inout) :: ab(ldab,*), c(ldc,*)
complex(${ck}$), intent(out) :: pt(ldpt,*), q(ldq,*), work(*)
! =====================================================================
! Local Scalars
logical(lk) :: wantb, wantc, wantpt, wantq
integer(${ik}$) :: i, inca, j, j1, j2, kb, kb1, kk, klm, klu1, kun, l, minmn, ml, ml0, mu,&
mu0, nr, nrt
real(${ck}$) :: abst, rc
complex(${ck}$) :: ra, rb, rs, t
! Intrinsic Functions
! Executable Statements
! test the input parameters
wantb = stdlib_lsame( vect, 'B' )
wantq = stdlib_lsame( vect, 'Q' ) .or. wantb
wantpt = stdlib_lsame( vect, 'P' ) .or. wantb
wantc = ncc>0_${ik}$
klu1 = kl + ku + 1_${ik}$
info = 0_${ik}$
if( .not.wantq .and. .not.wantpt .and. .not.stdlib_lsame( vect, 'N' ) )then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( n<0_${ik}$ ) then
info = -3_${ik}$
else if( ncc<0_${ik}$ ) then
info = -4_${ik}$
else if( kl<0_${ik}$ ) then
info = -5_${ik}$
else if( ku<0_${ik}$ ) then
info = -6_${ik}$
else if( ldab<klu1 ) then
info = -8_${ik}$
else if( ldq<1_${ik}$ .or. wantq .and. ldq<max( 1_${ik}$, m ) ) then
info = -12_${ik}$
else if( ldpt<1_${ik}$ .or. wantpt .and. ldpt<max( 1_${ik}$, n ) ) then
info = -14_${ik}$
else if( ldc<1_${ik}$ .or. wantc .and. ldc<max( 1_${ik}$, m ) ) then
info = -16_${ik}$
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZGBBRD', -info )
return
end if
! initialize q and p**h to the unit matrix, if needed
if( wantq )call stdlib${ii}$_${ci}$laset( 'FULL', m, m, czero, cone, q, ldq )
if( wantpt )call stdlib${ii}$_${ci}$laset( 'FULL', n, n, czero, cone, pt, ldpt )
! quick return if possible.
if( m==0 .or. n==0 )return
minmn = min( m, n )
if( kl+ku>1_${ik}$ ) then
! reduce to upper bidiagonal form if ku > 0; if ku = 0, reduce
! first to lower bidiagonal form and then transform to upper
! bidiagonal
if( ku>0_${ik}$ ) then
ml0 = 1_${ik}$
mu0 = 2_${ik}$
else
ml0 = 2_${ik}$
mu0 = 1_${ik}$
end if
! wherever possible, plane rotations are generated and applied in
! vector operations of length nr over the index set j1:j2:klu1.
! the complex sines of the plane rotations are stored in work,
! and the real cosines in rwork.
klm = min( m-1, kl )
kun = min( n-1, ku )
kb = klm + kun
kb1 = kb + 1_${ik}$
inca = kb1*ldab
nr = 0_${ik}$
j1 = klm + 2_${ik}$
j2 = 1_${ik}$ - kun
loop_90: do i = 1, minmn
! reduce i-th column and i-th row of matrix to bidiagonal form
ml = klm + 1_${ik}$
mu = kun + 1_${ik}$
loop_80: do kk = 1, kb
j1 = j1 + kb
j2 = j2 + kb
! generate plane rotations to annihilate nonzero elements
! which have been created below the band
if( nr>0_${ik}$ )call stdlib${ii}$_${ci}$largv( nr, ab( klu1, j1-klm-1 ), inca,work( j1 ), kb1, &
rwork( j1 ), kb1 )
! apply plane rotations from the left
do l = 1, kb
if( j2-klm+l-1>n ) then
nrt = nr - 1_${ik}$
else
nrt = nr
end if
if( nrt>0_${ik}$ )call stdlib${ii}$_${ci}$lartv( nrt, ab( klu1-l, j1-klm+l-1 ), inca,ab( &
klu1-l+1, j1-klm+l-1 ), inca,rwork( j1 ), work( j1 ), kb1 )
end do
if( ml>ml0 ) then
if( ml<=m-i+1 ) then
! generate plane rotation to annihilate a(i+ml-1,i)
! within the band, and apply rotation from the left
call stdlib${ii}$_${ci}$lartg( ab( ku+ml-1, i ), ab( ku+ml, i ),rwork( i+ml-1 ), &
work( i+ml-1 ), ra )
ab( ku+ml-1, i ) = ra
if( i<n )call stdlib${ii}$_${ci}$rot( min( ku+ml-2, n-i ),ab( ku+ml-2, i+1 ), ldab-&
1_${ik}$,ab( ku+ml-1, i+1 ), ldab-1,rwork( i+ml-1 ), work( i+ml-1 ) )
end if
nr = nr + 1_${ik}$
j1 = j1 - kb1
end if
if( wantq ) then
! accumulate product of plane rotations in q
do j = j1, j2, kb1
call stdlib${ii}$_${ci}$rot( m, q( 1_${ik}$, j-1 ), 1_${ik}$, q( 1_${ik}$, j ), 1_${ik}$,rwork( j ), conjg( &
work( j ) ) )
end do
end if
if( wantc ) then
! apply plane rotations to c
do j = j1, j2, kb1
call stdlib${ii}$_${ci}$rot( ncc, c( j-1, 1_${ik}$ ), ldc, c( j, 1_${ik}$ ), ldc,rwork( j ), &
work( j ) )
end do
end if
if( j2+kun>n ) then
! adjust j2 to keep within the bounds of the matrix
nr = nr - 1_${ik}$
j2 = j2 - kb1
end if
do j = j1, j2, kb1
! create nonzero element a(j-1,j+ku) above the band
! and store it in work(n+1:2*n)
work( j+kun ) = work( j )*ab( 1_${ik}$, j+kun )
ab( 1_${ik}$, j+kun ) = rwork( j )*ab( 1_${ik}$, j+kun )
end do
! generate plane rotations to annihilate nonzero elements
! which have been generated above the band
if( nr>0_${ik}$ )call stdlib${ii}$_${ci}$largv( nr, ab( 1_${ik}$, j1+kun-1 ), inca,work( j1+kun ), kb1,&
rwork( j1+kun ),kb1 )
! apply plane rotations from the right
do l = 1, kb
if( j2+l-1>m ) then
nrt = nr - 1_${ik}$
else
nrt = nr
end if
if( nrt>0_${ik}$ )call stdlib${ii}$_${ci}$lartv( nrt, ab( l+1, j1+kun-1 ), inca,ab( l, j1+&
kun ), inca,rwork( j1+kun ), work( j1+kun ), kb1 )
end do
if( ml==ml0 .and. mu>mu0 ) then
if( mu<=n-i+1 ) then
! generate plane rotation to annihilate a(i,i+mu-1)
! within the band, and apply rotation from the right
call stdlib${ii}$_${ci}$lartg( ab( ku-mu+3, i+mu-2 ),ab( ku-mu+2, i+mu-1 ),rwork( &
i+mu-1 ), work( i+mu-1 ), ra )
ab( ku-mu+3, i+mu-2 ) = ra
call stdlib${ii}$_${ci}$rot( min( kl+mu-2, m-i ),ab( ku-mu+4, i+mu-2 ), 1_${ik}$,ab( ku-&
mu+3, i+mu-1 ), 1_${ik}$,rwork( i+mu-1 ), work( i+mu-1 ) )
end if
nr = nr + 1_${ik}$
j1 = j1 - kb1
end if
if( wantpt ) then
! accumulate product of plane rotations in p**h
do j = j1, j2, kb1
call stdlib${ii}$_${ci}$rot( n, pt( j+kun-1, 1_${ik}$ ), ldpt,pt( j+kun, 1_${ik}$ ), ldpt, rwork(&
j+kun ),conjg( work( j+kun ) ) )
end do
end if
if( j2+kb>m ) then
! adjust j2 to keep within the bounds of the matrix
nr = nr - 1_${ik}$
j2 = j2 - kb1
end if
do j = j1, j2, kb1
! create nonzero element a(j+kl+ku,j+ku-1) below the
! band and store it in work(1:n)
work( j+kb ) = work( j+kun )*ab( klu1, j+kun )
ab( klu1, j+kun ) = rwork( j+kun )*ab( klu1, j+kun )
end do
if( ml>ml0 ) then
ml = ml - 1_${ik}$
else
mu = mu - 1_${ik}$
end if
end do loop_80
end do loop_90
end if
if( ku==0_${ik}$ .and. kl>0_${ik}$ ) then
! a has been reduced to complex lower bidiagonal form
! transform lower bidiagonal form to upper bidiagonal by applying
! plane rotations from the left, overwriting superdiagonal
! elements on subdiagonal elements
do i = 1, min( m-1, n )
call stdlib${ii}$_${ci}$lartg( ab( 1_${ik}$, i ), ab( 2_${ik}$, i ), rc, rs, ra )
ab( 1_${ik}$, i ) = ra
if( i<n ) then
ab( 2_${ik}$, i ) = rs*ab( 1_${ik}$, i+1 )
ab( 1_${ik}$, i+1 ) = rc*ab( 1_${ik}$, i+1 )
end if
if( wantq )call stdlib${ii}$_${ci}$rot( m, q( 1_${ik}$, i ), 1_${ik}$, q( 1_${ik}$, i+1 ), 1_${ik}$, rc,conjg( rs ) )
if( wantc )call stdlib${ii}$_${ci}$rot( ncc, c( i, 1_${ik}$ ), ldc, c( i+1, 1_${ik}$ ), ldc, rc,rs )
end do
else
! a has been reduced to complex upper bidiagonal form or is
! diagonal
if( ku>0_${ik}$ .and. m<n ) then
! annihilate a(m,m+1) by applying plane rotations from the
! right
rb = ab( ku, m+1 )
do i = m, 1, -1
call stdlib${ii}$_${ci}$lartg( ab( ku+1, i ), rb, rc, rs, ra )
ab( ku+1, i ) = ra
if( i>1_${ik}$ ) then
rb = -conjg( rs )*ab( ku, i )
ab( ku, i ) = rc*ab( ku, i )
end if
if( wantpt )call stdlib${ii}$_${ci}$rot( n, pt( i, 1_${ik}$ ), ldpt, pt( m+1, 1_${ik}$ ), ldpt,rc, &
conjg( rs ) )
end do
end if
end if
! make diagonal and superdiagonal elements real, storing them in d
! and e
t = ab( ku+1, 1_${ik}$ )
loop_120: do i = 1, minmn
abst = abs( t )
d( i ) = abst
if( abst/=zero ) then
t = t / abst
else
t = cone
end if
if( wantq )call stdlib${ii}$_${ci}$scal( m, t, q( 1_${ik}$, i ), 1_${ik}$ )
if( wantc )call stdlib${ii}$_${ci}$scal( ncc, conjg( t ), c( i, 1_${ik}$ ), ldc )
if( i<minmn ) then
if( ku==0_${ik}$ .and. kl==0_${ik}$ ) then
e( i ) = zero
t = ab( 1_${ik}$, i+1 )
else
if( ku==0_${ik}$ ) then
t = ab( 2_${ik}$, i )*conjg( t )
else
t = ab( ku, i+1 )*conjg( t )
end if
abst = abs( t )
e( i ) = abst
if( abst/=zero ) then
t = t / abst
else
t = cone
end if
if( wantpt )call stdlib${ii}$_${ci}$scal( n, t, pt( i+1, 1_${ik}$ ), ldpt )
t = ab( ku+1, i+1 )*conjg( t )
end if
end if
end do loop_120
return
end subroutine stdlib${ii}$_${ci}$gbbrd
#:endif
#:endfor
pure module subroutine stdlib${ii}$_sgsvj0( jobv, m, n, a, lda, d, sva, mv, v, ldv, eps,sfmin, tol, &
!! SGSVJ0 is called from SGESVJ as a pre-processor and that is its main
!! purpose. It applies Jacobi rotations in the same way as SGESVJ does, but
!! it does not check convergence (stopping criterion). Few tuning
!! parameters (marked by [TP]) are available for the implementer.
nsweep, work, lwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, ldv, lwork, m, mv, n, nsweep
real(sp), intent(in) :: eps, sfmin, tol
character, intent(in) :: jobv
! Array Arguments
real(sp), intent(inout) :: a(lda,*), sva(n), d(n), v(ldv,*)
real(sp), intent(out) :: work(lwork)
! =====================================================================
! Local Scalars
real(sp) :: aapp, aapp0, aapq, aaqq, apoaq, aqoap, big, bigtheta, cs, mxaapq, mxsinj, &
rootbig, rooteps, rootsfmin, roottol, small, sn, t, temp1, theta, thsign
integer(${ik}$) :: blskip, emptsw, i, ibr, ierr, igl, ijblsk, ir1, iswrot, jbc, jgl, kbl, &
lkahead, mvl, nbl, notrot, p, pskipped, q, rowskip, swband
logical(lk) :: applv, rotok, rsvec
! Local Arrays
real(sp) :: fastr(5_${ik}$)
! Intrinsic Functions
! Executable Statements
! test the input parameters.
applv = stdlib_lsame( jobv, 'A' )
rsvec = stdlib_lsame( jobv, 'V' )
if( .not.( rsvec .or. applv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( ( n<0_${ik}$ ) .or. ( n>m ) ) then
info = -3_${ik}$
else if( lda<m ) then
info = -5_${ik}$
else if( ( rsvec.or.applv ) .and. ( mv<0_${ik}$ ) ) then
info = -8_${ik}$
else if( ( rsvec.and.( ldv<n ) ).or.( applv.and.( ldv<mv ) ) ) then
info = -10_${ik}$
else if( tol<=eps ) then
info = -13_${ik}$
else if( nsweep<0_${ik}$ ) then
info = -14_${ik}$
else if( lwork<m ) then
info = -16_${ik}$
else
info = 0_${ik}$
end if
! #:(
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'SGSVJ0', -info )
return
end if
if( rsvec ) then
mvl = n
else if( applv ) then
mvl = mv
end if
rsvec = rsvec .or. applv
rooteps = sqrt( eps )
rootsfmin = sqrt( sfmin )
small = sfmin / eps
big = one / sfmin
rootbig = one / rootsfmin
bigtheta = one / rooteps
roottol = sqrt( tol )
! .. row-cyclic jacobi svd algorithm with column pivoting ..
emptsw = ( n*( n-1 ) ) / 2_${ik}$
notrot = 0_${ik}$
fastr( 1_${ik}$ ) = zero
! .. row-cyclic pivot strategy with de rijk's pivoting ..
swband = 0_${ik}$
! [tp] swband is a tuning parameter. it is meaningful and effective
! if stdlib${ii}$_sgesvj is used as a computational routine in the preconditioned
! jacobi svd algorithm stdlib${ii}$_sgesvj. for sweeps i=1:swband the procedure
! ......
kbl = min( 8_${ik}$, n )
! [tp] kbl is a tuning parameter that defines the tile size in the
! tiling of the p-q loops of pivot pairs. in general, an optimal
! value of kbl depends on the matrix dimensions and on the
! parameters of the computer's memory.
nbl = n / kbl
if( ( nbl*kbl )/=n )nbl = nbl + 1_${ik}$
blskip = ( kbl**2_${ik}$ ) + 1_${ik}$
! [tp] blkskip is a tuning parameter that depends on swband and kbl.
rowskip = min( 5_${ik}$, kbl )
! [tp] rowskip is a tuning parameter.
lkahead = 1_${ik}$
! [tp] lkahead is a tuning parameter.
swband = 0_${ik}$
pskipped = 0_${ik}$
loop_1993: do i = 1, nsweep
! .. go go go ...
mxaapq = zero
mxsinj = zero
iswrot = 0_${ik}$
notrot = 0_${ik}$
pskipped = 0_${ik}$
loop_2000: do ibr = 1, nbl
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_1002: do ir1 = 0, min( lkahead, nbl-ibr )
igl = igl + ir1*kbl
loop_2001: do p = igl, min( igl+kbl-1, n-1 )
! .. de rijk's pivoting
q = stdlib${ii}$_isamax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
call stdlib${ii}$_sswap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_sswap( mvl, v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
temp1 = d( p )
d( p ) = d( q )
d( q ) = temp1
end if
if( ir1==0_${ik}$ ) then
! column norms are periodically updated by explicit
! norm computation.
! caveat:
! some blas implementations compute stdlib${ii}$_snrm2(m,a(1,p),1)
! as sqrt(stdlib${ii}$_sdot(m,a(1,p),1,a(1,p),1)), which may result in
! overflow for ||a(:,p)||_2 > sqrt(overflow_threshold), and
! underflow for ||a(:,p)||_2 < sqrt(underflow_threshold).
! hence, stdlib${ii}$_snrm2 cannot be trusted, not even in the case when
! the true norm is far from the under(over)flow boundaries.
! if properly implemented stdlib${ii}$_snrm2 is available, the if-then-else
! below should read "aapp = stdlib${ii}$_snrm2( m, a(1,p), 1 ) * d(p)".
if( ( sva( p )<rootbig ) .and.( sva( p )>rootsfmin ) ) then
sva( p ) = stdlib${ii}$_snrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )*d( p )
else
temp1 = zero
aapp = one
call stdlib${ii}$_slassq( m, a( 1_${ik}$, p ), 1_${ik}$, temp1, aapp )
sva( p ) = temp1*sqrt( aapp )*d( p )
end if
aapp = sva( p )
else
aapp = sva( p )
end if
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2002: do q = p + 1, min( igl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
if( aaqq>=one ) then
rotok = ( small*aapp )<=aaqq
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_sdot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_scopy( m, a( 1_${ik}$, p ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aapp, d( p ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_sdot( m, work, 1_${ik}$, a( 1_${ik}$, q ),1_${ik}$ )*d( q ) / &
aaqq
end if
else
rotok = aapp<=( aaqq / small )
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_sdot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_scopy( m, a( 1_${ik}$, q ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, d( q ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_sdot( m, work, 1_${ik}$, a( 1_${ik}$, p ),1_${ik}$ )*d( p ) / &
aapp
end if
end if
mxaapq = max( mxaapq, abs( aapq ) )
! to rotate or not to rotate, that is the question ...
if( abs( aapq )>tol ) then
! Rotate
! rotated = rotated + one
if( ir1==0_${ik}$ ) then
notrot = 0_${ik}$
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
end if
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq ) / aapq
if( abs( theta )>bigtheta ) then
t = half / theta
fastr( 3_${ik}$ ) = t*d( p ) / d( q )
fastr( 4_${ik}$ ) = -t*d( q ) / d( p )
call stdlib${ii}$_srotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$, fastr )
if( rsvec )call stdlib${ii}$_srotm( mvl,v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ),&
1_${ik}$,fastr )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq )
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
apoaq = d( p ) / d( q )
aqoap = d( q ) / d( p )
if( d( p )>=one ) then
if( d( q )>=one ) then
fastr( 3_${ik}$ ) = t*apoaq
fastr( 4_${ik}$ ) = -t*aqoap
d( p ) = d( p )*cs
d( q ) = d( q )*cs
call stdlib${ii}$_srotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$,&
fastr )
if( rsvec )call stdlib${ii}$_srotm( mvl,v( 1_${ik}$, p ), 1_${ik}$, v( &
1_${ik}$, q ),1_${ik}$, fastr )
else
call stdlib${ii}$_saxpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( 1_${ik}$, &
p ), 1_${ik}$ )
call stdlib${ii}$_saxpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( &
1_${ik}$, q ), 1_${ik}$ )
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
if( rsvec ) then
call stdlib${ii}$_saxpy( mvl, -t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,v(&
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_saxpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ), 1_${ik}$,&
v( 1_${ik}$, q ), 1_${ik}$ )
end if
end if
else
if( d( q )>=one ) then
call stdlib${ii}$_saxpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q &
), 1_${ik}$ )
call stdlib${ii}$_saxpy( m, -cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
if( rsvec ) then
call stdlib${ii}$_saxpy( mvl, t*apoaq,v( 1_${ik}$, p ), 1_${ik}$,v( &
1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_saxpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q ), &
1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
else
if( d( p )>=d( q ) ) then
call stdlib${ii}$_saxpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_saxpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,&
a( 1_${ik}$, q ), 1_${ik}$ )
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
if( rsvec ) then
call stdlib${ii}$_saxpy( mvl,-t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,&
v( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_saxpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ),&
1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
end if
else
call stdlib${ii}$_saxpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$,&
q ), 1_${ik}$ )
call stdlib${ii}$_saxpy( m,-cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,&
a( 1_${ik}$, p ), 1_${ik}$ )
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
if( rsvec ) then
call stdlib${ii}$_saxpy( mvl,t*apoaq, v( 1_${ik}$, p ),1_${ik}$, &
v( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_saxpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q )&
, 1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
end if
end if
end if
end if
else
! .. have to use modified gram-schmidt like transformation
call stdlib${ii}$_scopy( m, a( 1_${ik}$, p ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one, m,1_${ik}$, work, lda, &
ierr )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one, m,1_${ik}$, a( 1_${ik}$, q ), &
lda, ierr )
temp1 = -aapq*d( p ) / d( q )
call stdlib${ii}$_saxpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq, m,1_${ik}$, a( 1_${ik}$, q ), &
lda, ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q), sva(p)
! recompute sva(q), sva(p).
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_snrm2( m, a( 1_${ik}$, q ), 1_${ik}$ )*d( q )
else
t = zero
aaqq = one
call stdlib${ii}$_slassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )*d( q )
end if
end if
if( ( aapp / aapp0 )<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_snrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )*d( p )
else
t = zero
aapp = one
call stdlib${ii}$_slassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )*d( p )
end if
sva( p ) = aapp
end if
else
! a(:,p) and a(:,q) already numerically orthogonal
if( ir1==0_${ik}$ )notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
end if
else
! a(:,q) is zero column
if( ir1==0_${ik}$ )notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
if( ir1==0_${ik}$ )aapp = -aapp
notrot = 0_${ik}$
go to 2103
end if
end do loop_2002
! end q-loop
2103 continue
! bailed out of q-loop
sva( p ) = aapp
else
sva( p ) = aapp
if( ( ir1==0_${ik}$ ) .and. ( aapp==zero ) )notrot = notrot + min( igl+kbl-1, &
n ) - p
end if
end do loop_2001
! end of the p-loop
! end of doing the block ( ibr, ibr )
end do loop_1002
! end of ir1-loop
! ........................................................
! ... go to the off diagonal blocks
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_2010: do jbc = ibr + 1, nbl
jgl = ( jbc-1 )*kbl + 1_${ik}$
! doing the block at ( ibr, jbc )
ijblsk = 0_${ik}$
loop_2100: do p = igl, min( igl+kbl-1, n )
aapp = sva( p )
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2200: do q = jgl, min( jgl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
! M X 2 Jacobi Svd
! Safe Gram Matrix Computation
if( aaqq>=one ) then
if( aapp>=aaqq ) then
rotok = ( small*aapp )<=aaqq
else
rotok = ( small*aaqq )<=aapp
end if
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_sdot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_scopy( m, a( 1_${ik}$, p ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aapp, d( p ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_sdot( m, work, 1_${ik}$, a( 1_${ik}$, q ),1_${ik}$ )*d( q ) / &
aaqq
end if
else
if( aapp>=aaqq ) then
rotok = aapp<=( aaqq / small )
else
rotok = aaqq<=( aapp / small )
end if
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_sdot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_scopy( m, a( 1_${ik}$, q ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, d( q ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_sdot( m, work, 1_${ik}$, a( 1_${ik}$, p ),1_${ik}$ )*d( p ) / &
aapp
end if
end if
mxaapq = max( mxaapq, abs( aapq ) )
! to rotate or not to rotate, that is the question ...
if( abs( aapq )>tol ) then
notrot = 0_${ik}$
! rotated = rotated + 1
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq ) / aapq
if( aaqq>aapp0 )theta = -theta
if( abs( theta )>bigtheta ) then
t = half / theta
fastr( 3_${ik}$ ) = t*d( p ) / d( q )
fastr( 4_${ik}$ ) = -t*d( q ) / d( p )
call stdlib${ii}$_srotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$, fastr )
if( rsvec )call stdlib${ii}$_srotm( mvl,v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ),&
1_${ik}$,fastr )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq )
if( aaqq>aapp0 )thsign = -thsign
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
apoaq = d( p ) / d( q )
aqoap = d( q ) / d( p )
if( d( p )>=one ) then
if( d( q )>=one ) then
fastr( 3_${ik}$ ) = t*apoaq
fastr( 4_${ik}$ ) = -t*aqoap
d( p ) = d( p )*cs
d( q ) = d( q )*cs
call stdlib${ii}$_srotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$,&
fastr )
if( rsvec )call stdlib${ii}$_srotm( mvl,v( 1_${ik}$, p ), 1_${ik}$, v( &
1_${ik}$, q ),1_${ik}$, fastr )
else
call stdlib${ii}$_saxpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( 1_${ik}$, &
p ), 1_${ik}$ )
call stdlib${ii}$_saxpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( &
1_${ik}$, q ), 1_${ik}$ )
if( rsvec ) then
call stdlib${ii}$_saxpy( mvl, -t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,v(&
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_saxpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ), 1_${ik}$,&
v( 1_${ik}$, q ), 1_${ik}$ )
end if
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
end if
else
if( d( q )>=one ) then
call stdlib${ii}$_saxpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q &
), 1_${ik}$ )
call stdlib${ii}$_saxpy( m, -cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
if( rsvec ) then
call stdlib${ii}$_saxpy( mvl, t*apoaq,v( 1_${ik}$, p ), 1_${ik}$,v( &
1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_saxpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q ), &
1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
else
if( d( p )>=d( q ) ) then
call stdlib${ii}$_saxpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_saxpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,&
a( 1_${ik}$, q ), 1_${ik}$ )
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
if( rsvec ) then
call stdlib${ii}$_saxpy( mvl,-t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,&
v( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_saxpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ),&
1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
end if
else
call stdlib${ii}$_saxpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$,&
q ), 1_${ik}$ )
call stdlib${ii}$_saxpy( m,-cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,&
a( 1_${ik}$, p ), 1_${ik}$ )
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
if( rsvec ) then
call stdlib${ii}$_saxpy( mvl,t*apoaq, v( 1_${ik}$, p ),1_${ik}$, &
v( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_saxpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q )&
, 1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
end if
end if
end if
end if
else
if( aapp>aaqq ) then
call stdlib${ii}$_scopy( m, a( 1_${ik}$, p ), 1_${ik}$, work,1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
temp1 = -aapq*d( p ) / d( q )
call stdlib${ii}$_saxpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
else
call stdlib${ii}$_scopy( m, a( 1_${ik}$, q ), 1_${ik}$, work,1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
temp1 = -aapq*d( q ) / d( p )
call stdlib${ii}$_saxpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, one, aapp,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
sva( p ) = aapp*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
end if
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q)
! .. recompute sva(q)
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_snrm2( m, a( 1_${ik}$, q ), 1_${ik}$ )*d( q )
else
t = zero
aaqq = one
call stdlib${ii}$_slassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )*d( q )
end if
end if
if( ( aapp / aapp0 )**2_${ik}$<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_snrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )*d( p )
else
t = zero
aapp = one
call stdlib${ii}$_slassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )*d( p )
end if
sva( p ) = aapp
end if
! end of ok rotation
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
if( ( i<=swband ) .and. ( ijblsk>=blskip ) )then
sva( p ) = aapp
notrot = 0_${ik}$
go to 2011
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
aapp = -aapp
notrot = 0_${ik}$
go to 2203
end if
end do loop_2200
! end of the q-loop
2203 continue
sva( p ) = aapp
else
if( aapp==zero )notrot = notrot +min( jgl+kbl-1, n ) - jgl + 1_${ik}$
if( aapp<zero )notrot = 0_${ik}$
end if
end do loop_2100
! end of the p-loop
end do loop_2010
! end of the jbc-loop
2011 continue
! 2011 bailed out of the jbc-loop
do p = igl, min( igl+kbl-1, n )
sva( p ) = abs( sva( p ) )
end do
end do loop_2000
! 2000 :: end of the ibr-loop
! .. update sva(n)
if( ( sva( n )<rootbig ) .and. ( sva( n )>rootsfmin ) )then
sva( n ) = stdlib${ii}$_snrm2( m, a( 1_${ik}$, n ), 1_${ik}$ )*d( n )
else
t = zero
aapp = one
call stdlib${ii}$_slassq( m, a( 1_${ik}$, n ), 1_${ik}$, t, aapp )
sva( n ) = t*sqrt( aapp )*d( n )
end if
! additional steering devices
if( ( i<swband ) .and. ( ( mxaapq<=roottol ) .or.( iswrot<=n ) ) )swband = i
if( ( i>swband+1 ) .and. ( mxaapq<real( n,KIND=sp)*tol ) .and.( real( n,KIND=sp)&
*mxaapq*mxsinj<tol ) ) then
go to 1994
end if
if( notrot>=emptsw )go to 1994
end do loop_1993
! end i=1:nsweep loop
! #:) reaching this point means that the procedure has completed the given
! number of iterations.
info = nsweep - 1_${ik}$
go to 1995
1994 continue
! #:) reaching this point means that during the i-th sweep all pivots were
! below the given tolerance, causing early exit.
info = 0_${ik}$
! #:) info = 0 confirms successful iterations.
1995 continue
! sort the vector d.
do p = 1, n - 1
q = stdlib${ii}$_isamax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
temp1 = d( p )
d( p ) = d( q )
d( q ) = temp1
call stdlib${ii}$_sswap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_sswap( mvl, v( 1_${ik}$, p ), 1_${ik}$, v( 1_${ik}$, q ), 1_${ik}$ )
end if
end do
return
end subroutine stdlib${ii}$_sgsvj0
pure module subroutine stdlib${ii}$_dgsvj0( jobv, m, n, a, lda, d, sva, mv, v, ldv, eps,sfmin, tol, &
!! DGSVJ0 is called from DGESVJ as a pre-processor and that is its main
!! purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
!! it does not check convergence (stopping criterion). Few tuning
!! parameters (marked by [TP]) are available for the implementer.
nsweep, work, lwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, ldv, lwork, m, mv, n, nsweep
real(dp), intent(in) :: eps, sfmin, tol
character, intent(in) :: jobv
! Array Arguments
real(dp), intent(inout) :: a(lda,*), sva(n), d(n), v(ldv,*)
real(dp), intent(out) :: work(lwork)
! =====================================================================
! Local Scalars
real(dp) :: aapp, aapp0, aapq, aaqq, apoaq, aqoap, big, bigtheta, cs, mxaapq, mxsinj, &
rootbig, rooteps, rootsfmin, roottol, small, sn, t, temp1, theta, thsign
integer(${ik}$) :: blskip, emptsw, i, ibr, ierr, igl, ijblsk, ir1, iswrot, jbc, jgl, kbl, &
lkahead, mvl, nbl, notrot, p, pskipped, q, rowskip, swband
logical(lk) :: applv, rotok, rsvec
! Local Arrays
real(dp) :: fastr(5_${ik}$)
! Intrinsic Functions
! Executable Statements
! test the input parameters.
applv = stdlib_lsame( jobv, 'A' )
rsvec = stdlib_lsame( jobv, 'V' )
if( .not.( rsvec .or. applv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( ( n<0_${ik}$ ) .or. ( n>m ) ) then
info = -3_${ik}$
else if( lda<m ) then
info = -5_${ik}$
else if( ( rsvec.or.applv ) .and. ( mv<0_${ik}$ ) ) then
info = -8_${ik}$
else if( ( rsvec.and.( ldv<n ) ).or.( applv.and.( ldv<mv ) ) ) then
info = -10_${ik}$
else if( tol<=eps ) then
info = -13_${ik}$
else if( nsweep<0_${ik}$ ) then
info = -14_${ik}$
else if( lwork<m ) then
info = -16_${ik}$
else
info = 0_${ik}$
end if
! #:(
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DGSVJ0', -info )
return
end if
if( rsvec ) then
mvl = n
else if( applv ) then
mvl = mv
end if
rsvec = rsvec .or. applv
rooteps = sqrt( eps )
rootsfmin = sqrt( sfmin )
small = sfmin / eps
big = one / sfmin
rootbig = one / rootsfmin
bigtheta = one / rooteps
roottol = sqrt( tol )
! -#- row-cyclic jacobi svd algorithm with column pivoting -#-
emptsw = ( n*( n-1 ) ) / 2_${ik}$
notrot = 0_${ik}$
fastr( 1_${ik}$ ) = zero
! -#- row-cyclic pivot strategy with de rijk's pivoting -#-
swband = 0_${ik}$
! [tp] swband is a tuning parameter. it is meaningful and effective
! if stdlib${ii}$_sgesvj is used as a computational routine in the preconditioned
! jacobi svd algorithm stdlib${ii}$_sgesvj. for sweeps i=1:swband the procedure
! ......
kbl = min( 8_${ik}$, n )
! [tp] kbl is a tuning parameter that defines the tile size in the
! tiling of the p-q loops of pivot pairs. in general, an optimal
! value of kbl depends on the matrix dimensions and on the
! parameters of the computer's memory.
nbl = n / kbl
if( ( nbl*kbl )/=n )nbl = nbl + 1_${ik}$
blskip = ( kbl**2_${ik}$ ) + 1_${ik}$
! [tp] blkskip is a tuning parameter that depends on swband and kbl.
rowskip = min( 5_${ik}$, kbl )
! [tp] rowskip is a tuning parameter.
lkahead = 1_${ik}$
! [tp] lkahead is a tuning parameter.
swband = 0_${ik}$
pskipped = 0_${ik}$
loop_1993: do i = 1, nsweep
! .. go go go ...
mxaapq = zero
mxsinj = zero
iswrot = 0_${ik}$
notrot = 0_${ik}$
pskipped = 0_${ik}$
loop_2000: do ibr = 1, nbl
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_1002: do ir1 = 0, min( lkahead, nbl-ibr )
igl = igl + ir1*kbl
loop_2001: do p = igl, min( igl+kbl-1, n-1 )
! .. de rijk's pivoting
q = stdlib${ii}$_idamax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
call stdlib${ii}$_dswap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_dswap( mvl, v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
temp1 = d( p )
d( p ) = d( q )
d( q ) = temp1
end if
if( ir1==0_${ik}$ ) then
! column norms are periodically updated by explicit
! norm computation.
! caveat:
! some blas implementations compute stdlib${ii}$_dnrm2(m,a(1,p),1)
! as sqrt(stdlib${ii}$_ddot(m,a(1,p),1,a(1,p),1)), which may result in
! overflow for ||a(:,p)||_2 > sqrt(overflow_threshold), and
! underflow for ||a(:,p)||_2 < sqrt(underflow_threshold).
! hence, stdlib${ii}$_dnrm2 cannot be trusted, not even in the case when
! the true norm is far from the under(over)flow boundaries.
! if properly implemented stdlib${ii}$_dnrm2 is available, the if-then-else
! below should read "aapp = stdlib${ii}$_dnrm2( m, a(1,p), 1 ) * d(p)".
if( ( sva( p )<rootbig ) .and.( sva( p )>rootsfmin ) ) then
sva( p ) = stdlib${ii}$_dnrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )*d( p )
else
temp1 = zero
aapp = one
call stdlib${ii}$_dlassq( m, a( 1_${ik}$, p ), 1_${ik}$, temp1, aapp )
sva( p ) = temp1*sqrt( aapp )*d( p )
end if
aapp = sva( p )
else
aapp = sva( p )
end if
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2002: do q = p + 1, min( igl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
if( aaqq>=one ) then
rotok = ( small*aapp )<=aaqq
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_ddot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_dcopy( m, a( 1_${ik}$, p ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aapp, d( p ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_ddot( m, work, 1_${ik}$, a( 1_${ik}$, q ),1_${ik}$ )*d( q ) / &
aaqq
end if
else
rotok = aapp<=( aaqq / small )
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_ddot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_dcopy( m, a( 1_${ik}$, q ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, d( q ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_ddot( m, work, 1_${ik}$, a( 1_${ik}$, p ),1_${ik}$ )*d( p ) / &
aapp
end if
end if
mxaapq = max( mxaapq, abs( aapq ) )
! to rotate or not to rotate, that is the question ...
if( abs( aapq )>tol ) then
! Rotate
! rotated = rotated + one
if( ir1==0_${ik}$ ) then
notrot = 0_${ik}$
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
end if
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq )/aapq
if( abs( theta )>bigtheta ) then
t = half / theta
fastr( 3_${ik}$ ) = t*d( p ) / d( q )
fastr( 4_${ik}$ ) = -t*d( q ) / d( p )
call stdlib${ii}$_drotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$, fastr )
if( rsvec )call stdlib${ii}$_drotm( mvl,v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ),&
1_${ik}$,fastr )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq )
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
apoaq = d( p ) / d( q )
aqoap = d( q ) / d( p )
if( d( p )>=one ) then
if( d( q )>=one ) then
fastr( 3_${ik}$ ) = t*apoaq
fastr( 4_${ik}$ ) = -t*aqoap
d( p ) = d( p )*cs
d( q ) = d( q )*cs
call stdlib${ii}$_drotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$,&
fastr )
if( rsvec )call stdlib${ii}$_drotm( mvl,v( 1_${ik}$, p ), 1_${ik}$, v( &
1_${ik}$, q ),1_${ik}$, fastr )
else
call stdlib${ii}$_daxpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( 1_${ik}$, &
p ), 1_${ik}$ )
call stdlib${ii}$_daxpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( &
1_${ik}$, q ), 1_${ik}$ )
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
if( rsvec ) then
call stdlib${ii}$_daxpy( mvl, -t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,v(&
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_daxpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ), 1_${ik}$,&
v( 1_${ik}$, q ), 1_${ik}$ )
end if
end if
else
if( d( q )>=one ) then
call stdlib${ii}$_daxpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q &
), 1_${ik}$ )
call stdlib${ii}$_daxpy( m, -cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
if( rsvec ) then
call stdlib${ii}$_daxpy( mvl, t*apoaq,v( 1_${ik}$, p ), 1_${ik}$,v( &
1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_daxpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q ), &
1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
else
if( d( p )>=d( q ) ) then
call stdlib${ii}$_daxpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_daxpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,&
a( 1_${ik}$, q ), 1_${ik}$ )
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
if( rsvec ) then
call stdlib${ii}$_daxpy( mvl,-t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,&
v( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_daxpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ),&
1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
end if
else
call stdlib${ii}$_daxpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$,&
q ), 1_${ik}$ )
call stdlib${ii}$_daxpy( m,-cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,&
a( 1_${ik}$, p ), 1_${ik}$ )
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
if( rsvec ) then
call stdlib${ii}$_daxpy( mvl,t*apoaq, v( 1_${ik}$, p ),1_${ik}$, &
v( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_daxpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q )&
, 1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
end if
end if
end if
end if
else
! .. have to use modified gram-schmidt like transformation
call stdlib${ii}$_dcopy( m, a( 1_${ik}$, p ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one, m,1_${ik}$, work, lda, &
ierr )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one, m,1_${ik}$, a( 1_${ik}$, q ), &
lda, ierr )
temp1 = -aapq*d( p ) / d( q )
call stdlib${ii}$_daxpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq, m,1_${ik}$, a( 1_${ik}$, q ), &
lda, ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q), sva(p)
! recompute sva(q), sva(p).
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_dnrm2( m, a( 1_${ik}$, q ), 1_${ik}$ )*d( q )
else
t = zero
aaqq = one
call stdlib${ii}$_dlassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )*d( q )
end if
end if
if( ( aapp / aapp0 )<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_dnrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )*d( p )
else
t = zero
aapp = one
call stdlib${ii}$_dlassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )*d( p )
end if
sva( p ) = aapp
end if
else
! a(:,p) and a(:,q) already numerically orthogonal
if( ir1==0_${ik}$ )notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
end if
else
! a(:,q) is zero column
if( ir1==0_${ik}$ )notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
if( ir1==0_${ik}$ )aapp = -aapp
notrot = 0_${ik}$
go to 2103
end if
end do loop_2002
! end q-loop
2103 continue
! bailed out of q-loop
sva( p ) = aapp
else
sva( p ) = aapp
if( ( ir1==0_${ik}$ ) .and. ( aapp==zero ) )notrot = notrot + min( igl+kbl-1, &
n ) - p
end if
end do loop_2001
! end of the p-loop
! end of doing the block ( ibr, ibr )
end do loop_1002
! end of ir1-loop
! ........................................................
! ... go to the off diagonal blocks
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_2010: do jbc = ibr + 1, nbl
jgl = ( jbc-1 )*kbl + 1_${ik}$
! doing the block at ( ibr, jbc )
ijblsk = 0_${ik}$
loop_2100: do p = igl, min( igl+kbl-1, n )
aapp = sva( p )
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2200: do q = jgl, min( jgl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
! -#- m x 2 jacobi svd -#-
! -#- safe gram matrix computation -#-
if( aaqq>=one ) then
if( aapp>=aaqq ) then
rotok = ( small*aapp )<=aaqq
else
rotok = ( small*aaqq )<=aapp
end if
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_ddot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_dcopy( m, a( 1_${ik}$, p ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aapp, d( p ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_ddot( m, work, 1_${ik}$, a( 1_${ik}$, q ),1_${ik}$ )*d( q ) / &
aaqq
end if
else
if( aapp>=aaqq ) then
rotok = aapp<=( aaqq / small )
else
rotok = aaqq<=( aapp / small )
end if
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_ddot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_dcopy( m, a( 1_${ik}$, q ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, d( q ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_ddot( m, work, 1_${ik}$, a( 1_${ik}$, p ),1_${ik}$ )*d( p ) / &
aapp
end if
end if
mxaapq = max( mxaapq, abs( aapq ) )
! to rotate or not to rotate, that is the question ...
if( abs( aapq )>tol ) then
notrot = 0_${ik}$
! rotated = rotated + 1
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq )/aapq
if( aaqq>aapp0 )theta = -theta
if( abs( theta )>bigtheta ) then
t = half / theta
fastr( 3_${ik}$ ) = t*d( p ) / d( q )
fastr( 4_${ik}$ ) = -t*d( q ) / d( p )
call stdlib${ii}$_drotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$, fastr )
if( rsvec )call stdlib${ii}$_drotm( mvl,v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ),&
1_${ik}$,fastr )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq )
if( aaqq>aapp0 )thsign = -thsign
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
apoaq = d( p ) / d( q )
aqoap = d( q ) / d( p )
if( d( p )>=one ) then
if( d( q )>=one ) then
fastr( 3_${ik}$ ) = t*apoaq
fastr( 4_${ik}$ ) = -t*aqoap
d( p ) = d( p )*cs
d( q ) = d( q )*cs
call stdlib${ii}$_drotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$,&
fastr )
if( rsvec )call stdlib${ii}$_drotm( mvl,v( 1_${ik}$, p ), 1_${ik}$, v( &
1_${ik}$, q ),1_${ik}$, fastr )
else
call stdlib${ii}$_daxpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( 1_${ik}$, &
p ), 1_${ik}$ )
call stdlib${ii}$_daxpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( &
1_${ik}$, q ), 1_${ik}$ )
if( rsvec ) then
call stdlib${ii}$_daxpy( mvl, -t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,v(&
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_daxpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ), 1_${ik}$,&
v( 1_${ik}$, q ), 1_${ik}$ )
end if
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
end if
else
if( d( q )>=one ) then
call stdlib${ii}$_daxpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q &
), 1_${ik}$ )
call stdlib${ii}$_daxpy( m, -cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
if( rsvec ) then
call stdlib${ii}$_daxpy( mvl, t*apoaq,v( 1_${ik}$, p ), 1_${ik}$,v( &
1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_daxpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q ), &
1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
else
if( d( p )>=d( q ) ) then
call stdlib${ii}$_daxpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_daxpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,&
a( 1_${ik}$, q ), 1_${ik}$ )
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
if( rsvec ) then
call stdlib${ii}$_daxpy( mvl,-t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,&
v( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_daxpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ),&
1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
end if
else
call stdlib${ii}$_daxpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$,&
q ), 1_${ik}$ )
call stdlib${ii}$_daxpy( m,-cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,&
a( 1_${ik}$, p ), 1_${ik}$ )
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
if( rsvec ) then
call stdlib${ii}$_daxpy( mvl,t*apoaq, v( 1_${ik}$, p ),1_${ik}$, &
v( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_daxpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q )&
, 1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
end if
end if
end if
end if
else
if( aapp>aaqq ) then
call stdlib${ii}$_dcopy( m, a( 1_${ik}$, p ), 1_${ik}$, work,1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
temp1 = -aapq*d( p ) / d( q )
call stdlib${ii}$_daxpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
else
call stdlib${ii}$_dcopy( m, a( 1_${ik}$, q ), 1_${ik}$, work,1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
temp1 = -aapq*d( q ) / d( p )
call stdlib${ii}$_daxpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, one, aapp,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
sva( p ) = aapp*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
end if
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q)
! .. recompute sva(q)
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_dnrm2( m, a( 1_${ik}$, q ), 1_${ik}$ )*d( q )
else
t = zero
aaqq = one
call stdlib${ii}$_dlassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )*d( q )
end if
end if
if( ( aapp / aapp0 )**2_${ik}$<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_dnrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )*d( p )
else
t = zero
aapp = one
call stdlib${ii}$_dlassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )*d( p )
end if
sva( p ) = aapp
end if
! end of ok rotation
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
if( ( i<=swband ) .and. ( ijblsk>=blskip ) )then
sva( p ) = aapp
notrot = 0_${ik}$
go to 2011
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
aapp = -aapp
notrot = 0_${ik}$
go to 2203
end if
end do loop_2200
! end of the q-loop
2203 continue
sva( p ) = aapp
else
if( aapp==zero )notrot = notrot +min( jgl+kbl-1, n ) - jgl + 1_${ik}$
if( aapp<zero )notrot = 0_${ik}$
end if
end do loop_2100
! end of the p-loop
end do loop_2010
! end of the jbc-loop
2011 continue
! 2011 bailed out of the jbc-loop
do p = igl, min( igl+kbl-1, n )
sva( p ) = abs( sva( p ) )
end do
end do loop_2000
! 2000 :: end of the ibr-loop
! .. update sva(n)
if( ( sva( n )<rootbig ) .and. ( sva( n )>rootsfmin ) )then
sva( n ) = stdlib${ii}$_dnrm2( m, a( 1_${ik}$, n ), 1_${ik}$ )*d( n )
else
t = zero
aapp = one
call stdlib${ii}$_dlassq( m, a( 1_${ik}$, n ), 1_${ik}$, t, aapp )
sva( n ) = t*sqrt( aapp )*d( n )
end if
! additional steering devices
if( ( i<swband ) .and. ( ( mxaapq<=roottol ) .or.( iswrot<=n ) ) )swband = i
if( ( i>swband+1 ) .and. ( mxaapq<real( n,KIND=dp)*tol ) .and.( real( n,KIND=dp)&
*mxaapq*mxsinj<tol ) ) then
go to 1994
end if
if( notrot>=emptsw )go to 1994
end do loop_1993
! end i=1:nsweep loop
! #:) reaching this point means that the procedure has completed the given
! number of iterations.
info = nsweep - 1_${ik}$
go to 1995
1994 continue
! #:) reaching this point means that during the i-th sweep all pivots were
! below the given tolerance, causing early exit.
info = 0_${ik}$
! #:) info = 0 confirms successful iterations.
1995 continue
! sort the vector d.
do p = 1, n - 1
q = stdlib${ii}$_idamax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
temp1 = d( p )
d( p ) = d( q )
d( q ) = temp1
call stdlib${ii}$_dswap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_dswap( mvl, v( 1_${ik}$, p ), 1_${ik}$, v( 1_${ik}$, q ), 1_${ik}$ )
end if
end do
return
end subroutine stdlib${ii}$_dgsvj0
#:for rk,rt,ri in REAL_KINDS_TYPES
#:if not rk in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ri}$gsvj0( jobv, m, n, a, lda, d, sva, mv, v, ldv, eps,sfmin, tol, &
!! DGSVJ0: is called from DGESVJ as a pre-processor and that is its main
!! purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
!! it does not check convergence (stopping criterion). Few tuning
!! parameters (marked by [TP]) are available for the implementer.
nsweep, work, lwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${rk}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, ldv, lwork, m, mv, n, nsweep
real(${rk}$), intent(in) :: eps, sfmin, tol
character, intent(in) :: jobv
! Array Arguments
real(${rk}$), intent(inout) :: a(lda,*), sva(n), d(n), v(ldv,*)
real(${rk}$), intent(out) :: work(lwork)
! =====================================================================
! Local Scalars
real(${rk}$) :: aapp, aapp0, aapq, aaqq, apoaq, aqoap, big, bigtheta, cs, mxaapq, mxsinj, &
rootbig, rooteps, rootsfmin, roottol, small, sn, t, temp1, theta, thsign
integer(${ik}$) :: blskip, emptsw, i, ibr, ierr, igl, ijblsk, ir1, iswrot, jbc, jgl, kbl, &
lkahead, mvl, nbl, notrot, p, pskipped, q, rowskip, swband
logical(lk) :: applv, rotok, rsvec
! Local Arrays
real(${rk}$) :: fastr(5_${ik}$)
! Intrinsic Functions
! Executable Statements
! test the input parameters.
applv = stdlib_lsame( jobv, 'A' )
rsvec = stdlib_lsame( jobv, 'V' )
if( .not.( rsvec .or. applv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( ( n<0_${ik}$ ) .or. ( n>m ) ) then
info = -3_${ik}$
else if( lda<m ) then
info = -5_${ik}$
else if( ( rsvec.or.applv ) .and. ( mv<0_${ik}$ ) ) then
info = -8_${ik}$
else if( ( rsvec.and.( ldv<n ) ).or.( applv.and.( ldv<mv ) ) ) then
info = -10_${ik}$
else if( tol<=eps ) then
info = -13_${ik}$
else if( nsweep<0_${ik}$ ) then
info = -14_${ik}$
else if( lwork<m ) then
info = -16_${ik}$
else
info = 0_${ik}$
end if
! #:(
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DGSVJ0', -info )
return
end if
if( rsvec ) then
mvl = n
else if( applv ) then
mvl = mv
end if
rsvec = rsvec .or. applv
rooteps = sqrt( eps )
rootsfmin = sqrt( sfmin )
small = sfmin / eps
big = one / sfmin
rootbig = one / rootsfmin
bigtheta = one / rooteps
roottol = sqrt( tol )
! -#- row-cyclic jacobi svd algorithm with column pivoting -#-
emptsw = ( n*( n-1 ) ) / 2_${ik}$
notrot = 0_${ik}$
fastr( 1_${ik}$ ) = zero
! -#- row-cyclic pivot strategy with de rijk's pivoting -#-
swband = 0_${ik}$
! [tp] swband is a tuning parameter. it is meaningful and effective
! if stdlib${ii}$_dgesvj is used as a computational routine in the preconditioned
! jacobi svd algorithm stdlib${ii}$_dgesvj. for sweeps i=1:swband the procedure
! ......
kbl = min( 8_${ik}$, n )
! [tp] kbl is a tuning parameter that defines the tile size in the
! tiling of the p-q loops of pivot pairs. in general, an optimal
! value of kbl depends on the matrix dimensions and on the
! parameters of the computer's memory.
nbl = n / kbl
if( ( nbl*kbl )/=n )nbl = nbl + 1_${ik}$
blskip = ( kbl**2_${ik}$ ) + 1_${ik}$
! [tp] blkskip is a tuning parameter that depends on swband and kbl.
rowskip = min( 5_${ik}$, kbl )
! [tp] rowskip is a tuning parameter.
lkahead = 1_${ik}$
! [tp] lkahead is a tuning parameter.
swband = 0_${ik}$
pskipped = 0_${ik}$
loop_1993: do i = 1, nsweep
! .. go go go ...
mxaapq = zero
mxsinj = zero
iswrot = 0_${ik}$
notrot = 0_${ik}$
pskipped = 0_${ik}$
loop_2000: do ibr = 1, nbl
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_1002: do ir1 = 0, min( lkahead, nbl-ibr )
igl = igl + ir1*kbl
loop_2001: do p = igl, min( igl+kbl-1, n-1 )
! .. de rijk's pivoting
q = stdlib${ii}$_i${ri}$amax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
call stdlib${ii}$_${ri}$swap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_${ri}$swap( mvl, v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
temp1 = d( p )
d( p ) = d( q )
d( q ) = temp1
end if
if( ir1==0_${ik}$ ) then
! column norms are periodically updated by explicit
! norm computation.
! caveat:
! some blas implementations compute stdlib${ii}$_${ri}$nrm2(m,a(1,p),1)
! as sqrt(stdlib${ii}$_${ri}$dot(m,a(1,p),1,a(1,p),1)), which may result in
! overflow for ||a(:,p)||_2 > sqrt(overflow_threshold), and
! underflow for ||a(:,p)||_2 < sqrt(underflow_threshold).
! hence, stdlib${ii}$_${ri}$nrm2 cannot be trusted, not even in the case when
! the true norm is far from the under(over)flow boundaries.
! if properly implemented stdlib${ii}$_${ri}$nrm2 is available, the if-then-else
! below should read "aapp = stdlib${ii}$_${ri}$nrm2( m, a(1,p), 1 ) * d(p)".
if( ( sva( p )<rootbig ) .and.( sva( p )>rootsfmin ) ) then
sva( p ) = stdlib${ii}$_${ri}$nrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )*d( p )
else
temp1 = zero
aapp = one
call stdlib${ii}$_${ri}$lassq( m, a( 1_${ik}$, p ), 1_${ik}$, temp1, aapp )
sva( p ) = temp1*sqrt( aapp )*d( p )
end if
aapp = sva( p )
else
aapp = sva( p )
end if
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2002: do q = p + 1, min( igl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
if( aaqq>=one ) then
rotok = ( small*aapp )<=aaqq
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_${ri}$dot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_${ri}$copy( m, a( 1_${ik}$, p ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp, d( p ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_${ri}$dot( m, work, 1_${ik}$, a( 1_${ik}$, q ),1_${ik}$ )*d( q ) / &
aaqq
end if
else
rotok = aapp<=( aaqq / small )
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_${ri}$dot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_${ri}$copy( m, a( 1_${ik}$, q ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, d( q ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_${ri}$dot( m, work, 1_${ik}$, a( 1_${ik}$, p ),1_${ik}$ )*d( p ) / &
aapp
end if
end if
mxaapq = max( mxaapq, abs( aapq ) )
! to rotate or not to rotate, that is the question ...
if( abs( aapq )>tol ) then
! Rotate
! rotated = rotated + one
if( ir1==0_${ik}$ ) then
notrot = 0_${ik}$
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
end if
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq )/aapq
if( abs( theta )>bigtheta ) then
t = half / theta
fastr( 3_${ik}$ ) = t*d( p ) / d( q )
fastr( 4_${ik}$ ) = -t*d( q ) / d( p )
call stdlib${ii}$_${ri}$rotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$, fastr )
if( rsvec )call stdlib${ii}$_${ri}$rotm( mvl,v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ),&
1_${ik}$,fastr )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq )
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
apoaq = d( p ) / d( q )
aqoap = d( q ) / d( p )
if( d( p )>=one ) then
if( d( q )>=one ) then
fastr( 3_${ik}$ ) = t*apoaq
fastr( 4_${ik}$ ) = -t*aqoap
d( p ) = d( p )*cs
d( q ) = d( q )*cs
call stdlib${ii}$_${ri}$rotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$,&
fastr )
if( rsvec )call stdlib${ii}$_${ri}$rotm( mvl,v( 1_${ik}$, p ), 1_${ik}$, v( &
1_${ik}$, q ),1_${ik}$, fastr )
else
call stdlib${ii}$_${ri}$axpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( 1_${ik}$, &
p ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( &
1_${ik}$, q ), 1_${ik}$ )
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
if( rsvec ) then
call stdlib${ii}$_${ri}$axpy( mvl, -t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,v(&
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ), 1_${ik}$,&
v( 1_${ik}$, q ), 1_${ik}$ )
end if
end if
else
if( d( q )>=one ) then
call stdlib${ii}$_${ri}$axpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q &
), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( m, -cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
if( rsvec ) then
call stdlib${ii}$_${ri}$axpy( mvl, t*apoaq,v( 1_${ik}$, p ), 1_${ik}$,v( &
1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q ), &
1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
else
if( d( p )>=d( q ) ) then
call stdlib${ii}$_${ri}$axpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,&
a( 1_${ik}$, q ), 1_${ik}$ )
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
if( rsvec ) then
call stdlib${ii}$_${ri}$axpy( mvl,-t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,&
v( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ),&
1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
end if
else
call stdlib${ii}$_${ri}$axpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$,&
q ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( m,-cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,&
a( 1_${ik}$, p ), 1_${ik}$ )
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
if( rsvec ) then
call stdlib${ii}$_${ri}$axpy( mvl,t*apoaq, v( 1_${ik}$, p ),1_${ik}$, &
v( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q )&
, 1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
end if
end if
end if
end if
else
! .. have to use modified gram-schmidt like transformation
call stdlib${ii}$_${ri}$copy( m, a( 1_${ik}$, p ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one, m,1_${ik}$, work, lda, &
ierr )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one, m,1_${ik}$, a( 1_${ik}$, q ), &
lda, ierr )
temp1 = -aapq*d( p ) / d( q )
call stdlib${ii}$_${ri}$axpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq, m,1_${ik}$, a( 1_${ik}$, q ), &
lda, ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q), sva(p)
! recompute sva(q), sva(p).
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_${ri}$nrm2( m, a( 1_${ik}$, q ), 1_${ik}$ )*d( q )
else
t = zero
aaqq = one
call stdlib${ii}$_${ri}$lassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )*d( q )
end if
end if
if( ( aapp / aapp0 )<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_${ri}$nrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )*d( p )
else
t = zero
aapp = one
call stdlib${ii}$_${ri}$lassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )*d( p )
end if
sva( p ) = aapp
end if
else
! a(:,p) and a(:,q) already numerically orthogonal
if( ir1==0_${ik}$ )notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
end if
else
! a(:,q) is zero column
if( ir1==0_${ik}$ )notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
if( ir1==0_${ik}$ )aapp = -aapp
notrot = 0_${ik}$
go to 2103
end if
end do loop_2002
! end q-loop
2103 continue
! bailed out of q-loop
sva( p ) = aapp
else
sva( p ) = aapp
if( ( ir1==0_${ik}$ ) .and. ( aapp==zero ) )notrot = notrot + min( igl+kbl-1, &
n ) - p
end if
end do loop_2001
! end of the p-loop
! end of doing the block ( ibr, ibr )
end do loop_1002
! end of ir1-loop
! ........................................................
! ... go to the off diagonal blocks
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_2010: do jbc = ibr + 1, nbl
jgl = ( jbc-1 )*kbl + 1_${ik}$
! doing the block at ( ibr, jbc )
ijblsk = 0_${ik}$
loop_2100: do p = igl, min( igl+kbl-1, n )
aapp = sva( p )
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2200: do q = jgl, min( jgl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
! -#- m x 2 jacobi svd -#-
! -#- safe gram matrix computation -#-
if( aaqq>=one ) then
if( aapp>=aaqq ) then
rotok = ( small*aapp )<=aaqq
else
rotok = ( small*aaqq )<=aapp
end if
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_${ri}$dot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_${ri}$copy( m, a( 1_${ik}$, p ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp, d( p ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_${ri}$dot( m, work, 1_${ik}$, a( 1_${ik}$, q ),1_${ik}$ )*d( q ) / &
aaqq
end if
else
if( aapp>=aaqq ) then
rotok = aapp<=( aaqq / small )
else
rotok = aaqq<=( aapp / small )
end if
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_${ri}$dot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_${ri}$copy( m, a( 1_${ik}$, q ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, d( q ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_${ri}$dot( m, work, 1_${ik}$, a( 1_${ik}$, p ),1_${ik}$ )*d( p ) / &
aapp
end if
end if
mxaapq = max( mxaapq, abs( aapq ) )
! to rotate or not to rotate, that is the question ...
if( abs( aapq )>tol ) then
notrot = 0_${ik}$
! rotated = rotated + 1
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq )/aapq
if( aaqq>aapp0 )theta = -theta
if( abs( theta )>bigtheta ) then
t = half / theta
fastr( 3_${ik}$ ) = t*d( p ) / d( q )
fastr( 4_${ik}$ ) = -t*d( q ) / d( p )
call stdlib${ii}$_${ri}$rotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$, fastr )
if( rsvec )call stdlib${ii}$_${ri}$rotm( mvl,v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ),&
1_${ik}$,fastr )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq )
if( aaqq>aapp0 )thsign = -thsign
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
apoaq = d( p ) / d( q )
aqoap = d( q ) / d( p )
if( d( p )>=one ) then
if( d( q )>=one ) then
fastr( 3_${ik}$ ) = t*apoaq
fastr( 4_${ik}$ ) = -t*aqoap
d( p ) = d( p )*cs
d( q ) = d( q )*cs
call stdlib${ii}$_${ri}$rotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$,&
fastr )
if( rsvec )call stdlib${ii}$_${ri}$rotm( mvl,v( 1_${ik}$, p ), 1_${ik}$, v( &
1_${ik}$, q ),1_${ik}$, fastr )
else
call stdlib${ii}$_${ri}$axpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( 1_${ik}$, &
p ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( &
1_${ik}$, q ), 1_${ik}$ )
if( rsvec ) then
call stdlib${ii}$_${ri}$axpy( mvl, -t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,v(&
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ), 1_${ik}$,&
v( 1_${ik}$, q ), 1_${ik}$ )
end if
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
end if
else
if( d( q )>=one ) then
call stdlib${ii}$_${ri}$axpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q &
), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( m, -cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
if( rsvec ) then
call stdlib${ii}$_${ri}$axpy( mvl, t*apoaq,v( 1_${ik}$, p ), 1_${ik}$,v( &
1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q ), &
1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
else
if( d( p )>=d( q ) ) then
call stdlib${ii}$_${ri}$axpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,&
a( 1_${ik}$, q ), 1_${ik}$ )
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
if( rsvec ) then
call stdlib${ii}$_${ri}$axpy( mvl,-t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,&
v( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ),&
1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
end if
else
call stdlib${ii}$_${ri}$axpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$,&
q ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( m,-cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,&
a( 1_${ik}$, p ), 1_${ik}$ )
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
if( rsvec ) then
call stdlib${ii}$_${ri}$axpy( mvl,t*apoaq, v( 1_${ik}$, p ),1_${ik}$, &
v( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q )&
, 1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
end if
end if
end if
end if
else
if( aapp>aaqq ) then
call stdlib${ii}$_${ri}$copy( m, a( 1_${ik}$, p ), 1_${ik}$, work,1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
temp1 = -aapq*d( p ) / d( q )
call stdlib${ii}$_${ri}$axpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
else
call stdlib${ii}$_${ri}$copy( m, a( 1_${ik}$, q ), 1_${ik}$, work,1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
temp1 = -aapq*d( q ) / d( p )
call stdlib${ii}$_${ri}$axpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, one, aapp,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
sva( p ) = aapp*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
end if
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q)
! .. recompute sva(q)
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_${ri}$nrm2( m, a( 1_${ik}$, q ), 1_${ik}$ )*d( q )
else
t = zero
aaqq = one
call stdlib${ii}$_${ri}$lassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )*d( q )
end if
end if
if( ( aapp / aapp0 )**2_${ik}$<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_${ri}$nrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )*d( p )
else
t = zero
aapp = one
call stdlib${ii}$_${ri}$lassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )*d( p )
end if
sva( p ) = aapp
end if
! end of ok rotation
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
if( ( i<=swband ) .and. ( ijblsk>=blskip ) )then
sva( p ) = aapp
notrot = 0_${ik}$
go to 2011
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
aapp = -aapp
notrot = 0_${ik}$
go to 2203
end if
end do loop_2200
! end of the q-loop
2203 continue
sva( p ) = aapp
else
if( aapp==zero )notrot = notrot +min( jgl+kbl-1, n ) - jgl + 1_${ik}$
if( aapp<zero )notrot = 0_${ik}$
end if
end do loop_2100
! end of the p-loop
end do loop_2010
! end of the jbc-loop
2011 continue
! 2011 bailed out of the jbc-loop
do p = igl, min( igl+kbl-1, n )
sva( p ) = abs( sva( p ) )
end do
end do loop_2000
! 2000 :: end of the ibr-loop
! .. update sva(n)
if( ( sva( n )<rootbig ) .and. ( sva( n )>rootsfmin ) )then
sva( n ) = stdlib${ii}$_${ri}$nrm2( m, a( 1_${ik}$, n ), 1_${ik}$ )*d( n )
else
t = zero
aapp = one
call stdlib${ii}$_${ri}$lassq( m, a( 1_${ik}$, n ), 1_${ik}$, t, aapp )
sva( n ) = t*sqrt( aapp )*d( n )
end if
! additional steering devices
if( ( i<swband ) .and. ( ( mxaapq<=roottol ) .or.( iswrot<=n ) ) )swband = i
if( ( i>swband+1 ) .and. ( mxaapq<real( n,KIND=${rk}$)*tol ) .and.( real( n,KIND=${rk}$)&
*mxaapq*mxsinj<tol ) ) then
go to 1994
end if
if( notrot>=emptsw )go to 1994
end do loop_1993
! end i=1:nsweep loop
! #:) reaching this point means that the procedure has completed the given
! number of iterations.
info = nsweep - 1_${ik}$
go to 1995
1994 continue
! #:) reaching this point means that during the i-th sweep all pivots were
! below the given tolerance, causing early exit.
info = 0_${ik}$
! #:) info = 0 confirms successful iterations.
1995 continue
! sort the vector d.
do p = 1, n - 1
q = stdlib${ii}$_i${ri}$amax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
temp1 = d( p )
d( p ) = d( q )
d( q ) = temp1
call stdlib${ii}$_${ri}$swap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_${ri}$swap( mvl, v( 1_${ik}$, p ), 1_${ik}$, v( 1_${ik}$, q ), 1_${ik}$ )
end if
end do
return
end subroutine stdlib${ii}$_${ri}$gsvj0
#:endif
#:endfor
pure module subroutine stdlib${ii}$_cgsvj0( jobv, m, n, a, lda, d, sva, mv, v, ldv, eps,sfmin, tol, &
!! CGSVJ0 is called from CGESVJ as a pre-processor and that is its main
!! purpose. It applies Jacobi rotations in the same way as CGESVJ does, but
!! it does not check convergence (stopping criterion). Few tuning
!! parameters (marked by [TP]) are available for the implementer.
nsweep, work, lwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, ldv, lwork, m, mv, n, nsweep
real(sp), intent(in) :: eps, sfmin, tol
character, intent(in) :: jobv
! Array Arguments
complex(sp), intent(inout) :: a(lda,*), d(n), v(ldv,*)
complex(sp), intent(out) :: work(lwork)
real(sp), intent(inout) :: sva(n)
! =====================================================================
! Local Scalars
complex(sp) :: aapq, ompq
real(sp) :: aapp, aapp0, aapq1, aaqq, apoaq, aqoap, big, bigtheta, cs, mxaapq, mxsinj, &
rootbig, rooteps, rootsfmin, roottol, small, sn, t, temp1, theta, thsign
integer(${ik}$) :: blskip, emptsw, i, ibr, ierr, igl, ijblsk, ir1, iswrot, jbc, jgl, kbl, &
lkahead, mvl, nbl, notrot, p, pskipped, q, rowskip, swband
logical(lk) :: applv, rotok, rsvec
! Intrinsic Functions
! from lapack
! Executable Statements
! test the input parameters.
applv = stdlib_lsame( jobv, 'A' )
rsvec = stdlib_lsame( jobv, 'V' )
if( .not.( rsvec .or. applv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( ( n<0_${ik}$ ) .or. ( n>m ) ) then
info = -3_${ik}$
else if( lda<m ) then
info = -5_${ik}$
else if( ( rsvec.or.applv ) .and. ( mv<0_${ik}$ ) ) then
info = -8_${ik}$
else if( ( rsvec.and.( ldv<n ) ).or.( applv.and.( ldv<mv ) ) ) then
info = -10_${ik}$
else if( tol<=eps ) then
info = -13_${ik}$
else if( nsweep<0_${ik}$ ) then
info = -14_${ik}$
else if( lwork<m ) then
info = -16_${ik}$
else
info = 0_${ik}$
end if
! #:(
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'CGSVJ0', -info )
return
end if
if( rsvec ) then
mvl = n
else if( applv ) then
mvl = mv
end if
rsvec = rsvec .or. applv
rooteps = sqrt( eps )
rootsfmin = sqrt( sfmin )
small = sfmin / eps
big = one / sfmin
rootbig = one / rootsfmin
bigtheta = one / rooteps
roottol = sqrt( tol )
! .. row-cyclic jacobi svd algorithm with column pivoting ..
emptsw = ( n*( n-1 ) ) / 2_${ik}$
notrot = 0_${ik}$
! .. row-cyclic pivot strategy with de rijk's pivoting ..
swband = 0_${ik}$
! [tp] swband is a tuning parameter [tp]. it is meaningful and effective
! if stdlib${ii}$_cgesvj is used as a computational routine in the preconditioned
! jacobi svd algorithm stdlib${ii}$_cgejsv. for sweeps i=1:swband the procedure
! works on pivots inside a band-like region around the diagonal.
! the boundaries are determined dynamically, based on the number of
! pivots above a threshold.
kbl = min( 8_${ik}$, n )
! [tp] kbl is a tuning parameter that defines the tile size in the
! tiling of the p-q loops of pivot pairs. in general, an optimal
! value of kbl depends on the matrix dimensions and on the
! parameters of the computer's memory.
nbl = n / kbl
if( ( nbl*kbl )/=n )nbl = nbl + 1_${ik}$
blskip = kbl**2_${ik}$
! [tp] blkskip is a tuning parameter that depends on swband and kbl.
rowskip = min( 5_${ik}$, kbl )
! [tp] rowskip is a tuning parameter.
lkahead = 1_${ik}$
! [tp] lkahead is a tuning parameter.
! quasi block transformations, using the lower (upper) triangular
! structure of the input matrix. the quasi-block-cycling usually
! invokes cubic convergence. big part of this cycle is done inside
! canonical subspaces of dimensions less than m.
! .. row-cyclic pivot strategy with de rijk's pivoting ..
loop_1993: do i = 1, nsweep
! .. go go go ...
mxaapq = zero
mxsinj = zero
iswrot = 0_${ik}$
notrot = 0_${ik}$
pskipped = 0_${ik}$
! each sweep is unrolled using kbl-by-kbl tiles over the pivot pairs
! 1 <= p < q <= n. this is the first step toward a blocked implementation
! of the rotations. new implementation, based on block transformations,
! is under development.
loop_2000: do ibr = 1, nbl
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_1002: do ir1 = 0, min( lkahead, nbl-ibr )
igl = igl + ir1*kbl
loop_2001: do p = igl, min( igl+kbl-1, n-1 )
! .. de rijk's pivoting
q = stdlib${ii}$_isamax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
call stdlib${ii}$_cswap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_cswap( mvl, v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
aapq = d(p)
d(p) = d(q)
d(q) = aapq
end if
if( ir1==0_${ik}$ ) then
! column norms are periodically updated by explicit
! norm computation.
! caveat:
! unfortunately, some blas implementations compute sncrm2(m,a(1,p),1)
! as sqrt(s=stdlib${ii}$_cdotc(m,a(1,p),1,a(1,p),1)), which may cause the result to
! overflow for ||a(:,p)||_2 > sqrt(overflow_threshold), and to
! underflow for ||a(:,p)||_2 < sqrt(underflow_threshold).
! hence, stdlib${ii}$_scnrm2 cannot be trusted, not even in the case when
! the true norm is far from the under(over)flow boundaries.
! if properly implemented stdlib${ii}$_scnrm2 is available, the if-then-else-end if
! below should be replaced with "aapp = stdlib${ii}$_scnrm2( m, a(1,p), 1 )".
if( ( sva( p )<rootbig ) .and.( sva( p )>rootsfmin ) ) then
sva( p ) = stdlib${ii}$_scnrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )
else
temp1 = zero
aapp = one
call stdlib${ii}$_classq( m, a( 1_${ik}$, p ), 1_${ik}$, temp1, aapp )
sva( p ) = temp1*sqrt( aapp )
end if
aapp = sva( p )
else
aapp = sva( p )
end if
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2002: do q = p + 1, min( igl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
if( aaqq>=one ) then
rotok = ( small*aapp )<=aaqq
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_cdotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq ) / aapp
else
call stdlib${ii}$_ccopy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work, lda, &
ierr )
aapq = stdlib${ii}$_cdotc( m, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq
end if
else
rotok = aapp<=( aaqq / small )
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_cdotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aapp ) / aaqq
else
call stdlib${ii}$_ccopy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aaqq,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_cdotc( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ ) / &
aapp
end if
end if
! aapq = aapq * conjg( cwork(p) ) * cwork(q)
aapq1 = -abs(aapq)
mxaapq = max( mxaapq, -aapq1 )
! to rotate or not to rotate, that is the question ...
if( abs( aapq1 )>tol ) then
ompq = aapq / abs(aapq)
! Rotate
! [rtd] rotated = rotated + one
if( ir1==0_${ik}$ ) then
notrot = 0_${ik}$
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
end if
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq )/aapq1
if( abs( theta )>bigtheta ) then
t = half / theta
cs = one
call stdlib${ii}$_crot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*t )
if ( rsvec ) then
call stdlib${ii}$_crot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*t )
end if
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq1 )
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
call stdlib${ii}$_crot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*sn )
if ( rsvec ) then
call stdlib${ii}$_crot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*sn )
end if
end if
d(p) = -d(q) * ompq
else
! .. have to use modified gram-schmidt like transformation
call stdlib${ii}$_ccopy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one, m,1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one, m,1_${ik}$, a( 1_${ik}$, q ), &
lda, ierr )
call stdlib${ii}$_caxpy( m, -aapq, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq, m,1_${ik}$, a( 1_${ik}$, q ), &
lda, ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q), sva(p)
! recompute sva(q), sva(p).
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_scnrm2( m, a( 1_${ik}$, q ), 1_${ik}$ )
else
t = zero
aaqq = one
call stdlib${ii}$_classq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )
end if
end if
if( ( aapp / aapp0 )<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_scnrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_classq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )
end if
sva( p ) = aapp
end if
else
! a(:,p) and a(:,q) already numerically orthogonal
if( ir1==0_${ik}$ )notrot = notrot + 1_${ik}$
! [rtd] skipped = skipped + 1
pskipped = pskipped + 1_${ik}$
end if
else
! a(:,q) is zero column
if( ir1==0_${ik}$ )notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
if( ir1==0_${ik}$ )aapp = -aapp
notrot = 0_${ik}$
go to 2103
end if
end do loop_2002
! end q-loop
2103 continue
! bailed out of q-loop
sva( p ) = aapp
else
sva( p ) = aapp
if( ( ir1==0_${ik}$ ) .and. ( aapp==zero ) )notrot = notrot + min( igl+kbl-1, &
n ) - p
end if
end do loop_2001
! end of the p-loop
! end of doing the block ( ibr, ibr )
end do loop_1002
! end of ir1-loop
! ... go to the off diagonal blocks
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_2010: do jbc = ibr + 1, nbl
jgl = ( jbc-1 )*kbl + 1_${ik}$
! doing the block at ( ibr, jbc )
ijblsk = 0_${ik}$
loop_2100: do p = igl, min( igl+kbl-1, n )
aapp = sva( p )
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2200: do q = jgl, min( jgl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
! M X 2 Jacobi Svd
! safe gram matrix computation
if( aaqq>=one ) then
if( aapp>=aaqq ) then
rotok = ( small*aapp )<=aaqq
else
rotok = ( small*aaqq )<=aapp
end if
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_cdotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq ) / aapp
else
call stdlib${ii}$_ccopy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aapp,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_cdotc( m, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq
end if
else
if( aapp>=aaqq ) then
rotok = aapp<=( aaqq / small )
else
rotok = aaqq<=( aapp / small )
end if
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_cdotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / max(&
aaqq,aapp) )/ min(aaqq,aapp)
else
call stdlib${ii}$_ccopy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aaqq,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_cdotc( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ ) / &
aapp
end if
end if
! aapq = aapq * conjg(cwork(p))*cwork(q)
aapq1 = -abs(aapq)
mxaapq = max( mxaapq, -aapq1 )
! to rotate or not to rotate, that is the question ...
if( abs( aapq1 )>tol ) then
ompq = aapq / abs(aapq)
notrot = 0_${ik}$
! [rtd] rotated = rotated + 1
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq )/ aapq1
if( aaqq>aapp0 )theta = -theta
if( abs( theta )>bigtheta ) then
t = half / theta
cs = one
call stdlib${ii}$_crot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*t )
if( rsvec ) then
call stdlib${ii}$_crot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*t )
end if
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq1 )
if( aaqq>aapp0 )thsign = -thsign
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
call stdlib${ii}$_crot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*sn )
if( rsvec ) then
call stdlib${ii}$_crot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*sn )
end if
end if
d(p) = -d(q) * ompq
else
! .. have to use modified gram-schmidt like transformation
if( aapp>aaqq ) then
call stdlib${ii}$_ccopy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work,lda,&
ierr )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
call stdlib${ii}$_caxpy( m, -aapq, work,1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
else
call stdlib${ii}$_ccopy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, work,lda,&
ierr )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
call stdlib${ii}$_caxpy( m, -conjg(aapq),work, 1_${ik}$, a( 1_${ik}$, p ), 1_${ik}$ &
)
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, one, aapp,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
sva( p ) = aapp*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
end if
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q), sva(p)
! .. recompute sva(q), sva(p)
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_scnrm2( m, a( 1_${ik}$, q ), 1_${ik}$)
else
t = zero
aaqq = one
call stdlib${ii}$_classq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )
end if
end if
if( ( aapp / aapp0 )**2_${ik}$<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_scnrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_classq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )
end if
sva( p ) = aapp
end if
! end of ok rotation
else
notrot = notrot + 1_${ik}$
! [rtd] skipped = skipped + 1
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
if( ( i<=swband ) .and. ( ijblsk>=blskip ) )then
sva( p ) = aapp
notrot = 0_${ik}$
go to 2011
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
aapp = -aapp
notrot = 0_${ik}$
go to 2203
end if
end do loop_2200
! end of the q-loop
2203 continue
sva( p ) = aapp
else
if( aapp==zero )notrot = notrot +min( jgl+kbl-1, n ) - jgl + 1_${ik}$
if( aapp<zero )notrot = 0_${ik}$
end if
end do loop_2100
! end of the p-loop
end do loop_2010
! end of the jbc-loop
2011 continue
! 2011 bailed out of the jbc-loop
do p = igl, min( igl+kbl-1, n )
sva( p ) = abs( sva( p ) )
end do
! **
end do loop_2000
! 2000 :: end of the ibr-loop
! .. update sva(n)
if( ( sva( n )<rootbig ) .and. ( sva( n )>rootsfmin ) )then
sva( n ) = stdlib${ii}$_scnrm2( m, a( 1_${ik}$, n ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_classq( m, a( 1_${ik}$, n ), 1_${ik}$, t, aapp )
sva( n ) = t*sqrt( aapp )
end if
! additional steering devices
if( ( i<swband ) .and. ( ( mxaapq<=roottol ) .or.( iswrot<=n ) ) )swband = i
if( ( i>swband+1 ) .and. ( mxaapq<sqrt( real( n,KIND=sp) )*tol ) .and. ( real( n,&
KIND=sp)*mxaapq*mxsinj<tol ) ) then
go to 1994
end if
if( notrot>=emptsw )go to 1994
end do loop_1993
! end i=1:nsweep loop
! #:( reaching this point means that the procedure has not converged.
info = nsweep - 1_${ik}$
go to 1995
1994 continue
! #:) reaching this point means numerical convergence after the i-th
! sweep.
info = 0_${ik}$
! #:) info = 0 confirms successful iterations.
1995 continue
! sort the vector sva() of column norms.
do p = 1, n - 1
q = stdlib${ii}$_isamax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
aapq = d( p )
d( p ) = d( q )
d( q ) = aapq
call stdlib${ii}$_cswap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_cswap( mvl, v( 1_${ik}$, p ), 1_${ik}$, v( 1_${ik}$, q ), 1_${ik}$ )
end if
end do
return
end subroutine stdlib${ii}$_cgsvj0
pure module subroutine stdlib${ii}$_zgsvj0( jobv, m, n, a, lda, d, sva, mv, v, ldv, eps,sfmin, tol, &
!! ZGSVJ0 is called from ZGESVJ as a pre-processor and that is its main
!! purpose. It applies Jacobi rotations in the same way as ZGESVJ does, but
!! it does not check convergence (stopping criterion). Few tuning
!! parameters (marked by [TP]) are available for the implementer.
nsweep, work, lwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, ldv, lwork, m, mv, n, nsweep
real(dp), intent(in) :: eps, sfmin, tol
character, intent(in) :: jobv
! Array Arguments
complex(dp), intent(inout) :: a(lda,*), d(n), v(ldv,*)
complex(dp), intent(out) :: work(lwork)
real(dp), intent(inout) :: sva(n)
! =====================================================================
! Local Scalars
complex(dp) :: aapq, ompq
real(dp) :: aapp, aapp0, aapq1, aaqq, apoaq, aqoap, big, bigtheta, cs, mxaapq, mxsinj, &
rootbig, rooteps, rootsfmin, roottol, small, sn, t, temp1, theta, thsign
integer(${ik}$) :: blskip, emptsw, i, ibr, ierr, igl, ijblsk, ir1, iswrot, jbc, jgl, kbl, &
lkahead, mvl, nbl, notrot, p, pskipped, q, rowskip, swband
logical(lk) :: applv, rotok, rsvec
! Intrinsic Functions
! from lapack
! Executable Statements
! test the input parameters.
applv = stdlib_lsame( jobv, 'A' )
rsvec = stdlib_lsame( jobv, 'V' )
if( .not.( rsvec .or. applv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( ( n<0_${ik}$ ) .or. ( n>m ) ) then
info = -3_${ik}$
else if( lda<m ) then
info = -5_${ik}$
else if( ( rsvec.or.applv ) .and. ( mv<0_${ik}$ ) ) then
info = -8_${ik}$
else if( ( rsvec.and.( ldv<n ) ).or.( applv.and.( ldv<mv ) ) ) then
info = -10_${ik}$
else if( tol<=eps ) then
info = -13_${ik}$
else if( nsweep<0_${ik}$ ) then
info = -14_${ik}$
else if( lwork<m ) then
info = -16_${ik}$
else
info = 0_${ik}$
end if
! #:(
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZGSVJ0', -info )
return
end if
if( rsvec ) then
mvl = n
else if( applv ) then
mvl = mv
end if
rsvec = rsvec .or. applv
rooteps = sqrt( eps )
rootsfmin = sqrt( sfmin )
small = sfmin / eps
big = one / sfmin
rootbig = one / rootsfmin
bigtheta = one / rooteps
roottol = sqrt( tol )
! .. row-cyclic jacobi svd algorithm with column pivoting ..
emptsw = ( n*( n-1 ) ) / 2_${ik}$
notrot = 0_${ik}$
! .. row-cyclic pivot strategy with de rijk's pivoting ..
swband = 0_${ik}$
! [tp] swband is a tuning parameter [tp]. it is meaningful and effective
! if stdlib${ii}$_zgesvj is used as a computational routine in the preconditioned
! jacobi svd algorithm stdlib${ii}$_zgejsv. for sweeps i=1:swband the procedure
! works on pivots inside a band-like region around the diagonal.
! the boundaries are determined dynamically, based on the number of
! pivots above a threshold.
kbl = min( 8_${ik}$, n )
! [tp] kbl is a tuning parameter that defines the tile size in the
! tiling of the p-q loops of pivot pairs. in general, an optimal
! value of kbl depends on the matrix dimensions and on the
! parameters of the computer's memory.
nbl = n / kbl
if( ( nbl*kbl )/=n )nbl = nbl + 1_${ik}$
blskip = kbl**2_${ik}$
! [tp] blkskip is a tuning parameter that depends on swband and kbl.
rowskip = min( 5_${ik}$, kbl )
! [tp] rowskip is a tuning parameter.
lkahead = 1_${ik}$
! [tp] lkahead is a tuning parameter.
! quasi block transformations, using the lower (upper) triangular
! structure of the input matrix. the quasi-block-cycling usually
! invokes cubic convergence. big part of this cycle is done inside
! canonical subspaces of dimensions less than m.
! .. row-cyclic pivot strategy with de rijk's pivoting ..
loop_1993: do i = 1, nsweep
! .. go go go ...
mxaapq = zero
mxsinj = zero
iswrot = 0_${ik}$
notrot = 0_${ik}$
pskipped = 0_${ik}$
! each sweep is unrolled using kbl-by-kbl tiles over the pivot pairs
! 1 <= p < q <= n. this is the first step toward a blocked implementation
! of the rotations. new implementation, based on block transformations,
! is under development.
loop_2000: do ibr = 1, nbl
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_1002: do ir1 = 0, min( lkahead, nbl-ibr )
igl = igl + ir1*kbl
loop_2001: do p = igl, min( igl+kbl-1, n-1 )
! .. de rijk's pivoting
q = stdlib${ii}$_idamax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
call stdlib${ii}$_zswap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_zswap( mvl, v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
aapq = d(p)
d(p) = d(q)
d(q) = aapq
end if
if( ir1==0_${ik}$ ) then
! column norms are periodically updated by explicit
! norm computation.
! caveat:
! unfortunately, some blas implementations compute sncrm2(m,a(1,p),1)
! as sqrt(s=stdlib${ii}$_zdotc(m,a(1,p),1,a(1,p),1)), which may cause the result to
! overflow for ||a(:,p)||_2 > sqrt(overflow_threshold), and to
! underflow for ||a(:,p)||_2 < sqrt(underflow_threshold).
! hence, stdlib${ii}$_dznrm2 cannot be trusted, not even in the case when
! the true norm is far from the under(over)flow boundaries.
! if properly implemented stdlib${ii}$_dznrm2 is available, the if-then-else-end if
! below should be replaced with "aapp = stdlib${ii}$_dznrm2( m, a(1,p), 1 )".
if( ( sva( p )<rootbig ) .and.( sva( p )>rootsfmin ) ) then
sva( p ) = stdlib${ii}$_dznrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )
else
temp1 = zero
aapp = one
call stdlib${ii}$_zlassq( m, a( 1_${ik}$, p ), 1_${ik}$, temp1, aapp )
sva( p ) = temp1*sqrt( aapp )
end if
aapp = sva( p )
else
aapp = sva( p )
end if
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2002: do q = p + 1, min( igl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
if( aaqq>=one ) then
rotok = ( small*aapp )<=aaqq
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_zdotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq ) / aapp
else
call stdlib${ii}$_zcopy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work, lda, &
ierr )
aapq = stdlib${ii}$_zdotc( m, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq
end if
else
rotok = aapp<=( aaqq / small )
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_zdotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aapp ) / aaqq
else
call stdlib${ii}$_zcopy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_zdotc( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ ) / &
aapp
end if
end if
! aapq = aapq * conjg( cwork(p) ) * cwork(q)
aapq1 = -abs(aapq)
mxaapq = max( mxaapq, -aapq1 )
! to rotate or not to rotate, that is the question ...
if( abs( aapq1 )>tol ) then
ompq = aapq / abs(aapq)
! Rotate
! [rtd] rotated = rotated + one
if( ir1==0_${ik}$ ) then
notrot = 0_${ik}$
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
end if
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq )/aapq1
if( abs( theta )>bigtheta ) then
t = half / theta
cs = one
call stdlib${ii}$_zrot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*t )
if ( rsvec ) then
call stdlib${ii}$_zrot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*t )
end if
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq1 )
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
call stdlib${ii}$_zrot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*sn )
if ( rsvec ) then
call stdlib${ii}$_zrot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*sn )
end if
end if
d(p) = -d(q) * ompq
else
! .. have to use modified gram-schmidt like transformation
call stdlib${ii}$_zcopy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one, m,1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one, m,1_${ik}$, a( 1_${ik}$, q ), &
lda, ierr )
call stdlib${ii}$_zaxpy( m, -aapq, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq, m,1_${ik}$, a( 1_${ik}$, q ), &
lda, ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q), sva(p)
! recompute sva(q), sva(p).
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_dznrm2( m, a( 1_${ik}$, q ), 1_${ik}$ )
else
t = zero
aaqq = one
call stdlib${ii}$_zlassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )
end if
end if
if( ( aapp / aapp0 )<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_dznrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_zlassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )
end if
sva( p ) = aapp
end if
else
! a(:,p) and a(:,q) already numerically orthogonal
if( ir1==0_${ik}$ )notrot = notrot + 1_${ik}$
! [rtd] skipped = skipped + 1
pskipped = pskipped + 1_${ik}$
end if
else
! a(:,q) is zero column
if( ir1==0_${ik}$ )notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
if( ir1==0_${ik}$ )aapp = -aapp
notrot = 0_${ik}$
go to 2103
end if
end do loop_2002
! end q-loop
2103 continue
! bailed out of q-loop
sva( p ) = aapp
else
sva( p ) = aapp
if( ( ir1==0_${ik}$ ) .and. ( aapp==zero ) )notrot = notrot + min( igl+kbl-1, &
n ) - p
end if
end do loop_2001
! end of the p-loop
! end of doing the block ( ibr, ibr )
end do loop_1002
! end of ir1-loop
! ... go to the off diagonal blocks
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_2010: do jbc = ibr + 1, nbl
jgl = ( jbc-1 )*kbl + 1_${ik}$
! doing the block at ( ibr, jbc )
ijblsk = 0_${ik}$
loop_2100: do p = igl, min( igl+kbl-1, n )
aapp = sva( p )
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2200: do q = jgl, min( jgl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
! M X 2 Jacobi Svd
! safe gram matrix computation
if( aaqq>=one ) then
if( aapp>=aaqq ) then
rotok = ( small*aapp )<=aaqq
else
rotok = ( small*aaqq )<=aapp
end if
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_zdotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq ) / aapp
else
call stdlib${ii}$_zcopy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aapp,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_zdotc( m, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq
end if
else
if( aapp>=aaqq ) then
rotok = aapp<=( aaqq / small )
else
rotok = aaqq<=( aapp / small )
end if
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_zdotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / max(&
aaqq,aapp) )/ min(aaqq,aapp)
else
call stdlib${ii}$_zcopy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_zdotc( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ ) / &
aapp
end if
end if
! aapq = aapq * conjg(cwork(p))*cwork(q)
aapq1 = -abs(aapq)
mxaapq = max( mxaapq, -aapq1 )
! to rotate or not to rotate, that is the question ...
if( abs( aapq1 )>tol ) then
ompq = aapq / abs(aapq)
notrot = 0_${ik}$
! [rtd] rotated = rotated + 1
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq )/ aapq1
if( aaqq>aapp0 )theta = -theta
if( abs( theta )>bigtheta ) then
t = half / theta
cs = one
call stdlib${ii}$_zrot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*t )
if( rsvec ) then
call stdlib${ii}$_zrot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*t )
end if
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq1 )
if( aaqq>aapp0 )thsign = -thsign
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
call stdlib${ii}$_zrot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*sn )
if( rsvec ) then
call stdlib${ii}$_zrot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*sn )
end if
end if
d(p) = -d(q) * ompq
else
! .. have to use modified gram-schmidt like transformation
if( aapp>aaqq ) then
call stdlib${ii}$_zcopy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work,lda,&
ierr )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
call stdlib${ii}$_zaxpy( m, -aapq, work,1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
else
call stdlib${ii}$_zcopy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, work,lda,&
ierr )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
call stdlib${ii}$_zaxpy( m, -conjg(aapq),work, 1_${ik}$, a( 1_${ik}$, p ), 1_${ik}$ &
)
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, one, aapp,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
sva( p ) = aapp*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
end if
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q), sva(p)
! .. recompute sva(q), sva(p)
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_dznrm2( m, a( 1_${ik}$, q ), 1_${ik}$)
else
t = zero
aaqq = one
call stdlib${ii}$_zlassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )
end if
end if
if( ( aapp / aapp0 )**2_${ik}$<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_dznrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_zlassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )
end if
sva( p ) = aapp
end if
! end of ok rotation
else
notrot = notrot + 1_${ik}$
! [rtd] skipped = skipped + 1
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
if( ( i<=swband ) .and. ( ijblsk>=blskip ) )then
sva( p ) = aapp
notrot = 0_${ik}$
go to 2011
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
aapp = -aapp
notrot = 0_${ik}$
go to 2203
end if
end do loop_2200
! end of the q-loop
2203 continue
sva( p ) = aapp
else
if( aapp==zero )notrot = notrot +min( jgl+kbl-1, n ) - jgl + 1_${ik}$
if( aapp<zero )notrot = 0_${ik}$
end if
end do loop_2100
! end of the p-loop
end do loop_2010
! end of the jbc-loop
2011 continue
! 2011 bailed out of the jbc-loop
do p = igl, min( igl+kbl-1, n )
sva( p ) = abs( sva( p ) )
end do
! **
end do loop_2000
! 2000 :: end of the ibr-loop
! .. update sva(n)
if( ( sva( n )<rootbig ) .and. ( sva( n )>rootsfmin ) )then
sva( n ) = stdlib${ii}$_dznrm2( m, a( 1_${ik}$, n ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_zlassq( m, a( 1_${ik}$, n ), 1_${ik}$, t, aapp )
sva( n ) = t*sqrt( aapp )
end if
! additional steering devices
if( ( i<swband ) .and. ( ( mxaapq<=roottol ) .or.( iswrot<=n ) ) )swband = i
if( ( i>swband+1 ) .and. ( mxaapq<sqrt( real( n,KIND=dp) )*tol ) .and. ( real( n,&
KIND=dp)*mxaapq*mxsinj<tol ) ) then
go to 1994
end if
if( notrot>=emptsw )go to 1994
end do loop_1993
! end i=1:nsweep loop
! #:( reaching this point means that the procedure has not converged.
info = nsweep - 1_${ik}$
go to 1995
1994 continue
! #:) reaching this point means numerical convergence after the i-th
! sweep.
info = 0_${ik}$
! #:) info = 0 confirms successful iterations.
1995 continue
! sort the vector sva() of column norms.
do p = 1, n - 1
q = stdlib${ii}$_idamax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
aapq = d( p )
d( p ) = d( q )
d( q ) = aapq
call stdlib${ii}$_zswap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_zswap( mvl, v( 1_${ik}$, p ), 1_${ik}$, v( 1_${ik}$, q ), 1_${ik}$ )
end if
end do
return
end subroutine stdlib${ii}$_zgsvj0
#:for ck,ct,ci in CMPLX_KINDS_TYPES
#:if not ck in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ci}$gsvj0( jobv, m, n, a, lda, d, sva, mv, v, ldv, eps,sfmin, tol, &
!! ZGSVJ0: is called from ZGESVJ as a pre-processor and that is its main
!! purpose. It applies Jacobi rotations in the same way as ZGESVJ does, but
!! it does not check convergence (stopping criterion). Few tuning
!! parameters (marked by [TP]) are available for the implementer.
nsweep, work, lwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${ck}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, ldv, lwork, m, mv, n, nsweep
real(${ck}$), intent(in) :: eps, sfmin, tol
character, intent(in) :: jobv
! Array Arguments
complex(${ck}$), intent(inout) :: a(lda,*), d(n), v(ldv,*)
complex(${ck}$), intent(out) :: work(lwork)
real(${ck}$), intent(inout) :: sva(n)
! =====================================================================
! Local Scalars
complex(${ck}$) :: aapq, ompq
real(${ck}$) :: aapp, aapp0, aapq1, aaqq, apoaq, aqoap, big, bigtheta, cs, mxaapq, mxsinj, &
rootbig, rooteps, rootsfmin, roottol, small, sn, t, temp1, theta, thsign
integer(${ik}$) :: blskip, emptsw, i, ibr, ierr, igl, ijblsk, ir1, iswrot, jbc, jgl, kbl, &
lkahead, mvl, nbl, notrot, p, pskipped, q, rowskip, swband
logical(lk) :: applv, rotok, rsvec
! Intrinsic Functions
! from lapack
! Executable Statements
! test the input parameters.
applv = stdlib_lsame( jobv, 'A' )
rsvec = stdlib_lsame( jobv, 'V' )
if( .not.( rsvec .or. applv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( ( n<0_${ik}$ ) .or. ( n>m ) ) then
info = -3_${ik}$
else if( lda<m ) then
info = -5_${ik}$
else if( ( rsvec.or.applv ) .and. ( mv<0_${ik}$ ) ) then
info = -8_${ik}$
else if( ( rsvec.and.( ldv<n ) ).or.( applv.and.( ldv<mv ) ) ) then
info = -10_${ik}$
else if( tol<=eps ) then
info = -13_${ik}$
else if( nsweep<0_${ik}$ ) then
info = -14_${ik}$
else if( lwork<m ) then
info = -16_${ik}$
else
info = 0_${ik}$
end if
! #:(
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZGSVJ0', -info )
return
end if
if( rsvec ) then
mvl = n
else if( applv ) then
mvl = mv
end if
rsvec = rsvec .or. applv
rooteps = sqrt( eps )
rootsfmin = sqrt( sfmin )
small = sfmin / eps
big = one / sfmin
rootbig = one / rootsfmin
bigtheta = one / rooteps
roottol = sqrt( tol )
! .. row-cyclic jacobi svd algorithm with column pivoting ..
emptsw = ( n*( n-1 ) ) / 2_${ik}$
notrot = 0_${ik}$
! .. row-cyclic pivot strategy with de rijk's pivoting ..
swband = 0_${ik}$
! [tp] swband is a tuning parameter [tp]. it is meaningful and effective
! if stdlib${ii}$_${ci}$gesvj is used as a computational routine in the preconditioned
! jacobi svd algorithm stdlib${ii}$_${ci}$gejsv. for sweeps i=1:swband the procedure
! works on pivots inside a band-like region around the diagonal.
! the boundaries are determined dynamically, based on the number of
! pivots above a threshold.
kbl = min( 8_${ik}$, n )
! [tp] kbl is a tuning parameter that defines the tile size in the
! tiling of the p-q loops of pivot pairs. in general, an optimal
! value of kbl depends on the matrix dimensions and on the
! parameters of the computer's memory.
nbl = n / kbl
if( ( nbl*kbl )/=n )nbl = nbl + 1_${ik}$
blskip = kbl**2_${ik}$
! [tp] blkskip is a tuning parameter that depends on swband and kbl.
rowskip = min( 5_${ik}$, kbl )
! [tp] rowskip is a tuning parameter.
lkahead = 1_${ik}$
! [tp] lkahead is a tuning parameter.
! quasi block transformations, using the lower (upper) triangular
! structure of the input matrix. the quasi-block-cycling usually
! invokes cubic convergence. big part of this cycle is done inside
! canonical subspaces of dimensions less than m.
! .. row-cyclic pivot strategy with de rijk's pivoting ..
loop_1993: do i = 1, nsweep
! .. go go go ...
mxaapq = zero
mxsinj = zero
iswrot = 0_${ik}$
notrot = 0_${ik}$
pskipped = 0_${ik}$
! each sweep is unrolled using kbl-by-kbl tiles over the pivot pairs
! 1 <= p < q <= n. this is the first step toward a blocked implementation
! of the rotations. new implementation, based on block transformations,
! is under development.
loop_2000: do ibr = 1, nbl
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_1002: do ir1 = 0, min( lkahead, nbl-ibr )
igl = igl + ir1*kbl
loop_2001: do p = igl, min( igl+kbl-1, n-1 )
! .. de rijk's pivoting
q = stdlib${ii}$_i${c2ri(ci)}$amax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
call stdlib${ii}$_${ci}$swap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_${ci}$swap( mvl, v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
aapq = d(p)
d(p) = d(q)
d(q) = aapq
end if
if( ir1==0_${ik}$ ) then
! column norms are periodically updated by explicit
! norm computation.
! caveat:
! unfortunately, some blas implementations compute sncrm2(m,a(1,p),1)
! as sqrt(s=stdlib${ii}$_${ci}$dotc(m,a(1,p),1,a(1,p),1)), which may cause the result to
! overflow for ||a(:,p)||_2 > sqrt(overflow_threshold), and to
! underflow for ||a(:,p)||_2 < sqrt(underflow_threshold).
! hence, stdlib${ii}$_${c2ri(ci)}$znrm2 cannot be trusted, not even in the case when
! the true norm is far from the under(over)flow boundaries.
! if properly implemented stdlib${ii}$_${c2ri(ci)}$znrm2 is available, the if-then-else-end if
! below should be replaced with "aapp = stdlib${ii}$_${c2ri(ci)}$znrm2( m, a(1,p), 1 )".
if( ( sva( p )<rootbig ) .and.( sva( p )>rootsfmin ) ) then
sva( p ) = stdlib${ii}$_${c2ri(ci)}$znrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )
else
temp1 = zero
aapp = one
call stdlib${ii}$_${ci}$lassq( m, a( 1_${ik}$, p ), 1_${ik}$, temp1, aapp )
sva( p ) = temp1*sqrt( aapp )
end if
aapp = sva( p )
else
aapp = sva( p )
end if
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2002: do q = p + 1, min( igl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
if( aaqq>=one ) then
rotok = ( small*aapp )<=aaqq
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_${ci}$dotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq ) / aapp
else
call stdlib${ii}$_${ci}$copy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work, lda, &
ierr )
aapq = stdlib${ii}$_${ci}$dotc( m, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq
end if
else
rotok = aapp<=( aaqq / small )
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_${ci}$dotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aapp ) / aaqq
else
call stdlib${ii}$_${ci}$copy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_${ci}$dotc( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ ) / &
aapp
end if
end if
! aapq = aapq * conjg( cwork(p) ) * cwork(q)
aapq1 = -abs(aapq)
mxaapq = max( mxaapq, -aapq1 )
! to rotate or not to rotate, that is the question ...
if( abs( aapq1 )>tol ) then
ompq = aapq / abs(aapq)
! Rotate
! [rtd] rotated = rotated + one
if( ir1==0_${ik}$ ) then
notrot = 0_${ik}$
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
end if
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq )/aapq1
if( abs( theta )>bigtheta ) then
t = half / theta
cs = one
call stdlib${ii}$_${ci}$rot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*t )
if ( rsvec ) then
call stdlib${ii}$_${ci}$rot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*t )
end if
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq1 )
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
call stdlib${ii}$_${ci}$rot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*sn )
if ( rsvec ) then
call stdlib${ii}$_${ci}$rot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*sn )
end if
end if
d(p) = -d(q) * ompq
else
! .. have to use modified gram-schmidt like transformation
call stdlib${ii}$_${ci}$copy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one, m,1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one, m,1_${ik}$, a( 1_${ik}$, q ), &
lda, ierr )
call stdlib${ii}$_${ci}$axpy( m, -aapq, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq, m,1_${ik}$, a( 1_${ik}$, q ), &
lda, ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q), sva(p)
! recompute sva(q), sva(p).
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_${c2ri(ci)}$znrm2( m, a( 1_${ik}$, q ), 1_${ik}$ )
else
t = zero
aaqq = one
call stdlib${ii}$_${ci}$lassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )
end if
end if
if( ( aapp / aapp0 )<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_${c2ri(ci)}$znrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_${ci}$lassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )
end if
sva( p ) = aapp
end if
else
! a(:,p) and a(:,q) already numerically orthogonal
if( ir1==0_${ik}$ )notrot = notrot + 1_${ik}$
! [rtd] skipped = skipped + 1
pskipped = pskipped + 1_${ik}$
end if
else
! a(:,q) is zero column
if( ir1==0_${ik}$ )notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
if( ir1==0_${ik}$ )aapp = -aapp
notrot = 0_${ik}$
go to 2103
end if
end do loop_2002
! end q-loop
2103 continue
! bailed out of q-loop
sva( p ) = aapp
else
sva( p ) = aapp
if( ( ir1==0_${ik}$ ) .and. ( aapp==zero ) )notrot = notrot + min( igl+kbl-1, &
n ) - p
end if
end do loop_2001
! end of the p-loop
! end of doing the block ( ibr, ibr )
end do loop_1002
! end of ir1-loop
! ... go to the off diagonal blocks
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_2010: do jbc = ibr + 1, nbl
jgl = ( jbc-1 )*kbl + 1_${ik}$
! doing the block at ( ibr, jbc )
ijblsk = 0_${ik}$
loop_2100: do p = igl, min( igl+kbl-1, n )
aapp = sva( p )
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2200: do q = jgl, min( jgl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
! M X 2 Jacobi Svd
! safe gram matrix computation
if( aaqq>=one ) then
if( aapp>=aaqq ) then
rotok = ( small*aapp )<=aaqq
else
rotok = ( small*aaqq )<=aapp
end if
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_${ci}$dotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq ) / aapp
else
call stdlib${ii}$_${ci}$copy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_${ci}$dotc( m, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq
end if
else
if( aapp>=aaqq ) then
rotok = aapp<=( aaqq / small )
else
rotok = aaqq<=( aapp / small )
end if
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_${ci}$dotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / max(&
aaqq,aapp) )/ min(aaqq,aapp)
else
call stdlib${ii}$_${ci}$copy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_${ci}$dotc( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ ) / &
aapp
end if
end if
! aapq = aapq * conjg(cwork(p))*cwork(q)
aapq1 = -abs(aapq)
mxaapq = max( mxaapq, -aapq1 )
! to rotate or not to rotate, that is the question ...
if( abs( aapq1 )>tol ) then
ompq = aapq / abs(aapq)
notrot = 0_${ik}$
! [rtd] rotated = rotated + 1
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq )/ aapq1
if( aaqq>aapp0 )theta = -theta
if( abs( theta )>bigtheta ) then
t = half / theta
cs = one
call stdlib${ii}$_${ci}$rot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*t )
if( rsvec ) then
call stdlib${ii}$_${ci}$rot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*t )
end if
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq1 )
if( aaqq>aapp0 )thsign = -thsign
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
call stdlib${ii}$_${ci}$rot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*sn )
if( rsvec ) then
call stdlib${ii}$_${ci}$rot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*sn )
end if
end if
d(p) = -d(q) * ompq
else
! .. have to use modified gram-schmidt like transformation
if( aapp>aaqq ) then
call stdlib${ii}$_${ci}$copy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work,lda,&
ierr )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
call stdlib${ii}$_${ci}$axpy( m, -aapq, work,1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
else
call stdlib${ii}$_${ci}$copy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, work,lda,&
ierr )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
call stdlib${ii}$_${ci}$axpy( m, -conjg(aapq),work, 1_${ik}$, a( 1_${ik}$, p ), 1_${ik}$ &
)
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, one, aapp,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
sva( p ) = aapp*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
end if
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q), sva(p)
! .. recompute sva(q), sva(p)
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_${c2ri(ci)}$znrm2( m, a( 1_${ik}$, q ), 1_${ik}$)
else
t = zero
aaqq = one
call stdlib${ii}$_${ci}$lassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )
end if
end if
if( ( aapp / aapp0 )**2_${ik}$<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_${c2ri(ci)}$znrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_${ci}$lassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )
end if
sva( p ) = aapp
end if
! end of ok rotation
else
notrot = notrot + 1_${ik}$
! [rtd] skipped = skipped + 1
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
if( ( i<=swband ) .and. ( ijblsk>=blskip ) )then
sva( p ) = aapp
notrot = 0_${ik}$
go to 2011
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
aapp = -aapp
notrot = 0_${ik}$
go to 2203
end if
end do loop_2200
! end of the q-loop
2203 continue
sva( p ) = aapp
else
if( aapp==zero )notrot = notrot +min( jgl+kbl-1, n ) - jgl + 1_${ik}$
if( aapp<zero )notrot = 0_${ik}$
end if
end do loop_2100
! end of the p-loop
end do loop_2010
! end of the jbc-loop
2011 continue
! 2011 bailed out of the jbc-loop
do p = igl, min( igl+kbl-1, n )
sva( p ) = abs( sva( p ) )
end do
! **
end do loop_2000
! 2000 :: end of the ibr-loop
! .. update sva(n)
if( ( sva( n )<rootbig ) .and. ( sva( n )>rootsfmin ) )then
sva( n ) = stdlib${ii}$_${c2ri(ci)}$znrm2( m, a( 1_${ik}$, n ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_${ci}$lassq( m, a( 1_${ik}$, n ), 1_${ik}$, t, aapp )
sva( n ) = t*sqrt( aapp )
end if
! additional steering devices
if( ( i<swband ) .and. ( ( mxaapq<=roottol ) .or.( iswrot<=n ) ) )swband = i
if( ( i>swband+1 ) .and. ( mxaapq<sqrt( real( n,KIND=${ck}$) )*tol ) .and. ( real( n,&
KIND=${ck}$)*mxaapq*mxsinj<tol ) ) then
go to 1994
end if
if( notrot>=emptsw )go to 1994
end do loop_1993
! end i=1:nsweep loop
! #:( reaching this point means that the procedure has not converged.
info = nsweep - 1_${ik}$
go to 1995
1994 continue
! #:) reaching this point means numerical convergence after the i-th
! sweep.
info = 0_${ik}$
! #:) info = 0 confirms successful iterations.
1995 continue
! sort the vector sva() of column norms.
do p = 1, n - 1
q = stdlib${ii}$_i${c2ri(ci)}$amax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
aapq = d( p )
d( p ) = d( q )
d( q ) = aapq
call stdlib${ii}$_${ci}$swap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_${ci}$swap( mvl, v( 1_${ik}$, p ), 1_${ik}$, v( 1_${ik}$, q ), 1_${ik}$ )
end if
end do
return
end subroutine stdlib${ii}$_${ci}$gsvj0
#:endif
#:endfor
pure module subroutine stdlib${ii}$_sgsvj1( jobv, m, n, n1, a, lda, d, sva, mv, v, ldv,eps, sfmin, tol, &
!! SGSVJ1 is called from SGESVJ as a pre-processor and that is its main
!! purpose. It applies Jacobi rotations in the same way as SGESVJ does, but
!! it targets only particular pivots and it does not check convergence
!! (stopping criterion). Few tuning parameters (marked by [TP]) are
!! available for the implementer.
!! Further Details
!! ~~~~~~~~~~~~~~~
!! SGSVJ1 applies few sweeps of Jacobi rotations in the column space of
!! the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
!! off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
!! block-entries (tiles) of the (1,2) off-diagonal block are marked by the
!! [x]'s in the following scheme:
!! | * * * [x] [x] [x]|
!! | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
!! | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
!! |[x] [x] [x] * * * |
!! |[x] [x] [x] * * * |
!! |[x] [x] [x] * * * |
!! In terms of the columns of A, the first N1 columns are rotated 'against'
!! the remaining N-N1 columns, trying to increase the angle between the
!! corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
!! tiled using quadratic tiles of side KBL. Here, KBL is a tuning parameter.
!! The number of sweeps is given in NSWEEP and the orthogonality threshold
!! is given in TOL.
nsweep, work, lwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
real(sp), intent(in) :: eps, sfmin, tol
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, ldv, lwork, m, mv, n, n1, nsweep
character, intent(in) :: jobv
! Array Arguments
real(sp), intent(inout) :: a(lda,*), d(n), sva(n), v(ldv,*)
real(sp), intent(out) :: work(lwork)
! =====================================================================
! Local Scalars
real(sp) :: aapp, aapp0, aapq, aaqq, apoaq, aqoap, big, bigtheta, cs, large, mxaapq, &
mxsinj, rootbig, rooteps, rootsfmin, roottol, small, sn, t, temp1, theta, &
thsign
integer(${ik}$) :: blskip, emptsw, i, ibr, igl, ierr, ijblsk, iswrot, jbc, jgl, kbl, mvl, &
notrot, nblc, nblr, p, pskipped, q, rowskip, swband
logical(lk) :: applv, rotok, rsvec
! Local Arrays
real(sp) :: fastr(5_${ik}$)
! Intrinsic Functions
! Executable Statements
! test the input parameters.
applv = stdlib_lsame( jobv, 'A' )
rsvec = stdlib_lsame( jobv, 'V' )
if( .not.( rsvec .or. applv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( ( n<0_${ik}$ ) .or. ( n>m ) ) then
info = -3_${ik}$
else if( n1<0_${ik}$ ) then
info = -4_${ik}$
else if( lda<m ) then
info = -6_${ik}$
else if( ( rsvec.or.applv ) .and. ( mv<0_${ik}$ ) ) then
info = -9_${ik}$
else if( ( rsvec.and.( ldv<n ) ).or.( applv.and.( ldv<mv ) ) ) then
info = -11_${ik}$
else if( tol<=eps ) then
info = -14_${ik}$
else if( nsweep<0_${ik}$ ) then
info = -15_${ik}$
else if( lwork<m ) then
info = -17_${ik}$
else
info = 0_${ik}$
end if
! #:(
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'SGSVJ1', -info )
return
end if
if( rsvec ) then
mvl = n
else if( applv ) then
mvl = mv
end if
rsvec = rsvec .or. applv
rooteps = sqrt( eps )
rootsfmin = sqrt( sfmin )
small = sfmin / eps
big = one / sfmin
rootbig = one / rootsfmin
large = big / sqrt( real( m*n,KIND=sp) )
bigtheta = one / rooteps
roottol = sqrt( tol )
! Initialize The Right Singular Vector Matrix
! rsvec = stdlib_lsame( jobv, 'y' )
emptsw = n1*( n-n1 )
notrot = 0_${ik}$
fastr( 1_${ik}$ ) = zero
! .. row-cyclic pivot strategy with de rijk's pivoting ..
kbl = min( 8_${ik}$, n )
nblr = n1 / kbl
if( ( nblr*kbl )/=n1 )nblr = nblr + 1_${ik}$
! .. the tiling is nblr-by-nblc [tiles]
nblc = ( n-n1 ) / kbl
if( ( nblc*kbl )/=( n-n1 ) )nblc = nblc + 1_${ik}$
blskip = ( kbl**2_${ik}$ ) + 1_${ik}$
! [tp] blkskip is a tuning parameter that depends on swband and kbl.
rowskip = min( 5_${ik}$, kbl )
! [tp] rowskip is a tuning parameter.
swband = 0_${ik}$
! [tp] swband is a tuning parameter. it is meaningful and effective
! if stdlib${ii}$_sgesvj is used as a computational routine in the preconditioned
! jacobi svd algorithm stdlib${ii}$_sgesvj.
! | * * * [x] [x] [x]|
! | * * * [x] [x] [x]| row-cycling in the nblr-by-nblc [x] blocks.
! | * * * [x] [x] [x]| row-cyclic pivoting inside each [x] block.
! |[x] [x] [x] * * * |
! |[x] [x] [x] * * * |
! |[x] [x] [x] * * * |
loop_1993: do i = 1, nsweep
! .. go go go ...
mxaapq = zero
mxsinj = zero
iswrot = 0_${ik}$
notrot = 0_${ik}$
pskipped = 0_${ik}$
loop_2000: do ibr = 1, nblr
igl = ( ibr-1 )*kbl + 1_${ik}$
! ........................................................
! ... go to the off diagonal blocks
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_2010: do jbc = 1, nblc
jgl = n1 + ( jbc-1 )*kbl + 1_${ik}$
! doing the block at ( ibr, jbc )
ijblsk = 0_${ik}$
loop_2100: do p = igl, min( igl+kbl-1, n1 )
aapp = sva( p )
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2200: do q = jgl, min( jgl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
! M X 2 Jacobi Svd
! Safe Gram Matrix Computation
if( aaqq>=one ) then
if( aapp>=aaqq ) then
rotok = ( small*aapp )<=aaqq
else
rotok = ( small*aaqq )<=aapp
end if
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_sdot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_scopy( m, a( 1_${ik}$, p ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aapp, d( p ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_sdot( m, work, 1_${ik}$, a( 1_${ik}$, q ),1_${ik}$ )*d( q ) / &
aaqq
end if
else
if( aapp>=aaqq ) then
rotok = aapp<=( aaqq / small )
else
rotok = aaqq<=( aapp / small )
end if
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_sdot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_scopy( m, a( 1_${ik}$, q ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, d( q ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_sdot( m, work, 1_${ik}$, a( 1_${ik}$, p ),1_${ik}$ )*d( p ) / &
aapp
end if
end if
mxaapq = max( mxaapq, abs( aapq ) )
! to rotate or not to rotate, that is the question ...
if( abs( aapq )>tol ) then
notrot = 0_${ik}$
! rotated = rotated + 1
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq ) / aapq
if( aaqq>aapp0 )theta = -theta
if( abs( theta )>bigtheta ) then
t = half / theta
fastr( 3_${ik}$ ) = t*d( p ) / d( q )
fastr( 4_${ik}$ ) = -t*d( q ) / d( p )
call stdlib${ii}$_srotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$, fastr )
if( rsvec )call stdlib${ii}$_srotm( mvl,v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ),&
1_${ik}$,fastr )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq )
if( aaqq>aapp0 )thsign = -thsign
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
apoaq = d( p ) / d( q )
aqoap = d( q ) / d( p )
if( d( p )>=one ) then
if( d( q )>=one ) then
fastr( 3_${ik}$ ) = t*apoaq
fastr( 4_${ik}$ ) = -t*aqoap
d( p ) = d( p )*cs
d( q ) = d( q )*cs
call stdlib${ii}$_srotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$,&
fastr )
if( rsvec )call stdlib${ii}$_srotm( mvl,v( 1_${ik}$, p ), 1_${ik}$, v( &
1_${ik}$, q ),1_${ik}$, fastr )
else
call stdlib${ii}$_saxpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( 1_${ik}$, &
p ), 1_${ik}$ )
call stdlib${ii}$_saxpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( &
1_${ik}$, q ), 1_${ik}$ )
if( rsvec ) then
call stdlib${ii}$_saxpy( mvl, -t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,v(&
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_saxpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ), 1_${ik}$,&
v( 1_${ik}$, q ), 1_${ik}$ )
end if
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
end if
else
if( d( q )>=one ) then
call stdlib${ii}$_saxpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q &
), 1_${ik}$ )
call stdlib${ii}$_saxpy( m, -cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
if( rsvec ) then
call stdlib${ii}$_saxpy( mvl, t*apoaq,v( 1_${ik}$, p ), 1_${ik}$,v( &
1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_saxpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q ), &
1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
else
if( d( p )>=d( q ) ) then
call stdlib${ii}$_saxpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_saxpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,&
a( 1_${ik}$, q ), 1_${ik}$ )
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
if( rsvec ) then
call stdlib${ii}$_saxpy( mvl,-t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,&
v( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_saxpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ),&
1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
end if
else
call stdlib${ii}$_saxpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$,&
q ), 1_${ik}$ )
call stdlib${ii}$_saxpy( m,-cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,&
a( 1_${ik}$, p ), 1_${ik}$ )
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
if( rsvec ) then
call stdlib${ii}$_saxpy( mvl,t*apoaq, v( 1_${ik}$, p ),1_${ik}$, &
v( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_saxpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q )&
, 1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
end if
end if
end if
end if
else
if( aapp>aaqq ) then
call stdlib${ii}$_scopy( m, a( 1_${ik}$, p ), 1_${ik}$, work,1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
temp1 = -aapq*d( p ) / d( q )
call stdlib${ii}$_saxpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
else
call stdlib${ii}$_scopy( m, a( 1_${ik}$, q ), 1_${ik}$, work,1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
temp1 = -aapq*d( q ) / d( p )
call stdlib${ii}$_saxpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_slascl( 'G', 0_${ik}$, 0_${ik}$, one, aapp,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
sva( p ) = aapp*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
end if
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q)
! .. recompute sva(q)
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_snrm2( m, a( 1_${ik}$, q ), 1_${ik}$ )*d( q )
else
t = zero
aaqq = one
call stdlib${ii}$_slassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )*d( q )
end if
end if
if( ( aapp / aapp0 )**2_${ik}$<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_snrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )*d( p )
else
t = zero
aapp = one
call stdlib${ii}$_slassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )*d( p )
end if
sva( p ) = aapp
end if
! end of ok rotation
else
notrot = notrot + 1_${ik}$
! skipped = skipped + 1
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
! if ( notrot >= emptsw ) go to 2011
if( ( i<=swband ) .and. ( ijblsk>=blskip ) )then
sva( p ) = aapp
notrot = 0_${ik}$
go to 2011
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
aapp = -aapp
notrot = 0_${ik}$
go to 2203
end if
end do loop_2200
! end of the q-loop
2203 continue
sva( p ) = aapp
else
if( aapp==zero )notrot = notrot +min( jgl+kbl-1, n ) - jgl + 1_${ik}$
if( aapp<zero )notrot = 0_${ik}$
! ** if ( notrot >= emptsw ) go to 2011
end if
end do loop_2100
! end of the p-loop
end do loop_2010
! end of the jbc-loop
2011 continue
! 2011 bailed out of the jbc-loop
do p = igl, min( igl+kbl-1, n )
sva( p ) = abs( sva( p ) )
end do
! ** if ( notrot >= emptsw ) go to 1994
end do loop_2000
! 2000 :: end of the ibr-loop
! .. update sva(n)
if( ( sva( n )<rootbig ) .and. ( sva( n )>rootsfmin ) )then
sva( n ) = stdlib${ii}$_snrm2( m, a( 1_${ik}$, n ), 1_${ik}$ )*d( n )
else
t = zero
aapp = one
call stdlib${ii}$_slassq( m, a( 1_${ik}$, n ), 1_${ik}$, t, aapp )
sva( n ) = t*sqrt( aapp )*d( n )
end if
! additional steering devices
if( ( i<swband ) .and. ( ( mxaapq<=roottol ) .or.( iswrot<=n ) ) )swband = i
if( ( i>swband+1 ) .and. ( mxaapq<real( n,KIND=sp)*tol ) .and.( real( n,KIND=sp)&
*mxaapq*mxsinj<tol ) ) then
go to 1994
end if
if( notrot>=emptsw )go to 1994
end do loop_1993
! end i=1:nsweep loop
! #:) reaching this point means that the procedure has completed the given
! number of sweeps.
info = nsweep - 1_${ik}$
go to 1995
1994 continue
! #:) reaching this point means that during the i-th sweep all pivots were
! below the given threshold, causing early exit.
info = 0_${ik}$
! #:) info = 0 confirms successful iterations.
1995 continue
! sort the vector d
do p = 1, n - 1
q = stdlib${ii}$_isamax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
temp1 = d( p )
d( p ) = d( q )
d( q ) = temp1
call stdlib${ii}$_sswap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_sswap( mvl, v( 1_${ik}$, p ), 1_${ik}$, v( 1_${ik}$, q ), 1_${ik}$ )
end if
end do
return
end subroutine stdlib${ii}$_sgsvj1
pure module subroutine stdlib${ii}$_dgsvj1( jobv, m, n, n1, a, lda, d, sva, mv, v, ldv,eps, sfmin, tol, &
!! DGSVJ1 is called from DGESVJ as a pre-processor and that is its main
!! purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
!! it targets only particular pivots and it does not check convergence
!! (stopping criterion). Few tuning parameters (marked by [TP]) are
!! available for the implementer.
!! Further Details
!! ~~~~~~~~~~~~~~~
!! DGSVJ1 applies few sweeps of Jacobi rotations in the column space of
!! the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
!! off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
!! block-entries (tiles) of the (1,2) off-diagonal block are marked by the
!! [x]'s in the following scheme:
!! | * * * [x] [x] [x]|
!! | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
!! | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
!! |[x] [x] [x] * * * |
!! |[x] [x] [x] * * * |
!! |[x] [x] [x] * * * |
!! In terms of the columns of A, the first N1 columns are rotated 'against'
!! the remaining N-N1 columns, trying to increase the angle between the
!! corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
!! tiled using quadratic tiles of side KBL. Here, KBL is a tuning parameter.
!! The number of sweeps is given in NSWEEP and the orthogonality threshold
!! is given in TOL.
nsweep, work, lwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
real(dp), intent(in) :: eps, sfmin, tol
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, ldv, lwork, m, mv, n, n1, nsweep
character, intent(in) :: jobv
! Array Arguments
real(dp), intent(inout) :: a(lda,*), d(n), sva(n), v(ldv,*)
real(dp), intent(out) :: work(lwork)
! =====================================================================
! Local Scalars
real(dp) :: aapp, aapp0, aapq, aaqq, apoaq, aqoap, big, bigtheta, cs, large, mxaapq, &
mxsinj, rootbig, rooteps, rootsfmin, roottol, small, sn, t, temp1, theta, &
thsign
integer(${ik}$) :: blskip, emptsw, i, ibr, igl, ierr, ijblsk, iswrot, jbc, jgl, kbl, mvl, &
notrot, nblc, nblr, p, pskipped, q, rowskip, swband
logical(lk) :: applv, rotok, rsvec
! Local Arrays
real(dp) :: fastr(5_${ik}$)
! Intrinsic Functions
! Executable Statements
! test the input parameters.
applv = stdlib_lsame( jobv, 'A' )
rsvec = stdlib_lsame( jobv, 'V' )
if( .not.( rsvec .or. applv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( ( n<0_${ik}$ ) .or. ( n>m ) ) then
info = -3_${ik}$
else if( n1<0_${ik}$ ) then
info = -4_${ik}$
else if( lda<m ) then
info = -6_${ik}$
else if( ( rsvec.or.applv ) .and. ( mv<0_${ik}$ ) ) then
info = -9_${ik}$
else if( ( rsvec.and.( ldv<n ) ).or.( applv.and.( ldv<mv ) ) ) then
info = -11_${ik}$
else if( tol<=eps ) then
info = -14_${ik}$
else if( nsweep<0_${ik}$ ) then
info = -15_${ik}$
else if( lwork<m ) then
info = -17_${ik}$
else
info = 0_${ik}$
end if
! #:(
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DGSVJ1', -info )
return
end if
if( rsvec ) then
mvl = n
else if( applv ) then
mvl = mv
end if
rsvec = rsvec .or. applv
rooteps = sqrt( eps )
rootsfmin = sqrt( sfmin )
small = sfmin / eps
big = one / sfmin
rootbig = one / rootsfmin
large = big / sqrt( real( m*n,KIND=dp) )
bigtheta = one / rooteps
roottol = sqrt( tol )
! Initialize The Right Singular Vector Matrix
! rsvec = stdlib_lsame( jobv, 'y' )
emptsw = n1*( n-n1 )
notrot = 0_${ik}$
fastr( 1_${ik}$ ) = zero
! .. row-cyclic pivot strategy with de rijk's pivoting ..
kbl = min( 8_${ik}$, n )
nblr = n1 / kbl
if( ( nblr*kbl )/=n1 )nblr = nblr + 1_${ik}$
! .. the tiling is nblr-by-nblc [tiles]
nblc = ( n-n1 ) / kbl
if( ( nblc*kbl )/=( n-n1 ) )nblc = nblc + 1_${ik}$
blskip = ( kbl**2_${ik}$ ) + 1_${ik}$
! [tp] blkskip is a tuning parameter that depends on swband and kbl.
rowskip = min( 5_${ik}$, kbl )
! [tp] rowskip is a tuning parameter.
swband = 0_${ik}$
! [tp] swband is a tuning parameter. it is meaningful and effective
! if stdlib${ii}$_sgesvj is used as a computational routine in the preconditioned
! jacobi svd algorithm stdlib${ii}$_sgesvj.
! | * * * [x] [x] [x]|
! | * * * [x] [x] [x]| row-cycling in the nblr-by-nblc [x] blocks.
! | * * * [x] [x] [x]| row-cyclic pivoting inside each [x] block.
! |[x] [x] [x] * * * |
! |[x] [x] [x] * * * |
! |[x] [x] [x] * * * |
loop_1993: do i = 1, nsweep
! .. go go go ...
mxaapq = zero
mxsinj = zero
iswrot = 0_${ik}$
notrot = 0_${ik}$
pskipped = 0_${ik}$
loop_2000: do ibr = 1, nblr
igl = ( ibr-1 )*kbl + 1_${ik}$
! ........................................................
! ... go to the off diagonal blocks
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_2010: do jbc = 1, nblc
jgl = n1 + ( jbc-1 )*kbl + 1_${ik}$
! doing the block at ( ibr, jbc )
ijblsk = 0_${ik}$
loop_2100: do p = igl, min( igl+kbl-1, n1 )
aapp = sva( p )
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2200: do q = jgl, min( jgl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
! M X 2 Jacobi Svd
! Safe Gram Matrix Computation
if( aaqq>=one ) then
if( aapp>=aaqq ) then
rotok = ( small*aapp )<=aaqq
else
rotok = ( small*aaqq )<=aapp
end if
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_ddot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_dcopy( m, a( 1_${ik}$, p ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aapp, d( p ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_ddot( m, work, 1_${ik}$, a( 1_${ik}$, q ),1_${ik}$ )*d( q ) / &
aaqq
end if
else
if( aapp>=aaqq ) then
rotok = aapp<=( aaqq / small )
else
rotok = aaqq<=( aapp / small )
end if
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_ddot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_dcopy( m, a( 1_${ik}$, q ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, d( q ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_ddot( m, work, 1_${ik}$, a( 1_${ik}$, p ),1_${ik}$ )*d( p ) / &
aapp
end if
end if
mxaapq = max( mxaapq, abs( aapq ) )
! to rotate or not to rotate, that is the question ...
if( abs( aapq )>tol ) then
notrot = 0_${ik}$
! rotated = rotated + 1
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs(aqoap-apoaq) / aapq
if( aaqq>aapp0 )theta = -theta
if( abs( theta )>bigtheta ) then
t = half / theta
fastr( 3_${ik}$ ) = t*d( p ) / d( q )
fastr( 4_${ik}$ ) = -t*d( q ) / d( p )
call stdlib${ii}$_drotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$, fastr )
if( rsvec )call stdlib${ii}$_drotm( mvl,v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ),&
1_${ik}$,fastr )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq )
if( aaqq>aapp0 )thsign = -thsign
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
apoaq = d( p ) / d( q )
aqoap = d( q ) / d( p )
if( d( p )>=one ) then
if( d( q )>=one ) then
fastr( 3_${ik}$ ) = t*apoaq
fastr( 4_${ik}$ ) = -t*aqoap
d( p ) = d( p )*cs
d( q ) = d( q )*cs
call stdlib${ii}$_drotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$,&
fastr )
if( rsvec )call stdlib${ii}$_drotm( mvl,v( 1_${ik}$, p ), 1_${ik}$, v( &
1_${ik}$, q ),1_${ik}$, fastr )
else
call stdlib${ii}$_daxpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( 1_${ik}$, &
p ), 1_${ik}$ )
call stdlib${ii}$_daxpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( &
1_${ik}$, q ), 1_${ik}$ )
if( rsvec ) then
call stdlib${ii}$_daxpy( mvl, -t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,v(&
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_daxpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ), 1_${ik}$,&
v( 1_${ik}$, q ), 1_${ik}$ )
end if
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
end if
else
if( d( q )>=one ) then
call stdlib${ii}$_daxpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q &
), 1_${ik}$ )
call stdlib${ii}$_daxpy( m, -cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
if( rsvec ) then
call stdlib${ii}$_daxpy( mvl, t*apoaq,v( 1_${ik}$, p ), 1_${ik}$,v( &
1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_daxpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q ), &
1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
else
if( d( p )>=d( q ) ) then
call stdlib${ii}$_daxpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_daxpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,&
a( 1_${ik}$, q ), 1_${ik}$ )
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
if( rsvec ) then
call stdlib${ii}$_daxpy( mvl,-t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,&
v( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_daxpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ),&
1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
end if
else
call stdlib${ii}$_daxpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$,&
q ), 1_${ik}$ )
call stdlib${ii}$_daxpy( m,-cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,&
a( 1_${ik}$, p ), 1_${ik}$ )
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
if( rsvec ) then
call stdlib${ii}$_daxpy( mvl,t*apoaq, v( 1_${ik}$, p ),1_${ik}$, &
v( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_daxpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q )&
, 1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
end if
end if
end if
end if
else
if( aapp>aaqq ) then
call stdlib${ii}$_dcopy( m, a( 1_${ik}$, p ), 1_${ik}$, work,1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
temp1 = -aapq*d( p ) / d( q )
call stdlib${ii}$_daxpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
else
call stdlib${ii}$_dcopy( m, a( 1_${ik}$, q ), 1_${ik}$, work,1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
temp1 = -aapq*d( q ) / d( p )
call stdlib${ii}$_daxpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_dlascl( 'G', 0_${ik}$, 0_${ik}$, one, aapp,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
sva( p ) = aapp*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
end if
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q)
! .. recompute sva(q)
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_dnrm2( m, a( 1_${ik}$, q ), 1_${ik}$ )*d( q )
else
t = zero
aaqq = one
call stdlib${ii}$_dlassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )*d( q )
end if
end if
if( ( aapp / aapp0 )**2_${ik}$<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_dnrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )*d( p )
else
t = zero
aapp = one
call stdlib${ii}$_dlassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )*d( p )
end if
sva( p ) = aapp
end if
! end of ok rotation
else
notrot = notrot + 1_${ik}$
! skipped = skipped + 1
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
! if ( notrot >= emptsw ) go to 2011
if( ( i<=swband ) .and. ( ijblsk>=blskip ) )then
sva( p ) = aapp
notrot = 0_${ik}$
go to 2011
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
aapp = -aapp
notrot = 0_${ik}$
go to 2203
end if
end do loop_2200
! end of the q-loop
2203 continue
sva( p ) = aapp
else
if( aapp==zero )notrot = notrot +min( jgl+kbl-1, n ) - jgl + 1_${ik}$
if( aapp<zero )notrot = 0_${ik}$
! ** if ( notrot >= emptsw ) go to 2011
end if
end do loop_2100
! end of the p-loop
end do loop_2010
! end of the jbc-loop
2011 continue
! 2011 bailed out of the jbc-loop
do p = igl, min( igl+kbl-1, n )
sva( p ) = abs( sva( p ) )
end do
! ** if ( notrot >= emptsw ) go to 1994
end do loop_2000
! 2000 :: end of the ibr-loop
! .. update sva(n)
if( ( sva( n )<rootbig ) .and. ( sva( n )>rootsfmin ) )then
sva( n ) = stdlib${ii}$_dnrm2( m, a( 1_${ik}$, n ), 1_${ik}$ )*d( n )
else
t = zero
aapp = one
call stdlib${ii}$_dlassq( m, a( 1_${ik}$, n ), 1_${ik}$, t, aapp )
sva( n ) = t*sqrt( aapp )*d( n )
end if
! additional steering devices
if( ( i<swband ) .and. ( ( mxaapq<=roottol ) .or.( iswrot<=n ) ) )swband = i
if( ( i>swband+1 ) .and. ( mxaapq<real( n,KIND=dp)*tol ) .and.( real( n,KIND=dp)&
*mxaapq*mxsinj<tol ) ) then
go to 1994
end if
if( notrot>=emptsw )go to 1994
end do loop_1993
! end i=1:nsweep loop
! #:) reaching this point means that the procedure has completed the given
! number of sweeps.
info = nsweep - 1_${ik}$
go to 1995
1994 continue
! #:) reaching this point means that during the i-th sweep all pivots were
! below the given threshold, causing early exit.
info = 0_${ik}$
! #:) info = 0 confirms successful iterations.
1995 continue
! sort the vector d
do p = 1, n - 1
q = stdlib${ii}$_idamax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
temp1 = d( p )
d( p ) = d( q )
d( q ) = temp1
call stdlib${ii}$_dswap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_dswap( mvl, v( 1_${ik}$, p ), 1_${ik}$, v( 1_${ik}$, q ), 1_${ik}$ )
end if
end do
return
end subroutine stdlib${ii}$_dgsvj1
#:for rk,rt,ri in REAL_KINDS_TYPES
#:if not rk in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ri}$gsvj1( jobv, m, n, n1, a, lda, d, sva, mv, v, ldv,eps, sfmin, tol, &
!! DGSVJ1: is called from DGESVJ as a pre-processor and that is its main
!! purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
!! it targets only particular pivots and it does not check convergence
!! (stopping criterion). Few tuning parameters (marked by [TP]) are
!! available for the implementer.
!! Further Details
!! ~~~~~~~~~~~~~~~
!! DGSVJ1 applies few sweeps of Jacobi rotations in the column space of
!! the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
!! off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
!! block-entries (tiles) of the (1,2) off-diagonal block are marked by the
!! [x]'s in the following scheme:
!! | * * * [x] [x] [x]|
!! | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
!! | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
!! |[x] [x] [x] * * * |
!! |[x] [x] [x] * * * |
!! |[x] [x] [x] * * * |
!! In terms of the columns of A, the first N1 columns are rotated 'against'
!! the remaining N-N1 columns, trying to increase the angle between the
!! corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
!! tiled using quadratic tiles of side KBL. Here, KBL is a tuning parameter.
!! The number of sweeps is given in NSWEEP and the orthogonality threshold
!! is given in TOL.
nsweep, work, lwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${rk}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
real(${rk}$), intent(in) :: eps, sfmin, tol
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, ldv, lwork, m, mv, n, n1, nsweep
character, intent(in) :: jobv
! Array Arguments
real(${rk}$), intent(inout) :: a(lda,*), d(n), sva(n), v(ldv,*)
real(${rk}$), intent(out) :: work(lwork)
! =====================================================================
! Local Scalars
real(${rk}$) :: aapp, aapp0, aapq, aaqq, apoaq, aqoap, big, bigtheta, cs, large, mxaapq, &
mxsinj, rootbig, rooteps, rootsfmin, roottol, small, sn, t, temp1, theta, &
thsign
integer(${ik}$) :: blskip, emptsw, i, ibr, igl, ierr, ijblsk, iswrot, jbc, jgl, kbl, mvl, &
notrot, nblc, nblr, p, pskipped, q, rowskip, swband
logical(lk) :: applv, rotok, rsvec
! Local Arrays
real(${rk}$) :: fastr(5_${ik}$)
! Intrinsic Functions
! Executable Statements
! test the input parameters.
applv = stdlib_lsame( jobv, 'A' )
rsvec = stdlib_lsame( jobv, 'V' )
if( .not.( rsvec .or. applv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( ( n<0_${ik}$ ) .or. ( n>m ) ) then
info = -3_${ik}$
else if( n1<0_${ik}$ ) then
info = -4_${ik}$
else if( lda<m ) then
info = -6_${ik}$
else if( ( rsvec.or.applv ) .and. ( mv<0_${ik}$ ) ) then
info = -9_${ik}$
else if( ( rsvec.and.( ldv<n ) ).or.( applv.and.( ldv<mv ) ) ) then
info = -11_${ik}$
else if( tol<=eps ) then
info = -14_${ik}$
else if( nsweep<0_${ik}$ ) then
info = -15_${ik}$
else if( lwork<m ) then
info = -17_${ik}$
else
info = 0_${ik}$
end if
! #:(
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DGSVJ1', -info )
return
end if
if( rsvec ) then
mvl = n
else if( applv ) then
mvl = mv
end if
rsvec = rsvec .or. applv
rooteps = sqrt( eps )
rootsfmin = sqrt( sfmin )
small = sfmin / eps
big = one / sfmin
rootbig = one / rootsfmin
large = big / sqrt( real( m*n,KIND=${rk}$) )
bigtheta = one / rooteps
roottol = sqrt( tol )
! Initialize The Right Singular Vector Matrix
! rsvec = stdlib_lsame( jobv, 'y' )
emptsw = n1*( n-n1 )
notrot = 0_${ik}$
fastr( 1_${ik}$ ) = zero
! .. row-cyclic pivot strategy with de rijk's pivoting ..
kbl = min( 8_${ik}$, n )
nblr = n1 / kbl
if( ( nblr*kbl )/=n1 )nblr = nblr + 1_${ik}$
! .. the tiling is nblr-by-nblc [tiles]
nblc = ( n-n1 ) / kbl
if( ( nblc*kbl )/=( n-n1 ) )nblc = nblc + 1_${ik}$
blskip = ( kbl**2_${ik}$ ) + 1_${ik}$
! [tp] blkskip is a tuning parameter that depends on swband and kbl.
rowskip = min( 5_${ik}$, kbl )
! [tp] rowskip is a tuning parameter.
swband = 0_${ik}$
! [tp] swband is a tuning parameter. it is meaningful and effective
! if stdlib${ii}$_dgesvj is used as a computational routine in the preconditioned
! jacobi svd algorithm stdlib${ii}$_dgesvj.
! | * * * [x] [x] [x]|
! | * * * [x] [x] [x]| row-cycling in the nblr-by-nblc [x] blocks.
! | * * * [x] [x] [x]| row-cyclic pivoting inside each [x] block.
! |[x] [x] [x] * * * |
! |[x] [x] [x] * * * |
! |[x] [x] [x] * * * |
loop_1993: do i = 1, nsweep
! .. go go go ...
mxaapq = zero
mxsinj = zero
iswrot = 0_${ik}$
notrot = 0_${ik}$
pskipped = 0_${ik}$
loop_2000: do ibr = 1, nblr
igl = ( ibr-1 )*kbl + 1_${ik}$
! ........................................................
! ... go to the off diagonal blocks
igl = ( ibr-1 )*kbl + 1_${ik}$
loop_2010: do jbc = 1, nblc
jgl = n1 + ( jbc-1 )*kbl + 1_${ik}$
! doing the block at ( ibr, jbc )
ijblsk = 0_${ik}$
loop_2100: do p = igl, min( igl+kbl-1, n1 )
aapp = sva( p )
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2200: do q = jgl, min( jgl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
! M X 2 Jacobi Svd
! Safe Gram Matrix Computation
if( aaqq>=one ) then
if( aapp>=aaqq ) then
rotok = ( small*aapp )<=aaqq
else
rotok = ( small*aaqq )<=aapp
end if
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_${ri}$dot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_${ri}$copy( m, a( 1_${ik}$, p ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp, d( p ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_${ri}$dot( m, work, 1_${ik}$, a( 1_${ik}$, q ),1_${ik}$ )*d( q ) / &
aaqq
end if
else
if( aapp>=aaqq ) then
rotok = aapp<=( aaqq / small )
else
rotok = aaqq<=( aapp / small )
end if
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_${ri}$dot( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$,q ), 1_${ik}$ )*d( p )&
*d( q ) / aaqq )/ aapp
else
call stdlib${ii}$_${ri}$copy( m, a( 1_${ik}$, q ), 1_${ik}$, work, 1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, d( q ),m, 1_${ik}$, work, lda,&
ierr )
aapq = stdlib${ii}$_${ri}$dot( m, work, 1_${ik}$, a( 1_${ik}$, p ),1_${ik}$ )*d( p ) / &
aapp
end if
end if
mxaapq = max( mxaapq, abs( aapq ) )
! to rotate or not to rotate, that is the question ...
if( abs( aapq )>tol ) then
notrot = 0_${ik}$
! rotated = rotated + 1
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs(aqoap-apoaq) / aapq
if( aaqq>aapp0 )theta = -theta
if( abs( theta )>bigtheta ) then
t = half / theta
fastr( 3_${ik}$ ) = t*d( p ) / d( q )
fastr( 4_${ik}$ ) = -t*d( q ) / d( p )
call stdlib${ii}$_${ri}$rotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$, fastr )
if( rsvec )call stdlib${ii}$_${ri}$rotm( mvl,v( 1_${ik}$, p ), 1_${ik}$,v( 1_${ik}$, q ),&
1_${ik}$,fastr )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq )
if( aaqq>aapp0 )thsign = -thsign
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq ) )
apoaq = d( p ) / d( q )
aqoap = d( q ) / d( p )
if( d( p )>=one ) then
if( d( q )>=one ) then
fastr( 3_${ik}$ ) = t*apoaq
fastr( 4_${ik}$ ) = -t*aqoap
d( p ) = d( p )*cs
d( q ) = d( q )*cs
call stdlib${ii}$_${ri}$rotm( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$,&
fastr )
if( rsvec )call stdlib${ii}$_${ri}$rotm( mvl,v( 1_${ik}$, p ), 1_${ik}$, v( &
1_${ik}$, q ),1_${ik}$, fastr )
else
call stdlib${ii}$_${ri}$axpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( 1_${ik}$, &
p ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( &
1_${ik}$, q ), 1_${ik}$ )
if( rsvec ) then
call stdlib${ii}$_${ri}$axpy( mvl, -t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,v(&
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ), 1_${ik}$,&
v( 1_${ik}$, q ), 1_${ik}$ )
end if
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
end if
else
if( d( q )>=one ) then
call stdlib${ii}$_${ri}$axpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q &
), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( m, -cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
if( rsvec ) then
call stdlib${ii}$_${ri}$axpy( mvl, t*apoaq,v( 1_${ik}$, p ), 1_${ik}$,v( &
1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q ), &
1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
else
if( d( p )>=d( q ) ) then
call stdlib${ii}$_${ri}$axpy( m, -t*aqoap,a( 1_${ik}$, q ), 1_${ik}$,a( &
1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( m, cs*sn*apoaq,a( 1_${ik}$, p ), 1_${ik}$,&
a( 1_${ik}$, q ), 1_${ik}$ )
d( p ) = d( p )*cs
d( q ) = d( q ) / cs
if( rsvec ) then
call stdlib${ii}$_${ri}$axpy( mvl,-t*aqoap,v( 1_${ik}$, q ), 1_${ik}$,&
v( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( mvl,cs*sn*apoaq,v( 1_${ik}$, p ),&
1_${ik}$,v( 1_${ik}$, q ), 1_${ik}$ )
end if
else
call stdlib${ii}$_${ri}$axpy( m, t*apoaq,a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$,&
q ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( m,-cs*sn*aqoap,a( 1_${ik}$, q ), 1_${ik}$,&
a( 1_${ik}$, p ), 1_${ik}$ )
d( p ) = d( p ) / cs
d( q ) = d( q )*cs
if( rsvec ) then
call stdlib${ii}$_${ri}$axpy( mvl,t*apoaq, v( 1_${ik}$, p ),1_${ik}$, &
v( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_${ri}$axpy( mvl,-cs*sn*aqoap,v( 1_${ik}$, q )&
, 1_${ik}$,v( 1_${ik}$, p ), 1_${ik}$ )
end if
end if
end if
end if
end if
else
if( aapp>aaqq ) then
call stdlib${ii}$_${ri}$copy( m, a( 1_${ik}$, p ), 1_${ik}$, work,1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
temp1 = -aapq*d( p ) / d( q )
call stdlib${ii}$_${ri}$axpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
else
call stdlib${ii}$_${ri}$copy( m, a( 1_${ik}$, q ), 1_${ik}$, work,1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, work, lda,&
ierr )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
temp1 = -aapq*d( q ) / d( p )
call stdlib${ii}$_${ri}$axpy( m, temp1, work, 1_${ik}$,a( 1_${ik}$, p ), 1_${ik}$ )
call stdlib${ii}$_${ri}$lascl( 'G', 0_${ik}$, 0_${ik}$, one, aapp,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
sva( p ) = aapp*sqrt( max( zero,one-aapq*aapq ) )
mxsinj = max( mxsinj, sfmin )
end if
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q)
! .. recompute sva(q)
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_${ri}$nrm2( m, a( 1_${ik}$, q ), 1_${ik}$ )*d( q )
else
t = zero
aaqq = one
call stdlib${ii}$_${ri}$lassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )*d( q )
end if
end if
if( ( aapp / aapp0 )**2_${ik}$<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_${ri}$nrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )*d( p )
else
t = zero
aapp = one
call stdlib${ii}$_${ri}$lassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )*d( p )
end if
sva( p ) = aapp
end if
! end of ok rotation
else
notrot = notrot + 1_${ik}$
! skipped = skipped + 1
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
! if ( notrot >= emptsw ) go to 2011
if( ( i<=swband ) .and. ( ijblsk>=blskip ) )then
sva( p ) = aapp
notrot = 0_${ik}$
go to 2011
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
aapp = -aapp
notrot = 0_${ik}$
go to 2203
end if
end do loop_2200
! end of the q-loop
2203 continue
sva( p ) = aapp
else
if( aapp==zero )notrot = notrot +min( jgl+kbl-1, n ) - jgl + 1_${ik}$
if( aapp<zero )notrot = 0_${ik}$
! ** if ( notrot >= emptsw ) go to 2011
end if
end do loop_2100
! end of the p-loop
end do loop_2010
! end of the jbc-loop
2011 continue
! 2011 bailed out of the jbc-loop
do p = igl, min( igl+kbl-1, n )
sva( p ) = abs( sva( p ) )
end do
! ** if ( notrot >= emptsw ) go to 1994
end do loop_2000
! 2000 :: end of the ibr-loop
! .. update sva(n)
if( ( sva( n )<rootbig ) .and. ( sva( n )>rootsfmin ) )then
sva( n ) = stdlib${ii}$_${ri}$nrm2( m, a( 1_${ik}$, n ), 1_${ik}$ )*d( n )
else
t = zero
aapp = one
call stdlib${ii}$_${ri}$lassq( m, a( 1_${ik}$, n ), 1_${ik}$, t, aapp )
sva( n ) = t*sqrt( aapp )*d( n )
end if
! additional steering devices
if( ( i<swband ) .and. ( ( mxaapq<=roottol ) .or.( iswrot<=n ) ) )swband = i
if( ( i>swband+1 ) .and. ( mxaapq<real( n,KIND=${rk}$)*tol ) .and.( real( n,KIND=${rk}$)&
*mxaapq*mxsinj<tol ) ) then
go to 1994
end if
if( notrot>=emptsw )go to 1994
end do loop_1993
! end i=1:nsweep loop
! #:) reaching this point means that the procedure has completed the given
! number of sweeps.
info = nsweep - 1_${ik}$
go to 1995
1994 continue
! #:) reaching this point means that during the i-th sweep all pivots were
! below the given threshold, causing early exit.
info = 0_${ik}$
! #:) info = 0 confirms successful iterations.
1995 continue
! sort the vector d
do p = 1, n - 1
q = stdlib${ii}$_i${ri}$amax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
temp1 = d( p )
d( p ) = d( q )
d( q ) = temp1
call stdlib${ii}$_${ri}$swap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_${ri}$swap( mvl, v( 1_${ik}$, p ), 1_${ik}$, v( 1_${ik}$, q ), 1_${ik}$ )
end if
end do
return
end subroutine stdlib${ii}$_${ri}$gsvj1
#:endif
#:endfor
pure module subroutine stdlib${ii}$_cgsvj1( jobv, m, n, n1, a, lda, d, sva, mv, v, ldv,eps, sfmin, tol, &
!! CGSVJ1 is called from CGESVJ as a pre-processor and that is its main
!! purpose. It applies Jacobi rotations in the same way as CGESVJ does, but
!! it targets only particular pivots and it does not check convergence
!! (stopping criterion). Few tuning parameters (marked by [TP]) are
!! available for the implementer.
!! Further Details
!! ~~~~~~~~~~~~~~~
!! CGSVJ1 applies few sweeps of Jacobi rotations in the column space of
!! the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
!! off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
!! block-entries (tiles) of the (1,2) off-diagonal block are marked by the
!! [x]'s in the following scheme:
!! | * * * [x] [x] [x]|
!! | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
!! | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
!! |[x] [x] [x] * * * |
!! |[x] [x] [x] * * * |
!! |[x] [x] [x] * * * |
!! In terms of the columns of A, the first N1 columns are rotated 'against'
!! the remaining N-N1 columns, trying to increase the angle between the
!! corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
!! tiled using quadratic tiles of side KBL. Here, KBL is a tuning parameter.
!! The number of sweeps is given in NSWEEP and the orthogonality threshold
!! is given in TOL.
nsweep, work, lwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
real(sp), intent(in) :: eps, sfmin, tol
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, ldv, lwork, m, mv, n, n1, nsweep
character, intent(in) :: jobv
! Array Arguments
complex(sp), intent(inout) :: a(lda,*), d(n), v(ldv,*)
complex(sp), intent(out) :: work(lwork)
real(sp), intent(inout) :: sva(n)
! =====================================================================
! Local Scalars
complex(sp) :: aapq, ompq
real(sp) :: aapp, aapp0, aapq1, aaqq, apoaq, aqoap, big, bigtheta, cs, mxaapq, mxsinj, &
rootbig, rooteps, rootsfmin, roottol, small, sn, t, temp1, theta, thsign
integer(${ik}$) :: blskip, emptsw, i, ibr, igl, ierr, ijblsk, iswrot, jbc, jgl, kbl, mvl, &
notrot, nblc, nblr, p, pskipped, q, rowskip, swband
logical(lk) :: applv, rotok, rsvec
! Intrinsic Functions
! From Lapack
! Executable Statements
! test the input parameters.
applv = stdlib_lsame( jobv, 'A' )
rsvec = stdlib_lsame( jobv, 'V' )
if( .not.( rsvec .or. applv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( ( n<0_${ik}$ ) .or. ( n>m ) ) then
info = -3_${ik}$
else if( n1<0_${ik}$ ) then
info = -4_${ik}$
else if( lda<m ) then
info = -6_${ik}$
else if( ( rsvec.or.applv ) .and. ( mv<0_${ik}$ ) ) then
info = -9_${ik}$
else if( ( rsvec.and.( ldv<n ) ).or.( applv.and.( ldv<mv ) ) ) then
info = -11_${ik}$
else if( tol<=eps ) then
info = -14_${ik}$
else if( nsweep<0_${ik}$ ) then
info = -15_${ik}$
else if( lwork<m ) then
info = -17_${ik}$
else
info = 0_${ik}$
end if
! #:(
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'CGSVJ1', -info )
return
end if
if( rsvec ) then
mvl = n
else if( applv ) then
mvl = mv
end if
rsvec = rsvec .or. applv
rooteps = sqrt( eps )
rootsfmin = sqrt( sfmin )
small = sfmin / eps
big = one / sfmin
rootbig = one / rootsfmin
! large = big / sqrt( real( m*n,KIND=sp) )
bigtheta = one / rooteps
roottol = sqrt( tol )
! Initialize The Right Singular Vector Matrix
! rsvec = stdlib_lsame( jobv, 'y' )
emptsw = n1*( n-n1 )
notrot = 0_${ik}$
! .. row-cyclic pivot strategy with de rijk's pivoting ..
kbl = min( 8_${ik}$, n )
nblr = n1 / kbl
if( ( nblr*kbl )/=n1 )nblr = nblr + 1_${ik}$
! .. the tiling is nblr-by-nblc [tiles]
nblc = ( n-n1 ) / kbl
if( ( nblc*kbl )/=( n-n1 ) )nblc = nblc + 1_${ik}$
blskip = ( kbl**2_${ik}$ ) + 1_${ik}$
! [tp] blkskip is a tuning parameter that depends on swband and kbl.
rowskip = min( 5_${ik}$, kbl )
! [tp] rowskip is a tuning parameter.
swband = 0_${ik}$
! [tp] swband is a tuning parameter. it is meaningful and effective
! if stdlib${ii}$_cgesvj is used as a computational routine in the preconditioned
! jacobi svd algorithm stdlib${ii}$_cgejsv.
! | * * * [x] [x] [x]|
! | * * * [x] [x] [x]| row-cycling in the nblr-by-nblc [x] blocks.
! | * * * [x] [x] [x]| row-cyclic pivoting inside each [x] block.
! |[x] [x] [x] * * * |
! |[x] [x] [x] * * * |
! |[x] [x] [x] * * * |
loop_1993: do i = 1, nsweep
! .. go go go ...
mxaapq = zero
mxsinj = zero
iswrot = 0_${ik}$
notrot = 0_${ik}$
pskipped = 0_${ik}$
! each sweep is unrolled using kbl-by-kbl tiles over the pivot pairs
! 1 <= p < q <= n. this is the first step toward a blocked implementation
! of the rotations. new implementation, based on block transformations,
! is under development.
loop_2000: do ibr = 1, nblr
igl = ( ibr-1 )*kbl + 1_${ik}$
! ... go to the off diagonal blocks
igl = ( ibr-1 )*kbl + 1_${ik}$
! do 2010 jbc = ibr + 1, nbl
loop_2010: do jbc = 1, nblc
jgl = ( jbc-1 )*kbl + n1 + 1_${ik}$
! doing the block at ( ibr, jbc )
ijblsk = 0_${ik}$
loop_2100: do p = igl, min( igl+kbl-1, n1 )
aapp = sva( p )
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2200: do q = jgl, min( jgl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
! M X 2 Jacobi Svd
! safe gram matrix computation
if( aaqq>=one ) then
if( aapp>=aaqq ) then
rotok = ( small*aapp )<=aaqq
else
rotok = ( small*aaqq )<=aapp
end if
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_cdotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq ) / aapp
else
call stdlib${ii}$_ccopy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aapp,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_cdotc( m, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq
end if
else
if( aapp>=aaqq ) then
rotok = aapp<=( aaqq / small )
else
rotok = aaqq<=( aapp / small )
end if
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_cdotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / max(&
aaqq,aapp) )/ min(aaqq,aapp)
else
call stdlib${ii}$_ccopy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aaqq,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_cdotc( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ ) / &
aapp
end if
end if
! aapq = aapq * conjg(cwork(p))*cwork(q)
aapq1 = -abs(aapq)
mxaapq = max( mxaapq, -aapq1 )
! to rotate or not to rotate, that is the question ...
if( abs( aapq1 )>tol ) then
ompq = aapq / abs(aapq)
notrot = 0_${ik}$
! [rtd] rotated = rotated + 1
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq )/ aapq1
if( aaqq>aapp0 )theta = -theta
if( abs( theta )>bigtheta ) then
t = half / theta
cs = one
call stdlib${ii}$_crot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*t )
if( rsvec ) then
call stdlib${ii}$_crot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*t )
end if
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq1 )
if( aaqq>aapp0 )thsign = -thsign
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
call stdlib${ii}$_crot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*sn )
if( rsvec ) then
call stdlib${ii}$_crot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*sn )
end if
end if
d(p) = -d(q) * ompq
else
! .. have to use modified gram-schmidt like transformation
if( aapp>aaqq ) then
call stdlib${ii}$_ccopy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work,lda,&
ierr )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
call stdlib${ii}$_caxpy( m, -aapq, work,1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
else
call stdlib${ii}$_ccopy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, work,lda,&
ierr )
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
call stdlib${ii}$_caxpy( m, -conjg(aapq),work, 1_${ik}$, a( 1_${ik}$, p ), 1_${ik}$ &
)
call stdlib${ii}$_clascl( 'G', 0_${ik}$, 0_${ik}$, one, aapp,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
sva( p ) = aapp*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
end if
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q), sva(p)
! .. recompute sva(q), sva(p)
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_scnrm2( m, a( 1_${ik}$, q ), 1_${ik}$)
else
t = zero
aaqq = one
call stdlib${ii}$_classq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )
end if
end if
if( ( aapp / aapp0 )**2_${ik}$<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_scnrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_classq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )
end if
sva( p ) = aapp
end if
! end of ok rotation
else
notrot = notrot + 1_${ik}$
! [rtd] skipped = skipped + 1
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
if( ( i<=swband ) .and. ( ijblsk>=blskip ) )then
sva( p ) = aapp
notrot = 0_${ik}$
go to 2011
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
aapp = -aapp
notrot = 0_${ik}$
go to 2203
end if
end do loop_2200
! end of the q-loop
2203 continue
sva( p ) = aapp
else
if( aapp==zero )notrot = notrot +min( jgl+kbl-1, n ) - jgl + 1_${ik}$
if( aapp<zero )notrot = 0_${ik}$
end if
end do loop_2100
! end of the p-loop
end do loop_2010
! end of the jbc-loop
2011 continue
! 2011 bailed out of the jbc-loop
do p = igl, min( igl+kbl-1, n )
sva( p ) = abs( sva( p ) )
end do
! **
end do loop_2000
! 2000 :: end of the ibr-loop
! .. update sva(n)
if( ( sva( n )<rootbig ) .and. ( sva( n )>rootsfmin ) )then
sva( n ) = stdlib${ii}$_scnrm2( m, a( 1_${ik}$, n ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_classq( m, a( 1_${ik}$, n ), 1_${ik}$, t, aapp )
sva( n ) = t*sqrt( aapp )
end if
! additional steering devices
if( ( i<swband ) .and. ( ( mxaapq<=roottol ) .or.( iswrot<=n ) ) )swband = i
if( ( i>swband+1 ) .and. ( mxaapq<sqrt( real( n,KIND=sp) )*tol ) .and. ( real( n,&
KIND=sp)*mxaapq*mxsinj<tol ) ) then
go to 1994
end if
if( notrot>=emptsw )go to 1994
end do loop_1993
! end i=1:nsweep loop
! #:( reaching this point means that the procedure has not converged.
info = nsweep - 1_${ik}$
go to 1995
1994 continue
! #:) reaching this point means numerical convergence after the i-th
! sweep.
info = 0_${ik}$
! #:) info = 0 confirms successful iterations.
1995 continue
! sort the vector sva() of column norms.
do p = 1, n - 1
q = stdlib${ii}$_isamax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
aapq = d( p )
d( p ) = d( q )
d( q ) = aapq
call stdlib${ii}$_cswap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_cswap( mvl, v( 1_${ik}$, p ), 1_${ik}$, v( 1_${ik}$, q ), 1_${ik}$ )
end if
end do
return
end subroutine stdlib${ii}$_cgsvj1
pure module subroutine stdlib${ii}$_zgsvj1( jobv, m, n, n1, a, lda, d, sva, mv, v, ldv,eps, sfmin, tol, &
!! ZGSVJ1 is called from ZGESVJ as a pre-processor and that is its main
!! purpose. It applies Jacobi rotations in the same way as ZGESVJ does, but
!! it targets only particular pivots and it does not check convergence
!! (stopping criterion). Few tuning parameters (marked by [TP]) are
!! available for the implementer.
!! Further Details
!! ~~~~~~~~~~~~~~~
!! ZGSVJ1 applies few sweeps of Jacobi rotations in the column space of
!! the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
!! off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
!! block-entries (tiles) of the (1,2) off-diagonal block are marked by the
!! [x]'s in the following scheme:
!! | * * * [x] [x] [x]|
!! | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
!! | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
!! |[x] [x] [x] * * * |
!! |[x] [x] [x] * * * |
!! |[x] [x] [x] * * * |
!! In terms of the columns of A, the first N1 columns are rotated 'against'
!! the remaining N-N1 columns, trying to increase the angle between the
!! corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
!! tiled using quadratic tiles of side KBL. Here, KBL is a tuning parameter.
!! The number of sweeps is given in NSWEEP and the orthogonality threshold
!! is given in TOL.
nsweep, work, lwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
real(dp), intent(in) :: eps, sfmin, tol
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, ldv, lwork, m, mv, n, n1, nsweep
character, intent(in) :: jobv
! Array Arguments
complex(dp), intent(inout) :: a(lda,*), d(n), v(ldv,*)
complex(dp), intent(out) :: work(lwork)
real(dp), intent(inout) :: sva(n)
! =====================================================================
! Local Scalars
complex(dp) :: aapq, ompq
real(dp) :: aapp, aapp0, aapq1, aaqq, apoaq, aqoap, big, bigtheta, cs, mxaapq, mxsinj, &
rootbig, rooteps, rootsfmin, roottol, small, sn, t, temp1, theta, thsign
integer(${ik}$) :: blskip, emptsw, i, ibr, igl, ierr, ijblsk, iswrot, jbc, jgl, kbl, mvl, &
notrot, nblc, nblr, p, pskipped, q, rowskip, swband
logical(lk) :: applv, rotok, rsvec
! Intrinsic Functions
! From Lapack
! Executable Statements
! test the input parameters.
applv = stdlib_lsame( jobv, 'A' )
rsvec = stdlib_lsame( jobv, 'V' )
if( .not.( rsvec .or. applv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( ( n<0_${ik}$ ) .or. ( n>m ) ) then
info = -3_${ik}$
else if( n1<0_${ik}$ ) then
info = -4_${ik}$
else if( lda<m ) then
info = -6_${ik}$
else if( ( rsvec.or.applv ) .and. ( mv<0_${ik}$ ) ) then
info = -9_${ik}$
else if( ( rsvec.and.( ldv<n ) ).or.( applv.and.( ldv<mv ) ) ) then
info = -11_${ik}$
else if( tol<=eps ) then
info = -14_${ik}$
else if( nsweep<0_${ik}$ ) then
info = -15_${ik}$
else if( lwork<m ) then
info = -17_${ik}$
else
info = 0_${ik}$
end if
! #:(
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZGSVJ1', -info )
return
end if
if( rsvec ) then
mvl = n
else if( applv ) then
mvl = mv
end if
rsvec = rsvec .or. applv
rooteps = sqrt( eps )
rootsfmin = sqrt( sfmin )
small = sfmin / eps
big = one / sfmin
rootbig = one / rootsfmin
! large = big / sqrt( real( m*n,KIND=dp) )
bigtheta = one / rooteps
roottol = sqrt( tol )
! Initialize The Right Singular Vector Matrix
! rsvec = stdlib_lsame( jobv, 'y' )
emptsw = n1*( n-n1 )
notrot = 0_${ik}$
! .. row-cyclic pivot strategy with de rijk's pivoting ..
kbl = min( 8_${ik}$, n )
nblr = n1 / kbl
if( ( nblr*kbl )/=n1 )nblr = nblr + 1_${ik}$
! .. the tiling is nblr-by-nblc [tiles]
nblc = ( n-n1 ) / kbl
if( ( nblc*kbl )/=( n-n1 ) )nblc = nblc + 1_${ik}$
blskip = ( kbl**2_${ik}$ ) + 1_${ik}$
! [tp] blkskip is a tuning parameter that depends on swband and kbl.
rowskip = min( 5_${ik}$, kbl )
! [tp] rowskip is a tuning parameter.
swband = 0_${ik}$
! [tp] swband is a tuning parameter. it is meaningful and effective
! if stdlib${ii}$_zgesvj is used as a computational routine in the preconditioned
! jacobi svd algorithm stdlib${ii}$_zgejsv.
! | * * * [x] [x] [x]|
! | * * * [x] [x] [x]| row-cycling in the nblr-by-nblc [x] blocks.
! | * * * [x] [x] [x]| row-cyclic pivoting inside each [x] block.
! |[x] [x] [x] * * * |
! |[x] [x] [x] * * * |
! |[x] [x] [x] * * * |
loop_1993: do i = 1, nsweep
! .. go go go ...
mxaapq = zero
mxsinj = zero
iswrot = 0_${ik}$
notrot = 0_${ik}$
pskipped = 0_${ik}$
! each sweep is unrolled using kbl-by-kbl tiles over the pivot pairs
! 1 <= p < q <= n. this is the first step toward a blocked implementation
! of the rotations. new implementation, based on block transformations,
! is under development.
loop_2000: do ibr = 1, nblr
igl = ( ibr-1 )*kbl + 1_${ik}$
! ... go to the off diagonal blocks
igl = ( ibr-1 )*kbl + 1_${ik}$
! do 2010 jbc = ibr + 1, nbl
loop_2010: do jbc = 1, nblc
jgl = ( jbc-1 )*kbl + n1 + 1_${ik}$
! doing the block at ( ibr, jbc )
ijblsk = 0_${ik}$
loop_2100: do p = igl, min( igl+kbl-1, n1 )
aapp = sva( p )
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2200: do q = jgl, min( jgl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
! M X 2 Jacobi Svd
! safe gram matrix computation
if( aaqq>=one ) then
if( aapp>=aaqq ) then
rotok = ( small*aapp )<=aaqq
else
rotok = ( small*aaqq )<=aapp
end if
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_zdotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq ) / aapp
else
call stdlib${ii}$_zcopy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aapp,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_zdotc( m, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq
end if
else
if( aapp>=aaqq ) then
rotok = aapp<=( aaqq / small )
else
rotok = aaqq<=( aapp / small )
end if
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_zdotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / max(&
aaqq,aapp) )/ min(aaqq,aapp)
else
call stdlib${ii}$_zcopy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_zdotc( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ ) / &
aapp
end if
end if
! aapq = aapq * conjg(cwork(p))*cwork(q)
aapq1 = -abs(aapq)
mxaapq = max( mxaapq, -aapq1 )
! to rotate or not to rotate, that is the question ...
if( abs( aapq1 )>tol ) then
ompq = aapq / abs(aapq)
notrot = 0_${ik}$
! [rtd] rotated = rotated + 1
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq )/ aapq1
if( aaqq>aapp0 )theta = -theta
if( abs( theta )>bigtheta ) then
t = half / theta
cs = one
call stdlib${ii}$_zrot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*t )
if( rsvec ) then
call stdlib${ii}$_zrot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*t )
end if
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq1 )
if( aaqq>aapp0 )thsign = -thsign
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
call stdlib${ii}$_zrot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*sn )
if( rsvec ) then
call stdlib${ii}$_zrot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*sn )
end if
end if
d(p) = -d(q) * ompq
else
! .. have to use modified gram-schmidt like transformation
if( aapp>aaqq ) then
call stdlib${ii}$_zcopy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work,lda,&
ierr )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
call stdlib${ii}$_zaxpy( m, -aapq, work,1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
else
call stdlib${ii}$_zcopy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, work,lda,&
ierr )
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
call stdlib${ii}$_zaxpy( m, -conjg(aapq),work, 1_${ik}$, a( 1_${ik}$, p ), 1_${ik}$ &
)
call stdlib${ii}$_zlascl( 'G', 0_${ik}$, 0_${ik}$, one, aapp,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
sva( p ) = aapp*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
end if
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q), sva(p)
! .. recompute sva(q), sva(p)
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_dznrm2( m, a( 1_${ik}$, q ), 1_${ik}$)
else
t = zero
aaqq = one
call stdlib${ii}$_zlassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )
end if
end if
if( ( aapp / aapp0 )**2_${ik}$<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_dznrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_zlassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )
end if
sva( p ) = aapp
end if
! end of ok rotation
else
notrot = notrot + 1_${ik}$
! [rtd] skipped = skipped + 1
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
if( ( i<=swband ) .and. ( ijblsk>=blskip ) )then
sva( p ) = aapp
notrot = 0_${ik}$
go to 2011
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
aapp = -aapp
notrot = 0_${ik}$
go to 2203
end if
end do loop_2200
! end of the q-loop
2203 continue
sva( p ) = aapp
else
if( aapp==zero )notrot = notrot +min( jgl+kbl-1, n ) - jgl + 1_${ik}$
if( aapp<zero )notrot = 0_${ik}$
end if
end do loop_2100
! end of the p-loop
end do loop_2010
! end of the jbc-loop
2011 continue
! 2011 bailed out of the jbc-loop
do p = igl, min( igl+kbl-1, n )
sva( p ) = abs( sva( p ) )
end do
! **
end do loop_2000
! 2000 :: end of the ibr-loop
! .. update sva(n)
if( ( sva( n )<rootbig ) .and. ( sva( n )>rootsfmin ) )then
sva( n ) = stdlib${ii}$_dznrm2( m, a( 1_${ik}$, n ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_zlassq( m, a( 1_${ik}$, n ), 1_${ik}$, t, aapp )
sva( n ) = t*sqrt( aapp )
end if
! additional steering devices
if( ( i<swband ) .and. ( ( mxaapq<=roottol ) .or.( iswrot<=n ) ) )swband = i
if( ( i>swband+1 ) .and. ( mxaapq<sqrt( real( n,KIND=dp) )*tol ) .and. ( real( n,&
KIND=dp)*mxaapq*mxsinj<tol ) ) then
go to 1994
end if
if( notrot>=emptsw )go to 1994
end do loop_1993
! end i=1:nsweep loop
! #:( reaching this point means that the procedure has not converged.
info = nsweep - 1_${ik}$
go to 1995
1994 continue
! #:) reaching this point means numerical convergence after the i-th
! sweep.
info = 0_${ik}$
! #:) info = 0 confirms successful iterations.
1995 continue
! sort the vector sva() of column norms.
do p = 1, n - 1
q = stdlib${ii}$_idamax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
aapq = d( p )
d( p ) = d( q )
d( q ) = aapq
call stdlib${ii}$_zswap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_zswap( mvl, v( 1_${ik}$, p ), 1_${ik}$, v( 1_${ik}$, q ), 1_${ik}$ )
end if
end do
return
end subroutine stdlib${ii}$_zgsvj1
#:for ck,ct,ci in CMPLX_KINDS_TYPES
#:if not ck in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ci}$gsvj1( jobv, m, n, n1, a, lda, d, sva, mv, v, ldv,eps, sfmin, tol, &
!! ZGSVJ1: is called from ZGESVJ as a pre-processor and that is its main
!! purpose. It applies Jacobi rotations in the same way as ZGESVJ does, but
!! it targets only particular pivots and it does not check convergence
!! (stopping criterion). Few tuning parameters (marked by [TP]) are
!! available for the implementer.
!! Further Details
!! ~~~~~~~~~~~~~~~
!! ZGSVJ1 applies few sweeps of Jacobi rotations in the column space of
!! the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
!! off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
!! block-entries (tiles) of the (1,2) off-diagonal block are marked by the
!! [x]'s in the following scheme:
!! | * * * [x] [x] [x]|
!! | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
!! | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
!! |[x] [x] [x] * * * |
!! |[x] [x] [x] * * * |
!! |[x] [x] [x] * * * |
!! In terms of the columns of A, the first N1 columns are rotated 'against'
!! the remaining N-N1 columns, trying to increase the angle between the
!! corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
!! tiled using quadratic tiles of side KBL. Here, KBL is a tuning parameter.
!! The number of sweeps is given in NSWEEP and the orthogonality threshold
!! is given in TOL.
nsweep, work, lwork, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${ck}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
real(${ck}$), intent(in) :: eps, sfmin, tol
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: lda, ldv, lwork, m, mv, n, n1, nsweep
character, intent(in) :: jobv
! Array Arguments
complex(${ck}$), intent(inout) :: a(lda,*), d(n), v(ldv,*)
complex(${ck}$), intent(out) :: work(lwork)
real(${ck}$), intent(inout) :: sva(n)
! =====================================================================
! Local Scalars
complex(${ck}$) :: aapq, ompq
real(${ck}$) :: aapp, aapp0, aapq1, aaqq, apoaq, aqoap, big, bigtheta, cs, mxaapq, mxsinj, &
rootbig, rooteps, rootsfmin, roottol, small, sn, t, temp1, theta, thsign
integer(${ik}$) :: blskip, emptsw, i, ibr, igl, ierr, ijblsk, iswrot, jbc, jgl, kbl, mvl, &
notrot, nblc, nblr, p, pskipped, q, rowskip, swband
logical(lk) :: applv, rotok, rsvec
! Intrinsic Functions
! From Lapack
! Executable Statements
! test the input parameters.
applv = stdlib_lsame( jobv, 'A' )
rsvec = stdlib_lsame( jobv, 'V' )
if( .not.( rsvec .or. applv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( ( n<0_${ik}$ ) .or. ( n>m ) ) then
info = -3_${ik}$
else if( n1<0_${ik}$ ) then
info = -4_${ik}$
else if( lda<m ) then
info = -6_${ik}$
else if( ( rsvec.or.applv ) .and. ( mv<0_${ik}$ ) ) then
info = -9_${ik}$
else if( ( rsvec.and.( ldv<n ) ).or.( applv.and.( ldv<mv ) ) ) then
info = -11_${ik}$
else if( tol<=eps ) then
info = -14_${ik}$
else if( nsweep<0_${ik}$ ) then
info = -15_${ik}$
else if( lwork<m ) then
info = -17_${ik}$
else
info = 0_${ik}$
end if
! #:(
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZGSVJ1', -info )
return
end if
if( rsvec ) then
mvl = n
else if( applv ) then
mvl = mv
end if
rsvec = rsvec .or. applv
rooteps = sqrt( eps )
rootsfmin = sqrt( sfmin )
small = sfmin / eps
big = one / sfmin
rootbig = one / rootsfmin
! large = big / sqrt( real( m*n,KIND=${ck}$) )
bigtheta = one / rooteps
roottol = sqrt( tol )
! Initialize The Right Singular Vector Matrix
! rsvec = stdlib_lsame( jobv, 'y' )
emptsw = n1*( n-n1 )
notrot = 0_${ik}$
! .. row-cyclic pivot strategy with de rijk's pivoting ..
kbl = min( 8_${ik}$, n )
nblr = n1 / kbl
if( ( nblr*kbl )/=n1 )nblr = nblr + 1_${ik}$
! .. the tiling is nblr-by-nblc [tiles]
nblc = ( n-n1 ) / kbl
if( ( nblc*kbl )/=( n-n1 ) )nblc = nblc + 1_${ik}$
blskip = ( kbl**2_${ik}$ ) + 1_${ik}$
! [tp] blkskip is a tuning parameter that depends on swband and kbl.
rowskip = min( 5_${ik}$, kbl )
! [tp] rowskip is a tuning parameter.
swband = 0_${ik}$
! [tp] swband is a tuning parameter. it is meaningful and effective
! if stdlib${ii}$_${ci}$gesvj is used as a computational routine in the preconditioned
! jacobi svd algorithm stdlib${ii}$_${ci}$gejsv.
! | * * * [x] [x] [x]|
! | * * * [x] [x] [x]| row-cycling in the nblr-by-nblc [x] blocks.
! | * * * [x] [x] [x]| row-cyclic pivoting inside each [x] block.
! |[x] [x] [x] * * * |
! |[x] [x] [x] * * * |
! |[x] [x] [x] * * * |
loop_1993: do i = 1, nsweep
! .. go go go ...
mxaapq = zero
mxsinj = zero
iswrot = 0_${ik}$
notrot = 0_${ik}$
pskipped = 0_${ik}$
! each sweep is unrolled using kbl-by-kbl tiles over the pivot pairs
! 1 <= p < q <= n. this is the first step toward a blocked implementation
! of the rotations. new implementation, based on block transformations,
! is under development.
loop_2000: do ibr = 1, nblr
igl = ( ibr-1 )*kbl + 1_${ik}$
! ... go to the off diagonal blocks
igl = ( ibr-1 )*kbl + 1_${ik}$
! do 2010 jbc = ibr + 1, nbl
loop_2010: do jbc = 1, nblc
jgl = ( jbc-1 )*kbl + n1 + 1_${ik}$
! doing the block at ( ibr, jbc )
ijblsk = 0_${ik}$
loop_2100: do p = igl, min( igl+kbl-1, n1 )
aapp = sva( p )
if( aapp>zero ) then
pskipped = 0_${ik}$
loop_2200: do q = jgl, min( jgl+kbl-1, n )
aaqq = sva( q )
if( aaqq>zero ) then
aapp0 = aapp
! M X 2 Jacobi Svd
! safe gram matrix computation
if( aaqq>=one ) then
if( aapp>=aaqq ) then
rotok = ( small*aapp )<=aaqq
else
rotok = ( small*aaqq )<=aapp
end if
if( aapp<( big / aaqq ) ) then
aapq = ( stdlib${ii}$_${ci}$dotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq ) / aapp
else
call stdlib${ii}$_${ci}$copy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_${ci}$dotc( m, work, 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / &
aaqq
end if
else
if( aapp>=aaqq ) then
rotok = aapp<=( aaqq / small )
else
rotok = aaqq<=( aapp / small )
end if
if( aapp>( small / aaqq ) ) then
aapq = ( stdlib${ii}$_${ci}$dotc( m, a( 1_${ik}$, p ), 1_${ik}$,a( 1_${ik}$, q ), 1_${ik}$ ) / max(&
aaqq,aapp) )/ min(aaqq,aapp)
else
call stdlib${ii}$_${ci}$copy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq,one, m, 1_${ik}$,work, lda, &
ierr )
aapq = stdlib${ii}$_${ci}$dotc( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ ) / &
aapp
end if
end if
! aapq = aapq * conjg(cwork(p))*cwork(q)
aapq1 = -abs(aapq)
mxaapq = max( mxaapq, -aapq1 )
! to rotate or not to rotate, that is the question ...
if( abs( aapq1 )>tol ) then
ompq = aapq / abs(aapq)
notrot = 0_${ik}$
! [rtd] rotated = rotated + 1
pskipped = 0_${ik}$
iswrot = iswrot + 1_${ik}$
if( rotok ) then
aqoap = aaqq / aapp
apoaq = aapp / aaqq
theta = -half*abs( aqoap-apoaq )/ aapq1
if( aaqq>aapp0 )theta = -theta
if( abs( theta )>bigtheta ) then
t = half / theta
cs = one
call stdlib${ii}$_${ci}$rot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*t )
if( rsvec ) then
call stdlib${ii}$_${ci}$rot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*t )
end if
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
mxsinj = max( mxsinj, abs( t ) )
else
! Choose Correct Signum For Theta And Rotate
thsign = -sign( one, aapq1 )
if( aaqq>aapp0 )thsign = -thsign
t = one / ( theta+thsign*sqrt( one+theta*theta ) )
cs = sqrt( one / ( one+t*t ) )
sn = t*cs
mxsinj = max( mxsinj, abs( sn ) )
sva( q ) = aaqq*sqrt( max( zero,one+t*apoaq*aapq1 ) )
aapp = aapp*sqrt( max( zero,one-t*aqoap*aapq1 ) )
call stdlib${ii}$_${ci}$rot( m, a(1_${ik}$,p), 1_${ik}$, a(1_${ik}$,q), 1_${ik}$,cs, conjg(ompq)&
*sn )
if( rsvec ) then
call stdlib${ii}$_${ci}$rot( mvl, v(1_${ik}$,p), 1_${ik}$,v(1_${ik}$,q), 1_${ik}$, cs, &
conjg(ompq)*sn )
end if
end if
d(p) = -d(q) * ompq
else
! .. have to use modified gram-schmidt like transformation
if( aapp>aaqq ) then
call stdlib${ii}$_${ci}$copy( m, a( 1_${ik}$, p ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, work,lda,&
ierr )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
call stdlib${ii}$_${ci}$axpy( m, -aapq, work,1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, one, aaqq,m, 1_${ik}$, a( 1_${ik}$, q ),&
lda,ierr )
sva( q ) = aaqq*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
else
call stdlib${ii}$_${ci}$copy( m, a( 1_${ik}$, q ), 1_${ik}$,work, 1_${ik}$ )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aaqq, one,m, 1_${ik}$, work,lda,&
ierr )
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, aapp, one,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
call stdlib${ii}$_${ci}$axpy( m, -conjg(aapq),work, 1_${ik}$, a( 1_${ik}$, p ), 1_${ik}$ &
)
call stdlib${ii}$_${ci}$lascl( 'G', 0_${ik}$, 0_${ik}$, one, aapp,m, 1_${ik}$, a( 1_${ik}$, p ),&
lda,ierr )
sva( p ) = aapp*sqrt( max( zero,one-aapq1*aapq1 ) )
mxsinj = max( mxsinj, sfmin )
end if
end if
! end if rotok then ... else
! in the case of cancellation in updating sva(q), sva(p)
! .. recompute sva(q), sva(p)
if( ( sva( q ) / aaqq )**2_${ik}$<=rooteps )then
if( ( aaqq<rootbig ) .and.( aaqq>rootsfmin ) ) then
sva( q ) = stdlib${ii}$_${c2ri(ci)}$znrm2( m, a( 1_${ik}$, q ), 1_${ik}$)
else
t = zero
aaqq = one
call stdlib${ii}$_${ci}$lassq( m, a( 1_${ik}$, q ), 1_${ik}$, t,aaqq )
sva( q ) = t*sqrt( aaqq )
end if
end if
if( ( aapp / aapp0 )**2_${ik}$<=rooteps ) then
if( ( aapp<rootbig ) .and.( aapp>rootsfmin ) ) then
aapp = stdlib${ii}$_${c2ri(ci)}$znrm2( m, a( 1_${ik}$, p ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_${ci}$lassq( m, a( 1_${ik}$, p ), 1_${ik}$, t,aapp )
aapp = t*sqrt( aapp )
end if
sva( p ) = aapp
end if
! end of ok rotation
else
notrot = notrot + 1_${ik}$
! [rtd] skipped = skipped + 1
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
else
notrot = notrot + 1_${ik}$
pskipped = pskipped + 1_${ik}$
ijblsk = ijblsk + 1_${ik}$
end if
if( ( i<=swband ) .and. ( ijblsk>=blskip ) )then
sva( p ) = aapp
notrot = 0_${ik}$
go to 2011
end if
if( ( i<=swband ) .and.( pskipped>rowskip ) ) then
aapp = -aapp
notrot = 0_${ik}$
go to 2203
end if
end do loop_2200
! end of the q-loop
2203 continue
sva( p ) = aapp
else
if( aapp==zero )notrot = notrot +min( jgl+kbl-1, n ) - jgl + 1_${ik}$
if( aapp<zero )notrot = 0_${ik}$
end if
end do loop_2100
! end of the p-loop
end do loop_2010
! end of the jbc-loop
2011 continue
! 2011 bailed out of the jbc-loop
do p = igl, min( igl+kbl-1, n )
sva( p ) = abs( sva( p ) )
end do
! **
end do loop_2000
! 2000 :: end of the ibr-loop
! .. update sva(n)
if( ( sva( n )<rootbig ) .and. ( sva( n )>rootsfmin ) )then
sva( n ) = stdlib${ii}$_${c2ri(ci)}$znrm2( m, a( 1_${ik}$, n ), 1_${ik}$ )
else
t = zero
aapp = one
call stdlib${ii}$_${ci}$lassq( m, a( 1_${ik}$, n ), 1_${ik}$, t, aapp )
sva( n ) = t*sqrt( aapp )
end if
! additional steering devices
if( ( i<swband ) .and. ( ( mxaapq<=roottol ) .or.( iswrot<=n ) ) )swband = i
if( ( i>swband+1 ) .and. ( mxaapq<sqrt( real( n,KIND=${ck}$) )*tol ) .and. ( real( n,&
KIND=${ck}$)*mxaapq*mxsinj<tol ) ) then
go to 1994
end if
if( notrot>=emptsw )go to 1994
end do loop_1993
! end i=1:nsweep loop
! #:( reaching this point means that the procedure has not converged.
info = nsweep - 1_${ik}$
go to 1995
1994 continue
! #:) reaching this point means numerical convergence after the i-th
! sweep.
info = 0_${ik}$
! #:) info = 0 confirms successful iterations.
1995 continue
! sort the vector sva() of column norms.
do p = 1, n - 1
q = stdlib${ii}$_i${c2ri(ci)}$amax( n-p+1, sva( p ), 1_${ik}$ ) + p - 1_${ik}$
if( p/=q ) then
temp1 = sva( p )
sva( p ) = sva( q )
sva( q ) = temp1
aapq = d( p )
d( p ) = d( q )
d( q ) = aapq
call stdlib${ii}$_${ci}$swap( m, a( 1_${ik}$, p ), 1_${ik}$, a( 1_${ik}$, q ), 1_${ik}$ )
if( rsvec )call stdlib${ii}$_${ci}$swap( mvl, v( 1_${ik}$, p ), 1_${ik}$, v( 1_${ik}$, q ), 1_${ik}$ )
end if
end do
return
end subroutine stdlib${ii}$_${ci}$gsvj1
#:endif
#:endfor
pure module subroutine stdlib${ii}$_stgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b,ldb, tola, tolb, &
!! STGSJA computes the generalized singular value decomposition (GSVD)
!! of two real upper triangular (or trapezoidal) matrices A and B.
!! On entry, it is assumed that matrices A and B have the following
!! forms, which may be obtained by the preprocessing subroutine SGGSVP
!! from a general M-by-N matrix A and P-by-N matrix B:
!! N-K-L K L
!! A = K ( 0 A12 A13 ) if M-K-L >= 0;
!! L ( 0 0 A23 )
!! M-K-L ( 0 0 0 )
!! N-K-L K L
!! A = K ( 0 A12 A13 ) if M-K-L < 0;
!! M-K ( 0 0 A23 )
!! N-K-L K L
!! B = L ( 0 0 B13 )
!! P-L ( 0 0 0 )
!! where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
!! upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
!! otherwise A23 is (M-K)-by-L upper trapezoidal.
!! On exit,
!! U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ),
!! where U, V and Q are orthogonal matrices.
!! R is a nonsingular upper triangular matrix, and D1 and D2 are
!! ``diagonal'' matrices, which are of the following structures:
!! If M-K-L >= 0,
!! K L
!! D1 = K ( I 0 )
!! L ( 0 C )
!! M-K-L ( 0 0 )
!! K L
!! D2 = L ( 0 S )
!! P-L ( 0 0 )
!! N-K-L K L
!! ( 0 R ) = K ( 0 R11 R12 ) K
!! L ( 0 0 R22 ) L
!! where
!! C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
!! S = diag( BETA(K+1), ... , BETA(K+L) ),
!! C**2 + S**2 = I.
!! R is stored in A(1:K+L,N-K-L+1:N) on exit.
!! If M-K-L < 0,
!! K M-K K+L-M
!! D1 = K ( I 0 0 )
!! M-K ( 0 C 0 )
!! K M-K K+L-M
!! D2 = M-K ( 0 S 0 )
!! K+L-M ( 0 0 I )
!! P-L ( 0 0 0 )
!! N-K-L K M-K K+L-M
!! ( 0 R ) = K ( 0 R11 R12 R13 )
!! M-K ( 0 0 R22 R23 )
!! K+L-M ( 0 0 0 R33 )
!! where
!! C = diag( ALPHA(K+1), ... , ALPHA(M) ),
!! S = diag( BETA(K+1), ... , BETA(M) ),
!! C**2 + S**2 = I.
!! R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
!! ( 0 R22 R23 )
!! in B(M-K+1:L,N+M-K-L+1:N) on exit.
!! The computation of the orthogonal transformation matrices U, V or Q
!! is optional. These matrices may either be formed explicitly, or they
!! may be postmultiplied into input matrices U1, V1, or Q1.
alpha, beta, u, ldu, v, ldv,q, ldq, work, ncycle, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: jobq, jobu, jobv
integer(${ik}$), intent(out) :: info, ncycle
integer(${ik}$), intent(in) :: k, l, lda, ldb, ldq, ldu, ldv, m, n, p
real(sp), intent(in) :: tola, tolb
! Array Arguments
real(sp), intent(inout) :: a(lda,*), b(ldb,*), q(ldq,*), u(ldu,*), v(ldv,*)
real(sp), intent(out) :: alpha(*), beta(*), work(*)
! =====================================================================
! Parameters
integer(${ik}$), parameter :: maxit = 40_${ik}$
real(sp), parameter :: hugenum = huge(zero)
! Local Scalars
logical(lk) :: initq, initu, initv, upper, wantq, wantu, wantv
integer(${ik}$) :: i, j, kcycle
real(sp) :: a1, a2, a3, b1, b2, b3, csq, csu, csv, error, gamma, rwk, snq, snu, snv, &
ssmin
! Intrinsic Functions
! Executable Statements
! decode and test the input parameters
initu = stdlib_lsame( jobu, 'I' )
wantu = initu .or. stdlib_lsame( jobu, 'U' )
initv = stdlib_lsame( jobv, 'I' )
wantv = initv .or. stdlib_lsame( jobv, 'V' )
initq = stdlib_lsame( jobq, 'I' )
wantq = initq .or. stdlib_lsame( jobq, 'Q' )
info = 0_${ik}$
if( .not.( initu .or. wantu .or. stdlib_lsame( jobu, 'N' ) ) ) then
info = -1_${ik}$
else if( .not.( initv .or. wantv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -2_${ik}$
else if( .not.( initq .or. wantq .or. stdlib_lsame( jobq, 'N' ) ) ) then
info = -3_${ik}$
else if( m<0_${ik}$ ) then
info = -4_${ik}$
else if( p<0_${ik}$ ) then
info = -5_${ik}$
else if( n<0_${ik}$ ) then
info = -6_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -10_${ik}$
else if( ldb<max( 1_${ik}$, p ) ) then
info = -12_${ik}$
else if( ldu<1_${ik}$ .or. ( wantu .and. ldu<m ) ) then
info = -18_${ik}$
else if( ldv<1_${ik}$ .or. ( wantv .and. ldv<p ) ) then
info = -20_${ik}$
else if( ldq<1_${ik}$ .or. ( wantq .and. ldq<n ) ) then
info = -22_${ik}$
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'STGSJA', -info )
return
end if
! initialize u, v and q, if necessary
if( initu )call stdlib${ii}$_slaset( 'FULL', m, m, zero, one, u, ldu )
if( initv )call stdlib${ii}$_slaset( 'FULL', p, p, zero, one, v, ldv )
if( initq )call stdlib${ii}$_slaset( 'FULL', n, n, zero, one, q, ldq )
! loop until convergence
upper = .false.
loop_40: do kcycle = 1, maxit
upper = .not.upper
loop_20: do i = 1, l - 1
loop_10: do j = i + 1, l
a1 = zero
a2 = zero
a3 = zero
if( k+i<=m )a1 = a( k+i, n-l+i )
if( k+j<=m )a3 = a( k+j, n-l+j )
b1 = b( i, n-l+i )
b3 = b( j, n-l+j )
if( upper ) then
if( k+i<=m )a2 = a( k+i, n-l+j )
b2 = b( i, n-l+j )
else
if( k+j<=m )a2 = a( k+j, n-l+i )
b2 = b( j, n-l+i )
end if
call stdlib${ii}$_slags2( upper, a1, a2, a3, b1, b2, b3, csu, snu,csv, snv, csq, &
snq )
! update (k+i)-th and (k+j)-th rows of matrix a: u**t *a
if( k+j<=m )call stdlib${ii}$_srot( l, a( k+j, n-l+1 ), lda, a( k+i, n-l+1 ),lda, &
csu, snu )
! update i-th and j-th rows of matrix b: v**t *b
call stdlib${ii}$_srot( l, b( j, n-l+1 ), ldb, b( i, n-l+1 ), ldb,csv, snv )
! update (n-l+i)-th and (n-l+j)-th columns of matrices
! a and b: a*q and b*q
call stdlib${ii}$_srot( min( k+l, m ), a( 1_${ik}$, n-l+j ), 1_${ik}$,a( 1_${ik}$, n-l+i ), 1_${ik}$, csq, snq )
call stdlib${ii}$_srot( l, b( 1_${ik}$, n-l+j ), 1_${ik}$, b( 1_${ik}$, n-l+i ), 1_${ik}$, csq,snq )
if( upper ) then
if( k+i<=m )a( k+i, n-l+j ) = zero
b( i, n-l+j ) = zero
else
if( k+j<=m )a( k+j, n-l+i ) = zero
b( j, n-l+i ) = zero
end if
! update orthogonal matrices u, v, q, if desired.
if( wantu .and. k+j<=m )call stdlib${ii}$_srot( m, u( 1_${ik}$, k+j ), 1_${ik}$, u( 1_${ik}$, k+i ), 1_${ik}$, &
csu,snu )
if( wantv )call stdlib${ii}$_srot( p, v( 1_${ik}$, j ), 1_${ik}$, v( 1_${ik}$, i ), 1_${ik}$, csv, snv )
if( wantq )call stdlib${ii}$_srot( n, q( 1_${ik}$, n-l+j ), 1_${ik}$, q( 1_${ik}$, n-l+i ), 1_${ik}$, csq,snq )
end do loop_10
end do loop_20
if( .not.upper ) then
! the matrices a13 and b13 were lower triangular at the start
! of the cycle, and are now upper triangular.
! convergence test: test the parallelism of the corresponding
! rows of a and b.
error = zero
do i = 1, min( l, m-k )
call stdlib${ii}$_scopy( l-i+1, a( k+i, n-l+i ), lda, work, 1_${ik}$ )
call stdlib${ii}$_scopy( l-i+1, b( i, n-l+i ), ldb, work( l+1 ), 1_${ik}$ )
call stdlib${ii}$_slapll( l-i+1, work, 1_${ik}$, work( l+1 ), 1_${ik}$, ssmin )
error = max( error, ssmin )
end do
if( abs( error )<=min( tola, tolb ) )go to 50
end if
! end of cycle loop
end do loop_40
! the algorithm has not converged after maxit cycles.
info = 1_${ik}$
go to 100
50 continue
! if error <= min(tola,tolb), then the algorithm has converged.
! compute the generalized singular value pairs (alpha, beta), and
! set the triangular matrix r to array a.
do i = 1, k
alpha( i ) = one
beta( i ) = zero
end do
do i = 1, min( l, m-k )
a1 = a( k+i, n-l+i )
b1 = b( i, n-l+i )
gamma = b1 / a1
if( (gamma<=hugenum).and.(gamma>=-hugenum) ) then
! change sign if necessary
if( gamma<zero ) then
call stdlib${ii}$_sscal( l-i+1, -one, b( i, n-l+i ), ldb )
if( wantv )call stdlib${ii}$_sscal( p, -one, v( 1_${ik}$, i ), 1_${ik}$ )
end if
call stdlib${ii}$_slartg( abs( gamma ), one, beta( k+i ), alpha( k+i ),rwk )
if( alpha( k+i )>=beta( k+i ) ) then
call stdlib${ii}$_sscal( l-i+1, one / alpha( k+i ), a( k+i, n-l+i ),lda )
else
call stdlib${ii}$_sscal( l-i+1, one / beta( k+i ), b( i, n-l+i ),ldb )
call stdlib${ii}$_scopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),lda )
end if
else
alpha( k+i ) = zero
beta( k+i ) = one
call stdlib${ii}$_scopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),lda )
end if
end do
! post-assignment
do i = m + 1, k + l
alpha( i ) = zero
beta( i ) = one
end do
if( k+l<n ) then
do i = k + l + 1, n
alpha( i ) = zero
beta( i ) = zero
end do
end if
100 continue
ncycle = kcycle
return
end subroutine stdlib${ii}$_stgsja
pure module subroutine stdlib${ii}$_dtgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b,ldb, tola, tolb, &
!! DTGSJA computes the generalized singular value decomposition (GSVD)
!! of two real upper triangular (or trapezoidal) matrices A and B.
!! On entry, it is assumed that matrices A and B have the following
!! forms, which may be obtained by the preprocessing subroutine DGGSVP
!! from a general M-by-N matrix A and P-by-N matrix B:
!! N-K-L K L
!! A = K ( 0 A12 A13 ) if M-K-L >= 0;
!! L ( 0 0 A23 )
!! M-K-L ( 0 0 0 )
!! N-K-L K L
!! A = K ( 0 A12 A13 ) if M-K-L < 0;
!! M-K ( 0 0 A23 )
!! N-K-L K L
!! B = L ( 0 0 B13 )
!! P-L ( 0 0 0 )
!! where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
!! upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
!! otherwise A23 is (M-K)-by-L upper trapezoidal.
!! On exit,
!! U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ),
!! where U, V and Q are orthogonal matrices.
!! R is a nonsingular upper triangular matrix, and D1 and D2 are
!! ``diagonal'' matrices, which are of the following structures:
!! If M-K-L >= 0,
!! K L
!! D1 = K ( I 0 )
!! L ( 0 C )
!! M-K-L ( 0 0 )
!! K L
!! D2 = L ( 0 S )
!! P-L ( 0 0 )
!! N-K-L K L
!! ( 0 R ) = K ( 0 R11 R12 ) K
!! L ( 0 0 R22 ) L
!! where
!! C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
!! S = diag( BETA(K+1), ... , BETA(K+L) ),
!! C**2 + S**2 = I.
!! R is stored in A(1:K+L,N-K-L+1:N) on exit.
!! If M-K-L < 0,
!! K M-K K+L-M
!! D1 = K ( I 0 0 )
!! M-K ( 0 C 0 )
!! K M-K K+L-M
!! D2 = M-K ( 0 S 0 )
!! K+L-M ( 0 0 I )
!! P-L ( 0 0 0 )
!! N-K-L K M-K K+L-M
!! ( 0 R ) = K ( 0 R11 R12 R13 )
!! M-K ( 0 0 R22 R23 )
!! K+L-M ( 0 0 0 R33 )
!! where
!! C = diag( ALPHA(K+1), ... , ALPHA(M) ),
!! S = diag( BETA(K+1), ... , BETA(M) ),
!! C**2 + S**2 = I.
!! R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
!! ( 0 R22 R23 )
!! in B(M-K+1:L,N+M-K-L+1:N) on exit.
!! The computation of the orthogonal transformation matrices U, V or Q
!! is optional. These matrices may either be formed explicitly, or they
!! may be postmultiplied into input matrices U1, V1, or Q1.
alpha, beta, u, ldu, v, ldv,q, ldq, work, ncycle, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: jobq, jobu, jobv
integer(${ik}$), intent(out) :: info, ncycle
integer(${ik}$), intent(in) :: k, l, lda, ldb, ldq, ldu, ldv, m, n, p
real(dp), intent(in) :: tola, tolb
! Array Arguments
real(dp), intent(inout) :: a(lda,*), b(ldb,*), q(ldq,*), u(ldu,*), v(ldv,*)
real(dp), intent(out) :: alpha(*), beta(*), work(*)
! =====================================================================
! Parameters
integer(${ik}$), parameter :: maxit = 40_${ik}$
real(dp), parameter :: hugenum = huge(zero)
! Local Scalars
logical(lk) :: initq, initu, initv, upper, wantq, wantu, wantv
integer(${ik}$) :: i, j, kcycle
real(dp) :: a1, a2, a3, b1, b2, b3, csq, csu, csv, error, gamma, rwk, snq, snu, snv, &
ssmin
! Intrinsic Functions
! Executable Statements
! decode and test the input parameters
initu = stdlib_lsame( jobu, 'I' )
wantu = initu .or. stdlib_lsame( jobu, 'U' )
initv = stdlib_lsame( jobv, 'I' )
wantv = initv .or. stdlib_lsame( jobv, 'V' )
initq = stdlib_lsame( jobq, 'I' )
wantq = initq .or. stdlib_lsame( jobq, 'Q' )
info = 0_${ik}$
if( .not.( initu .or. wantu .or. stdlib_lsame( jobu, 'N' ) ) ) then
info = -1_${ik}$
else if( .not.( initv .or. wantv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -2_${ik}$
else if( .not.( initq .or. wantq .or. stdlib_lsame( jobq, 'N' ) ) ) then
info = -3_${ik}$
else if( m<0_${ik}$ ) then
info = -4_${ik}$
else if( p<0_${ik}$ ) then
info = -5_${ik}$
else if( n<0_${ik}$ ) then
info = -6_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -10_${ik}$
else if( ldb<max( 1_${ik}$, p ) ) then
info = -12_${ik}$
else if( ldu<1_${ik}$ .or. ( wantu .and. ldu<m ) ) then
info = -18_${ik}$
else if( ldv<1_${ik}$ .or. ( wantv .and. ldv<p ) ) then
info = -20_${ik}$
else if( ldq<1_${ik}$ .or. ( wantq .and. ldq<n ) ) then
info = -22_${ik}$
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DTGSJA', -info )
return
end if
! initialize u, v and q, if necessary
if( initu )call stdlib${ii}$_dlaset( 'FULL', m, m, zero, one, u, ldu )
if( initv )call stdlib${ii}$_dlaset( 'FULL', p, p, zero, one, v, ldv )
if( initq )call stdlib${ii}$_dlaset( 'FULL', n, n, zero, one, q, ldq )
! loop until convergence
upper = .false.
loop_40: do kcycle = 1, maxit
upper = .not.upper
loop_20: do i = 1, l - 1
loop_10: do j = i + 1, l
a1 = zero
a2 = zero
a3 = zero
if( k+i<=m )a1 = a( k+i, n-l+i )
if( k+j<=m )a3 = a( k+j, n-l+j )
b1 = b( i, n-l+i )
b3 = b( j, n-l+j )
if( upper ) then
if( k+i<=m )a2 = a( k+i, n-l+j )
b2 = b( i, n-l+j )
else
if( k+j<=m )a2 = a( k+j, n-l+i )
b2 = b( j, n-l+i )
end if
call stdlib${ii}$_dlags2( upper, a1, a2, a3, b1, b2, b3, csu, snu,csv, snv, csq, &
snq )
! update (k+i)-th and (k+j)-th rows of matrix a: u**t *a
if( k+j<=m )call stdlib${ii}$_drot( l, a( k+j, n-l+1 ), lda, a( k+i, n-l+1 ),lda, &
csu, snu )
! update i-th and j-th rows of matrix b: v**t *b
call stdlib${ii}$_drot( l, b( j, n-l+1 ), ldb, b( i, n-l+1 ), ldb,csv, snv )
! update (n-l+i)-th and (n-l+j)-th columns of matrices
! a and b: a*q and b*q
call stdlib${ii}$_drot( min( k+l, m ), a( 1_${ik}$, n-l+j ), 1_${ik}$,a( 1_${ik}$, n-l+i ), 1_${ik}$, csq, snq )
call stdlib${ii}$_drot( l, b( 1_${ik}$, n-l+j ), 1_${ik}$, b( 1_${ik}$, n-l+i ), 1_${ik}$, csq,snq )
if( upper ) then
if( k+i<=m )a( k+i, n-l+j ) = zero
b( i, n-l+j ) = zero
else
if( k+j<=m )a( k+j, n-l+i ) = zero
b( j, n-l+i ) = zero
end if
! update orthogonal matrices u, v, q, if desired.
if( wantu .and. k+j<=m )call stdlib${ii}$_drot( m, u( 1_${ik}$, k+j ), 1_${ik}$, u( 1_${ik}$, k+i ), 1_${ik}$, &
csu,snu )
if( wantv )call stdlib${ii}$_drot( p, v( 1_${ik}$, j ), 1_${ik}$, v( 1_${ik}$, i ), 1_${ik}$, csv, snv )
if( wantq )call stdlib${ii}$_drot( n, q( 1_${ik}$, n-l+j ), 1_${ik}$, q( 1_${ik}$, n-l+i ), 1_${ik}$, csq,snq )
end do loop_10
end do loop_20
if( .not.upper ) then
! the matrices a13 and b13 were lower triangular at the start
! of the cycle, and are now upper triangular.
! convergence test: test the parallelism of the corresponding
! rows of a and b.
error = zero
do i = 1, min( l, m-k )
call stdlib${ii}$_dcopy( l-i+1, a( k+i, n-l+i ), lda, work, 1_${ik}$ )
call stdlib${ii}$_dcopy( l-i+1, b( i, n-l+i ), ldb, work( l+1 ), 1_${ik}$ )
call stdlib${ii}$_dlapll( l-i+1, work, 1_${ik}$, work( l+1 ), 1_${ik}$, ssmin )
error = max( error, ssmin )
end do
if( abs( error )<=min( tola, tolb ) )go to 50
end if
! end of cycle loop
end do loop_40
! the algorithm has not converged after maxit cycles.
info = 1_${ik}$
go to 100
50 continue
! if error <= min(tola,tolb), then the algorithm has converged.
! compute the generalized singular value pairs (alpha, beta), and
! set the triangular matrix r to array a.
do i = 1, k
alpha( i ) = one
beta( i ) = zero
end do
do i = 1, min( l, m-k )
a1 = a( k+i, n-l+i )
b1 = b( i, n-l+i )
gamma = b1 / a1
if( (gamma<=hugenum).and.(gamma>=-hugenum) ) then
! change sign if necessary
if( gamma<zero ) then
call stdlib${ii}$_dscal( l-i+1, -one, b( i, n-l+i ), ldb )
if( wantv )call stdlib${ii}$_dscal( p, -one, v( 1_${ik}$, i ), 1_${ik}$ )
end if
call stdlib${ii}$_dlartg( abs( gamma ), one, beta( k+i ), alpha( k+i ),rwk )
if( alpha( k+i )>=beta( k+i ) ) then
call stdlib${ii}$_dscal( l-i+1, one / alpha( k+i ), a( k+i, n-l+i ),lda )
else
call stdlib${ii}$_dscal( l-i+1, one / beta( k+i ), b( i, n-l+i ),ldb )
call stdlib${ii}$_dcopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),lda )
end if
else
alpha( k+i ) = zero
beta( k+i ) = one
call stdlib${ii}$_dcopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),lda )
end if
end do
! post-assignment
do i = m + 1, k + l
alpha( i ) = zero
beta( i ) = one
end do
if( k+l<n ) then
do i = k + l + 1, n
alpha( i ) = zero
beta( i ) = zero
end do
end if
100 continue
ncycle = kcycle
return
end subroutine stdlib${ii}$_dtgsja
#:for rk,rt,ri in REAL_KINDS_TYPES
#:if not rk in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ri}$tgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b,ldb, tola, tolb, &
!! DTGSJA: computes the generalized singular value decomposition (GSVD)
!! of two real upper triangular (or trapezoidal) matrices A and B.
!! On entry, it is assumed that matrices A and B have the following
!! forms, which may be obtained by the preprocessing subroutine DGGSVP
!! from a general M-by-N matrix A and P-by-N matrix B:
!! N-K-L K L
!! A = K ( 0 A12 A13 ) if M-K-L >= 0;
!! L ( 0 0 A23 )
!! M-K-L ( 0 0 0 )
!! N-K-L K L
!! A = K ( 0 A12 A13 ) if M-K-L < 0;
!! M-K ( 0 0 A23 )
!! N-K-L K L
!! B = L ( 0 0 B13 )
!! P-L ( 0 0 0 )
!! where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
!! upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
!! otherwise A23 is (M-K)-by-L upper trapezoidal.
!! On exit,
!! U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ),
!! where U, V and Q are orthogonal matrices.
!! R is a nonsingular upper triangular matrix, and D1 and D2 are
!! ``diagonal'' matrices, which are of the following structures:
!! If M-K-L >= 0,
!! K L
!! D1 = K ( I 0 )
!! L ( 0 C )
!! M-K-L ( 0 0 )
!! K L
!! D2 = L ( 0 S )
!! P-L ( 0 0 )
!! N-K-L K L
!! ( 0 R ) = K ( 0 R11 R12 ) K
!! L ( 0 0 R22 ) L
!! where
!! C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
!! S = diag( BETA(K+1), ... , BETA(K+L) ),
!! C**2 + S**2 = I.
!! R is stored in A(1:K+L,N-K-L+1:N) on exit.
!! If M-K-L < 0,
!! K M-K K+L-M
!! D1 = K ( I 0 0 )
!! M-K ( 0 C 0 )
!! K M-K K+L-M
!! D2 = M-K ( 0 S 0 )
!! K+L-M ( 0 0 I )
!! P-L ( 0 0 0 )
!! N-K-L K M-K K+L-M
!! ( 0 R ) = K ( 0 R11 R12 R13 )
!! M-K ( 0 0 R22 R23 )
!! K+L-M ( 0 0 0 R33 )
!! where
!! C = diag( ALPHA(K+1), ... , ALPHA(M) ),
!! S = diag( BETA(K+1), ... , BETA(M) ),
!! C**2 + S**2 = I.
!! R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
!! ( 0 R22 R23 )
!! in B(M-K+1:L,N+M-K-L+1:N) on exit.
!! The computation of the orthogonal transformation matrices U, V or Q
!! is optional. These matrices may either be formed explicitly, or they
!! may be postmultiplied into input matrices U1, V1, or Q1.
alpha, beta, u, ldu, v, ldv,q, ldq, work, ncycle, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${rk}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: jobq, jobu, jobv
integer(${ik}$), intent(out) :: info, ncycle
integer(${ik}$), intent(in) :: k, l, lda, ldb, ldq, ldu, ldv, m, n, p
real(${rk}$), intent(in) :: tola, tolb
! Array Arguments
real(${rk}$), intent(inout) :: a(lda,*), b(ldb,*), q(ldq,*), u(ldu,*), v(ldv,*)
real(${rk}$), intent(out) :: alpha(*), beta(*), work(*)
! =====================================================================
! Parameters
integer(${ik}$), parameter :: maxit = 40_${ik}$
real(${rk}$), parameter :: hugenum = huge(zero)
! Local Scalars
logical(lk) :: initq, initu, initv, upper, wantq, wantu, wantv
integer(${ik}$) :: i, j, kcycle
real(${rk}$) :: a1, a2, a3, b1, b2, b3, csq, csu, csv, error, gamma, rwk, snq, snu, snv, &
ssmin
! Intrinsic Functions
! Executable Statements
! decode and test the input parameters
initu = stdlib_lsame( jobu, 'I' )
wantu = initu .or. stdlib_lsame( jobu, 'U' )
initv = stdlib_lsame( jobv, 'I' )
wantv = initv .or. stdlib_lsame( jobv, 'V' )
initq = stdlib_lsame( jobq, 'I' )
wantq = initq .or. stdlib_lsame( jobq, 'Q' )
info = 0_${ik}$
if( .not.( initu .or. wantu .or. stdlib_lsame( jobu, 'N' ) ) ) then
info = -1_${ik}$
else if( .not.( initv .or. wantv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -2_${ik}$
else if( .not.( initq .or. wantq .or. stdlib_lsame( jobq, 'N' ) ) ) then
info = -3_${ik}$
else if( m<0_${ik}$ ) then
info = -4_${ik}$
else if( p<0_${ik}$ ) then
info = -5_${ik}$
else if( n<0_${ik}$ ) then
info = -6_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -10_${ik}$
else if( ldb<max( 1_${ik}$, p ) ) then
info = -12_${ik}$
else if( ldu<1_${ik}$ .or. ( wantu .and. ldu<m ) ) then
info = -18_${ik}$
else if( ldv<1_${ik}$ .or. ( wantv .and. ldv<p ) ) then
info = -20_${ik}$
else if( ldq<1_${ik}$ .or. ( wantq .and. ldq<n ) ) then
info = -22_${ik}$
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DTGSJA', -info )
return
end if
! initialize u, v and q, if necessary
if( initu )call stdlib${ii}$_${ri}$laset( 'FULL', m, m, zero, one, u, ldu )
if( initv )call stdlib${ii}$_${ri}$laset( 'FULL', p, p, zero, one, v, ldv )
if( initq )call stdlib${ii}$_${ri}$laset( 'FULL', n, n, zero, one, q, ldq )
! loop until convergence
upper = .false.
loop_40: do kcycle = 1, maxit
upper = .not.upper
loop_20: do i = 1, l - 1
loop_10: do j = i + 1, l
a1 = zero
a2 = zero
a3 = zero
if( k+i<=m )a1 = a( k+i, n-l+i )
if( k+j<=m )a3 = a( k+j, n-l+j )
b1 = b( i, n-l+i )
b3 = b( j, n-l+j )
if( upper ) then
if( k+i<=m )a2 = a( k+i, n-l+j )
b2 = b( i, n-l+j )
else
if( k+j<=m )a2 = a( k+j, n-l+i )
b2 = b( j, n-l+i )
end if
call stdlib${ii}$_${ri}$lags2( upper, a1, a2, a3, b1, b2, b3, csu, snu,csv, snv, csq, &
snq )
! update (k+i)-th and (k+j)-th rows of matrix a: u**t *a
if( k+j<=m )call stdlib${ii}$_${ri}$rot( l, a( k+j, n-l+1 ), lda, a( k+i, n-l+1 ),lda, &
csu, snu )
! update i-th and j-th rows of matrix b: v**t *b
call stdlib${ii}$_${ri}$rot( l, b( j, n-l+1 ), ldb, b( i, n-l+1 ), ldb,csv, snv )
! update (n-l+i)-th and (n-l+j)-th columns of matrices
! a and b: a*q and b*q
call stdlib${ii}$_${ri}$rot( min( k+l, m ), a( 1_${ik}$, n-l+j ), 1_${ik}$,a( 1_${ik}$, n-l+i ), 1_${ik}$, csq, snq )
call stdlib${ii}$_${ri}$rot( l, b( 1_${ik}$, n-l+j ), 1_${ik}$, b( 1_${ik}$, n-l+i ), 1_${ik}$, csq,snq )
if( upper ) then
if( k+i<=m )a( k+i, n-l+j ) = zero
b( i, n-l+j ) = zero
else
if( k+j<=m )a( k+j, n-l+i ) = zero
b( j, n-l+i ) = zero
end if
! update orthogonal matrices u, v, q, if desired.
if( wantu .and. k+j<=m )call stdlib${ii}$_${ri}$rot( m, u( 1_${ik}$, k+j ), 1_${ik}$, u( 1_${ik}$, k+i ), 1_${ik}$, &
csu,snu )
if( wantv )call stdlib${ii}$_${ri}$rot( p, v( 1_${ik}$, j ), 1_${ik}$, v( 1_${ik}$, i ), 1_${ik}$, csv, snv )
if( wantq )call stdlib${ii}$_${ri}$rot( n, q( 1_${ik}$, n-l+j ), 1_${ik}$, q( 1_${ik}$, n-l+i ), 1_${ik}$, csq,snq )
end do loop_10
end do loop_20
if( .not.upper ) then
! the matrices a13 and b13 were lower triangular at the start
! of the cycle, and are now upper triangular.
! convergence test: test the parallelism of the corresponding
! rows of a and b.
error = zero
do i = 1, min( l, m-k )
call stdlib${ii}$_${ri}$copy( l-i+1, a( k+i, n-l+i ), lda, work, 1_${ik}$ )
call stdlib${ii}$_${ri}$copy( l-i+1, b( i, n-l+i ), ldb, work( l+1 ), 1_${ik}$ )
call stdlib${ii}$_${ri}$lapll( l-i+1, work, 1_${ik}$, work( l+1 ), 1_${ik}$, ssmin )
error = max( error, ssmin )
end do
if( abs( error )<=min( tola, tolb ) )go to 50
end if
! end of cycle loop
end do loop_40
! the algorithm has not converged after maxit cycles.
info = 1_${ik}$
go to 100
50 continue
! if error <= min(tola,tolb), then the algorithm has converged.
! compute the generalized singular value pairs (alpha, beta), and
! set the triangular matrix r to array a.
do i = 1, k
alpha( i ) = one
beta( i ) = zero
end do
do i = 1, min( l, m-k )
a1 = a( k+i, n-l+i )
b1 = b( i, n-l+i )
gamma = b1 / a1
if( (gamma<=hugenum).and.(gamma>=-hugenum) ) then
! change sign if necessary
if( gamma<zero ) then
call stdlib${ii}$_${ri}$scal( l-i+1, -one, b( i, n-l+i ), ldb )
if( wantv )call stdlib${ii}$_${ri}$scal( p, -one, v( 1_${ik}$, i ), 1_${ik}$ )
end if
call stdlib${ii}$_${ri}$lartg( abs( gamma ), one, beta( k+i ), alpha( k+i ),rwk )
if( alpha( k+i )>=beta( k+i ) ) then
call stdlib${ii}$_${ri}$scal( l-i+1, one / alpha( k+i ), a( k+i, n-l+i ),lda )
else
call stdlib${ii}$_${ri}$scal( l-i+1, one / beta( k+i ), b( i, n-l+i ),ldb )
call stdlib${ii}$_${ri}$copy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),lda )
end if
else
alpha( k+i ) = zero
beta( k+i ) = one
call stdlib${ii}$_${ri}$copy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),lda )
end if
end do
! post-assignment
do i = m + 1, k + l
alpha( i ) = zero
beta( i ) = one
end do
if( k+l<n ) then
do i = k + l + 1, n
alpha( i ) = zero
beta( i ) = zero
end do
end if
100 continue
ncycle = kcycle
return
end subroutine stdlib${ii}$_${ri}$tgsja
#:endif
#:endfor
pure module subroutine stdlib${ii}$_ctgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b,ldb, tola, tolb, &
!! CTGSJA computes the generalized singular value decomposition (GSVD)
!! of two complex upper triangular (or trapezoidal) matrices A and B.
!! On entry, it is assumed that matrices A and B have the following
!! forms, which may be obtained by the preprocessing subroutine CGGSVP
!! from a general M-by-N matrix A and P-by-N matrix B:
!! N-K-L K L
!! A = K ( 0 A12 A13 ) if M-K-L >= 0;
!! L ( 0 0 A23 )
!! M-K-L ( 0 0 0 )
!! N-K-L K L
!! A = K ( 0 A12 A13 ) if M-K-L < 0;
!! M-K ( 0 0 A23 )
!! N-K-L K L
!! B = L ( 0 0 B13 )
!! P-L ( 0 0 0 )
!! where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
!! upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
!! otherwise A23 is (M-K)-by-L upper trapezoidal.
!! On exit,
!! U**H *A*Q = D1*( 0 R ), V**H *B*Q = D2*( 0 R ),
!! where U, V and Q are unitary matrices.
!! R is a nonsingular upper triangular matrix, and D1
!! and D2 are ``diagonal'' matrices, which are of the following
!! structures:
!! If M-K-L >= 0,
!! K L
!! D1 = K ( I 0 )
!! L ( 0 C )
!! M-K-L ( 0 0 )
!! K L
!! D2 = L ( 0 S )
!! P-L ( 0 0 )
!! N-K-L K L
!! ( 0 R ) = K ( 0 R11 R12 ) K
!! L ( 0 0 R22 ) L
!! where
!! C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
!! S = diag( BETA(K+1), ... , BETA(K+L) ),
!! C**2 + S**2 = I.
!! R is stored in A(1:K+L,N-K-L+1:N) on exit.
!! If M-K-L < 0,
!! K M-K K+L-M
!! D1 = K ( I 0 0 )
!! M-K ( 0 C 0 )
!! K M-K K+L-M
!! D2 = M-K ( 0 S 0 )
!! K+L-M ( 0 0 I )
!! P-L ( 0 0 0 )
!! N-K-L K M-K K+L-M
!! ( 0 R ) = K ( 0 R11 R12 R13 )
!! M-K ( 0 0 R22 R23 )
!! K+L-M ( 0 0 0 R33 )
!! where
!! C = diag( ALPHA(K+1), ... , ALPHA(M) ),
!! S = diag( BETA(K+1), ... , BETA(M) ),
!! C**2 + S**2 = I.
!! R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
!! ( 0 R22 R23 )
!! in B(M-K+1:L,N+M-K-L+1:N) on exit.
!! The computation of the unitary transformation matrices U, V or Q
!! is optional. These matrices may either be formed explicitly, or they
!! may be postmultiplied into input matrices U1, V1, or Q1.
alpha, beta, u, ldu, v, ldv,q, ldq, work, ncycle, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: jobq, jobu, jobv
integer(${ik}$), intent(out) :: info, ncycle
integer(${ik}$), intent(in) :: k, l, lda, ldb, ldq, ldu, ldv, m, n, p
real(sp), intent(in) :: tola, tolb
! Array Arguments
real(sp), intent(out) :: alpha(*), beta(*)
complex(sp), intent(inout) :: a(lda,*), b(ldb,*), q(ldq,*), u(ldu,*), v(ldv,*)
complex(sp), intent(out) :: work(*)
! =====================================================================
! Parameters
integer(${ik}$), parameter :: maxit = 40_${ik}$
real(sp), parameter :: hugenum = huge(zero)
! Local Scalars
logical(lk) :: initq, initu, initv, upper, wantq, wantu, wantv
integer(${ik}$) :: i, j, kcycle
real(sp) :: a1, a3, b1, b3, csq, csu, csv, error, gamma, rwk, ssmin
complex(sp) :: a2, b2, snq, snu, snv
! Intrinsic Functions
! Executable Statements
! decode and test the input parameters
initu = stdlib_lsame( jobu, 'I' )
wantu = initu .or. stdlib_lsame( jobu, 'U' )
initv = stdlib_lsame( jobv, 'I' )
wantv = initv .or. stdlib_lsame( jobv, 'V' )
initq = stdlib_lsame( jobq, 'I' )
wantq = initq .or. stdlib_lsame( jobq, 'Q' )
info = 0_${ik}$
if( .not.( initu .or. wantu .or. stdlib_lsame( jobu, 'N' ) ) ) then
info = -1_${ik}$
else if( .not.( initv .or. wantv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -2_${ik}$
else if( .not.( initq .or. wantq .or. stdlib_lsame( jobq, 'N' ) ) ) then
info = -3_${ik}$
else if( m<0_${ik}$ ) then
info = -4_${ik}$
else if( p<0_${ik}$ ) then
info = -5_${ik}$
else if( n<0_${ik}$ ) then
info = -6_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -10_${ik}$
else if( ldb<max( 1_${ik}$, p ) ) then
info = -12_${ik}$
else if( ldu<1_${ik}$ .or. ( wantu .and. ldu<m ) ) then
info = -18_${ik}$
else if( ldv<1_${ik}$ .or. ( wantv .and. ldv<p ) ) then
info = -20_${ik}$
else if( ldq<1_${ik}$ .or. ( wantq .and. ldq<n ) ) then
info = -22_${ik}$
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'CTGSJA', -info )
return
end if
! initialize u, v and q, if necessary
if( initu )call stdlib${ii}$_claset( 'FULL', m, m, czero, cone, u, ldu )
if( initv )call stdlib${ii}$_claset( 'FULL', p, p, czero, cone, v, ldv )
if( initq )call stdlib${ii}$_claset( 'FULL', n, n, czero, cone, q, ldq )
! loop until convergence
upper = .false.
loop_40: do kcycle = 1, maxit
upper = .not.upper
loop_20: do i = 1, l - 1
loop_10: do j = i + 1, l
a1 = zero
a2 = czero
a3 = zero
if( k+i<=m )a1 = real( a( k+i, n-l+i ),KIND=sp)
if( k+j<=m )a3 = real( a( k+j, n-l+j ),KIND=sp)
b1 = real( b( i, n-l+i ),KIND=sp)
b3 = real( b( j, n-l+j ),KIND=sp)
if( upper ) then
if( k+i<=m )a2 = a( k+i, n-l+j )
b2 = b( i, n-l+j )
else
if( k+j<=m )a2 = a( k+j, n-l+i )
b2 = b( j, n-l+i )
end if
call stdlib${ii}$_clags2( upper, a1, a2, a3, b1, b2, b3, csu, snu,csv, snv, csq, &
snq )
! update (k+i)-th and (k+j)-th rows of matrix a: u**h *a
if( k+j<=m )call stdlib${ii}$_crot( l, a( k+j, n-l+1 ), lda, a( k+i, n-l+1 ),lda, &
csu, conjg( snu ) )
! update i-th and j-th rows of matrix b: v**h *b
call stdlib${ii}$_crot( l, b( j, n-l+1 ), ldb, b( i, n-l+1 ), ldb,csv, conjg( snv ) &
)
! update (n-l+i)-th and (n-l+j)-th columns of matrices
! a and b: a*q and b*q
call stdlib${ii}$_crot( min( k+l, m ), a( 1_${ik}$, n-l+j ), 1_${ik}$,a( 1_${ik}$, n-l+i ), 1_${ik}$, csq, snq )
call stdlib${ii}$_crot( l, b( 1_${ik}$, n-l+j ), 1_${ik}$, b( 1_${ik}$, n-l+i ), 1_${ik}$, csq,snq )
if( upper ) then
if( k+i<=m )a( k+i, n-l+j ) = czero
b( i, n-l+j ) = czero
else
if( k+j<=m )a( k+j, n-l+i ) = czero
b( j, n-l+i ) = czero
end if
! ensure that the diagonal elements of a and b are real.
if( k+i<=m )a( k+i, n-l+i ) = real( a( k+i, n-l+i ),KIND=sp)
if( k+j<=m )a( k+j, n-l+j ) = real( a( k+j, n-l+j ),KIND=sp)
b( i, n-l+i ) = real( b( i, n-l+i ),KIND=sp)
b( j, n-l+j ) = real( b( j, n-l+j ),KIND=sp)
! update unitary matrices u, v, q, if desired.
if( wantu .and. k+j<=m )call stdlib${ii}$_crot( m, u( 1_${ik}$, k+j ), 1_${ik}$, u( 1_${ik}$, k+i ), 1_${ik}$, &
csu,snu )
if( wantv )call stdlib${ii}$_crot( p, v( 1_${ik}$, j ), 1_${ik}$, v( 1_${ik}$, i ), 1_${ik}$, csv, snv )
if( wantq )call stdlib${ii}$_crot( n, q( 1_${ik}$, n-l+j ), 1_${ik}$, q( 1_${ik}$, n-l+i ), 1_${ik}$, csq,snq )
end do loop_10
end do loop_20
if( .not.upper ) then
! the matrices a13 and b13 were lower triangular at the start
! of the cycle, and are now upper triangular.
! convergence test: test the parallelism of the corresponding
! rows of a and b.
error = zero
do i = 1, min( l, m-k )
call stdlib${ii}$_ccopy( l-i+1, a( k+i, n-l+i ), lda, work, 1_${ik}$ )
call stdlib${ii}$_ccopy( l-i+1, b( i, n-l+i ), ldb, work( l+1 ), 1_${ik}$ )
call stdlib${ii}$_clapll( l-i+1, work, 1_${ik}$, work( l+1 ), 1_${ik}$, ssmin )
error = max( error, ssmin )
end do
if( abs( error )<=min( tola, tolb ) )go to 50
end if
! end of cycle loop
end do loop_40
! the algorithm has not converged after maxit cycles.
info = 1_${ik}$
go to 100
50 continue
! if error <= min(tola,tolb), then the algorithm has converged.
! compute the generalized singular value pairs (alpha, beta), and
! set the triangular matrix r to array a.
do i = 1, k
alpha( i ) = one
beta( i ) = zero
end do
do i = 1, min( l, m-k )
a1 = real( a( k+i, n-l+i ),KIND=sp)
b1 = real( b( i, n-l+i ),KIND=sp)
gamma = b1 / a1
if( (gamma<=hugenum).and.(gamma>=-hugenum) ) then
if( gamma<zero ) then
call stdlib${ii}$_csscal( l-i+1, -one, b( i, n-l+i ), ldb )
if( wantv )call stdlib${ii}$_csscal( p, -one, v( 1_${ik}$, i ), 1_${ik}$ )
end if
call stdlib${ii}$_slartg( abs( gamma ), one, beta( k+i ), alpha( k+i ),rwk )
if( alpha( k+i )>=beta( k+i ) ) then
call stdlib${ii}$_csscal( l-i+1, one / alpha( k+i ), a( k+i, n-l+i ),lda )
else
call stdlib${ii}$_csscal( l-i+1, one / beta( k+i ), b( i, n-l+i ),ldb )
call stdlib${ii}$_ccopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),lda )
end if
else
alpha( k+i ) = zero
beta( k+i ) = one
call stdlib${ii}$_ccopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),lda )
end if
end do
! post-assignment
do i = m + 1, k + l
alpha( i ) = zero
beta( i ) = one
end do
if( k+l<n ) then
do i = k + l + 1, n
alpha( i ) = zero
beta( i ) = zero
end do
end if
100 continue
ncycle = kcycle
return
end subroutine stdlib${ii}$_ctgsja
pure module subroutine stdlib${ii}$_ztgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b,ldb, tola, tolb, &
!! ZTGSJA computes the generalized singular value decomposition (GSVD)
!! of two complex upper triangular (or trapezoidal) matrices A and B.
!! On entry, it is assumed that matrices A and B have the following
!! forms, which may be obtained by the preprocessing subroutine ZGGSVP
!! from a general M-by-N matrix A and P-by-N matrix B:
!! N-K-L K L
!! A = K ( 0 A12 A13 ) if M-K-L >= 0;
!! L ( 0 0 A23 )
!! M-K-L ( 0 0 0 )
!! N-K-L K L
!! A = K ( 0 A12 A13 ) if M-K-L < 0;
!! M-K ( 0 0 A23 )
!! N-K-L K L
!! B = L ( 0 0 B13 )
!! P-L ( 0 0 0 )
!! where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
!! upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
!! otherwise A23 is (M-K)-by-L upper trapezoidal.
!! On exit,
!! U**H *A*Q = D1*( 0 R ), V**H *B*Q = D2*( 0 R ),
!! where U, V and Q are unitary matrices.
!! R is a nonsingular upper triangular matrix, and D1
!! and D2 are ``diagonal'' matrices, which are of the following
!! structures:
!! If M-K-L >= 0,
!! K L
!! D1 = K ( I 0 )
!! L ( 0 C )
!! M-K-L ( 0 0 )
!! K L
!! D2 = L ( 0 S )
!! P-L ( 0 0 )
!! N-K-L K L
!! ( 0 R ) = K ( 0 R11 R12 ) K
!! L ( 0 0 R22 ) L
!! where
!! C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
!! S = diag( BETA(K+1), ... , BETA(K+L) ),
!! C**2 + S**2 = I.
!! R is stored in A(1:K+L,N-K-L+1:N) on exit.
!! If M-K-L < 0,
!! K M-K K+L-M
!! D1 = K ( I 0 0 )
!! M-K ( 0 C 0 )
!! K M-K K+L-M
!! D2 = M-K ( 0 S 0 )
!! K+L-M ( 0 0 I )
!! P-L ( 0 0 0 )
!! N-K-L K M-K K+L-M
!! ( 0 R ) = K ( 0 R11 R12 R13 )
!! M-K ( 0 0 R22 R23 )
!! K+L-M ( 0 0 0 R33 )
!! where
!! C = diag( ALPHA(K+1), ... , ALPHA(M) ),
!! S = diag( BETA(K+1), ... , BETA(M) ),
!! C**2 + S**2 = I.
!! R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
!! ( 0 R22 R23 )
!! in B(M-K+1:L,N+M-K-L+1:N) on exit.
!! The computation of the unitary transformation matrices U, V or Q
!! is optional. These matrices may either be formed explicitly, or they
!! may be postmultiplied into input matrices U1, V1, or Q1.
alpha, beta, u, ldu, v, ldv,q, ldq, work, ncycle, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: jobq, jobu, jobv
integer(${ik}$), intent(out) :: info, ncycle
integer(${ik}$), intent(in) :: k, l, lda, ldb, ldq, ldu, ldv, m, n, p
real(dp), intent(in) :: tola, tolb
! Array Arguments
real(dp), intent(out) :: alpha(*), beta(*)
complex(dp), intent(inout) :: a(lda,*), b(ldb,*), q(ldq,*), u(ldu,*), v(ldv,*)
complex(dp), intent(out) :: work(*)
! =====================================================================
! Parameters
integer(${ik}$), parameter :: maxit = 40_${ik}$
real(dp), parameter :: hugenum = huge(zero)
! Local Scalars
logical(lk) :: initq, initu, initv, upper, wantq, wantu, wantv
integer(${ik}$) :: i, j, kcycle
real(dp) :: a1, a3, b1, b3, csq, csu, csv, error, gamma, rwk, ssmin
complex(dp) :: a2, b2, snq, snu, snv
! Intrinsic Functions
! Executable Statements
! decode and test the input parameters
initu = stdlib_lsame( jobu, 'I' )
wantu = initu .or. stdlib_lsame( jobu, 'U' )
initv = stdlib_lsame( jobv, 'I' )
wantv = initv .or. stdlib_lsame( jobv, 'V' )
initq = stdlib_lsame( jobq, 'I' )
wantq = initq .or. stdlib_lsame( jobq, 'Q' )
info = 0_${ik}$
if( .not.( initu .or. wantu .or. stdlib_lsame( jobu, 'N' ) ) ) then
info = -1_${ik}$
else if( .not.( initv .or. wantv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -2_${ik}$
else if( .not.( initq .or. wantq .or. stdlib_lsame( jobq, 'N' ) ) ) then
info = -3_${ik}$
else if( m<0_${ik}$ ) then
info = -4_${ik}$
else if( p<0_${ik}$ ) then
info = -5_${ik}$
else if( n<0_${ik}$ ) then
info = -6_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -10_${ik}$
else if( ldb<max( 1_${ik}$, p ) ) then
info = -12_${ik}$
else if( ldu<1_${ik}$ .or. ( wantu .and. ldu<m ) ) then
info = -18_${ik}$
else if( ldv<1_${ik}$ .or. ( wantv .and. ldv<p ) ) then
info = -20_${ik}$
else if( ldq<1_${ik}$ .or. ( wantq .and. ldq<n ) ) then
info = -22_${ik}$
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZTGSJA', -info )
return
end if
! initialize u, v and q, if necessary
if( initu )call stdlib${ii}$_zlaset( 'FULL', m, m, czero, cone, u, ldu )
if( initv )call stdlib${ii}$_zlaset( 'FULL', p, p, czero, cone, v, ldv )
if( initq )call stdlib${ii}$_zlaset( 'FULL', n, n, czero, cone, q, ldq )
! loop until convergence
upper = .false.
loop_40: do kcycle = 1, maxit
upper = .not.upper
loop_20: do i = 1, l - 1
loop_10: do j = i + 1, l
a1 = zero
a2 = czero
a3 = zero
if( k+i<=m )a1 = real( a( k+i, n-l+i ),KIND=dp)
if( k+j<=m )a3 = real( a( k+j, n-l+j ),KIND=dp)
b1 = real( b( i, n-l+i ),KIND=dp)
b3 = real( b( j, n-l+j ),KIND=dp)
if( upper ) then
if( k+i<=m )a2 = a( k+i, n-l+j )
b2 = b( i, n-l+j )
else
if( k+j<=m )a2 = a( k+j, n-l+i )
b2 = b( j, n-l+i )
end if
call stdlib${ii}$_zlags2( upper, a1, a2, a3, b1, b2, b3, csu, snu,csv, snv, csq, &
snq )
! update (k+i)-th and (k+j)-th rows of matrix a: u**h *a
if( k+j<=m )call stdlib${ii}$_zrot( l, a( k+j, n-l+1 ), lda, a( k+i, n-l+1 ),lda, &
csu, conjg( snu ) )
! update i-th and j-th rows of matrix b: v**h *b
call stdlib${ii}$_zrot( l, b( j, n-l+1 ), ldb, b( i, n-l+1 ), ldb,csv, conjg( snv ) &
)
! update (n-l+i)-th and (n-l+j)-th columns of matrices
! a and b: a*q and b*q
call stdlib${ii}$_zrot( min( k+l, m ), a( 1_${ik}$, n-l+j ), 1_${ik}$,a( 1_${ik}$, n-l+i ), 1_${ik}$, csq, snq )
call stdlib${ii}$_zrot( l, b( 1_${ik}$, n-l+j ), 1_${ik}$, b( 1_${ik}$, n-l+i ), 1_${ik}$, csq,snq )
if( upper ) then
if( k+i<=m )a( k+i, n-l+j ) = czero
b( i, n-l+j ) = czero
else
if( k+j<=m )a( k+j, n-l+i ) = czero
b( j, n-l+i ) = czero
end if
! ensure that the diagonal elements of a and b are real.
if( k+i<=m )a( k+i, n-l+i ) = real( a( k+i, n-l+i ),KIND=dp)
if( k+j<=m )a( k+j, n-l+j ) = real( a( k+j, n-l+j ),KIND=dp)
b( i, n-l+i ) = real( b( i, n-l+i ),KIND=dp)
b( j, n-l+j ) = real( b( j, n-l+j ),KIND=dp)
! update unitary matrices u, v, q, if desired.
if( wantu .and. k+j<=m )call stdlib${ii}$_zrot( m, u( 1_${ik}$, k+j ), 1_${ik}$, u( 1_${ik}$, k+i ), 1_${ik}$, &
csu,snu )
if( wantv )call stdlib${ii}$_zrot( p, v( 1_${ik}$, j ), 1_${ik}$, v( 1_${ik}$, i ), 1_${ik}$, csv, snv )
if( wantq )call stdlib${ii}$_zrot( n, q( 1_${ik}$, n-l+j ), 1_${ik}$, q( 1_${ik}$, n-l+i ), 1_${ik}$, csq,snq )
end do loop_10
end do loop_20
if( .not.upper ) then
! the matrices a13 and b13 were lower triangular at the start
! of the cycle, and are now upper triangular.
! convergence test: test the parallelism of the corresponding
! rows of a and b.
error = zero
do i = 1, min( l, m-k )
call stdlib${ii}$_zcopy( l-i+1, a( k+i, n-l+i ), lda, work, 1_${ik}$ )
call stdlib${ii}$_zcopy( l-i+1, b( i, n-l+i ), ldb, work( l+1 ), 1_${ik}$ )
call stdlib${ii}$_zlapll( l-i+1, work, 1_${ik}$, work( l+1 ), 1_${ik}$, ssmin )
error = max( error, ssmin )
end do
if( abs( error )<=min( tola, tolb ) )go to 50
end if
! end of cycle loop
end do loop_40
! the algorithm has not converged after maxit cycles.
info = 1_${ik}$
go to 100
50 continue
! if error <= min(tola,tolb), then the algorithm has converged.
! compute the generalized singular value pairs (alpha, beta), and
! set the triangular matrix r to array a.
do i = 1, k
alpha( i ) = one
beta( i ) = zero
end do
do i = 1, min( l, m-k )
a1 = real( a( k+i, n-l+i ),KIND=dp)
b1 = real( b( i, n-l+i ),KIND=dp)
gamma = b1 / a1
if( (gamma<=hugenum).and.(gamma>=-hugenum) ) then
if( gamma<zero ) then
call stdlib${ii}$_zdscal( l-i+1, -one, b( i, n-l+i ), ldb )
if( wantv )call stdlib${ii}$_zdscal( p, -one, v( 1_${ik}$, i ), 1_${ik}$ )
end if
call stdlib${ii}$_dlartg( abs( gamma ), one, beta( k+i ), alpha( k+i ),rwk )
if( alpha( k+i )>=beta( k+i ) ) then
call stdlib${ii}$_zdscal( l-i+1, one / alpha( k+i ), a( k+i, n-l+i ),lda )
else
call stdlib${ii}$_zdscal( l-i+1, one / beta( k+i ), b( i, n-l+i ),ldb )
call stdlib${ii}$_zcopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),lda )
end if
else
alpha( k+i ) = zero
beta( k+i ) = one
call stdlib${ii}$_zcopy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),lda )
end if
end do
! post-assignment
do i = m + 1, k + l
alpha( i ) = zero
beta( i ) = one
end do
if( k+l<n ) then
do i = k + l + 1, n
alpha( i ) = zero
beta( i ) = zero
end do
end if
100 continue
ncycle = kcycle
return
end subroutine stdlib${ii}$_ztgsja
#:for ck,ct,ci in CMPLX_KINDS_TYPES
#:if not ck in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ci}$tgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b,ldb, tola, tolb, &
!! ZTGSJA: computes the generalized singular value decomposition (GSVD)
!! of two complex upper triangular (or trapezoidal) matrices A and B.
!! On entry, it is assumed that matrices A and B have the following
!! forms, which may be obtained by the preprocessing subroutine ZGGSVP
!! from a general M-by-N matrix A and P-by-N matrix B:
!! N-K-L K L
!! A = K ( 0 A12 A13 ) if M-K-L >= 0;
!! L ( 0 0 A23 )
!! M-K-L ( 0 0 0 )
!! N-K-L K L
!! A = K ( 0 A12 A13 ) if M-K-L < 0;
!! M-K ( 0 0 A23 )
!! N-K-L K L
!! B = L ( 0 0 B13 )
!! P-L ( 0 0 0 )
!! where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
!! upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
!! otherwise A23 is (M-K)-by-L upper trapezoidal.
!! On exit,
!! U**H *A*Q = D1*( 0 R ), V**H *B*Q = D2*( 0 R ),
!! where U, V and Q are unitary matrices.
!! R is a nonsingular upper triangular matrix, and D1
!! and D2 are ``diagonal'' matrices, which are of the following
!! structures:
!! If M-K-L >= 0,
!! K L
!! D1 = K ( I 0 )
!! L ( 0 C )
!! M-K-L ( 0 0 )
!! K L
!! D2 = L ( 0 S )
!! P-L ( 0 0 )
!! N-K-L K L
!! ( 0 R ) = K ( 0 R11 R12 ) K
!! L ( 0 0 R22 ) L
!! where
!! C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
!! S = diag( BETA(K+1), ... , BETA(K+L) ),
!! C**2 + S**2 = I.
!! R is stored in A(1:K+L,N-K-L+1:N) on exit.
!! If M-K-L < 0,
!! K M-K K+L-M
!! D1 = K ( I 0 0 )
!! M-K ( 0 C 0 )
!! K M-K K+L-M
!! D2 = M-K ( 0 S 0 )
!! K+L-M ( 0 0 I )
!! P-L ( 0 0 0 )
!! N-K-L K M-K K+L-M
!! ( 0 R ) = K ( 0 R11 R12 R13 )
!! M-K ( 0 0 R22 R23 )
!! K+L-M ( 0 0 0 R33 )
!! where
!! C = diag( ALPHA(K+1), ... , ALPHA(M) ),
!! S = diag( BETA(K+1), ... , BETA(M) ),
!! C**2 + S**2 = I.
!! R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
!! ( 0 R22 R23 )
!! in B(M-K+1:L,N+M-K-L+1:N) on exit.
!! The computation of the unitary transformation matrices U, V or Q
!! is optional. These matrices may either be formed explicitly, or they
!! may be postmultiplied into input matrices U1, V1, or Q1.
alpha, beta, u, ldu, v, ldv,q, ldq, work, ncycle, info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${ck}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: jobq, jobu, jobv
integer(${ik}$), intent(out) :: info, ncycle
integer(${ik}$), intent(in) :: k, l, lda, ldb, ldq, ldu, ldv, m, n, p
real(${ck}$), intent(in) :: tola, tolb
! Array Arguments
real(${ck}$), intent(out) :: alpha(*), beta(*)
complex(${ck}$), intent(inout) :: a(lda,*), b(ldb,*), q(ldq,*), u(ldu,*), v(ldv,*)
complex(${ck}$), intent(out) :: work(*)
! =====================================================================
! Parameters
integer(${ik}$), parameter :: maxit = 40_${ik}$
real(${ck}$), parameter :: hugenum = huge(zero)
! Local Scalars
logical(lk) :: initq, initu, initv, upper, wantq, wantu, wantv
integer(${ik}$) :: i, j, kcycle
real(${ck}$) :: a1, a3, b1, b3, csq, csu, csv, error, gamma, rwk, ssmin
complex(${ck}$) :: a2, b2, snq, snu, snv
! Intrinsic Functions
! Executable Statements
! decode and test the input parameters
initu = stdlib_lsame( jobu, 'I' )
wantu = initu .or. stdlib_lsame( jobu, 'U' )
initv = stdlib_lsame( jobv, 'I' )
wantv = initv .or. stdlib_lsame( jobv, 'V' )
initq = stdlib_lsame( jobq, 'I' )
wantq = initq .or. stdlib_lsame( jobq, 'Q' )
info = 0_${ik}$
if( .not.( initu .or. wantu .or. stdlib_lsame( jobu, 'N' ) ) ) then
info = -1_${ik}$
else if( .not.( initv .or. wantv .or. stdlib_lsame( jobv, 'N' ) ) ) then
info = -2_${ik}$
else if( .not.( initq .or. wantq .or. stdlib_lsame( jobq, 'N' ) ) ) then
info = -3_${ik}$
else if( m<0_${ik}$ ) then
info = -4_${ik}$
else if( p<0_${ik}$ ) then
info = -5_${ik}$
else if( n<0_${ik}$ ) then
info = -6_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -10_${ik}$
else if( ldb<max( 1_${ik}$, p ) ) then
info = -12_${ik}$
else if( ldu<1_${ik}$ .or. ( wantu .and. ldu<m ) ) then
info = -18_${ik}$
else if( ldv<1_${ik}$ .or. ( wantv .and. ldv<p ) ) then
info = -20_${ik}$
else if( ldq<1_${ik}$ .or. ( wantq .and. ldq<n ) ) then
info = -22_${ik}$
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZTGSJA', -info )
return
end if
! initialize u, v and q, if necessary
if( initu )call stdlib${ii}$_${ci}$laset( 'FULL', m, m, czero, cone, u, ldu )
if( initv )call stdlib${ii}$_${ci}$laset( 'FULL', p, p, czero, cone, v, ldv )
if( initq )call stdlib${ii}$_${ci}$laset( 'FULL', n, n, czero, cone, q, ldq )
! loop until convergence
upper = .false.
loop_40: do kcycle = 1, maxit
upper = .not.upper
loop_20: do i = 1, l - 1
loop_10: do j = i + 1, l
a1 = zero
a2 = czero
a3 = zero
if( k+i<=m )a1 = real( a( k+i, n-l+i ),KIND=${ck}$)
if( k+j<=m )a3 = real( a( k+j, n-l+j ),KIND=${ck}$)
b1 = real( b( i, n-l+i ),KIND=${ck}$)
b3 = real( b( j, n-l+j ),KIND=${ck}$)
if( upper ) then
if( k+i<=m )a2 = a( k+i, n-l+j )
b2 = b( i, n-l+j )
else
if( k+j<=m )a2 = a( k+j, n-l+i )
b2 = b( j, n-l+i )
end if
call stdlib${ii}$_${ci}$lags2( upper, a1, a2, a3, b1, b2, b3, csu, snu,csv, snv, csq, &
snq )
! update (k+i)-th and (k+j)-th rows of matrix a: u**h *a
if( k+j<=m )call stdlib${ii}$_${ci}$rot( l, a( k+j, n-l+1 ), lda, a( k+i, n-l+1 ),lda, &
csu, conjg( snu ) )
! update i-th and j-th rows of matrix b: v**h *b
call stdlib${ii}$_${ci}$rot( l, b( j, n-l+1 ), ldb, b( i, n-l+1 ), ldb,csv, conjg( snv ) &
)
! update (n-l+i)-th and (n-l+j)-th columns of matrices
! a and b: a*q and b*q
call stdlib${ii}$_${ci}$rot( min( k+l, m ), a( 1_${ik}$, n-l+j ), 1_${ik}$,a( 1_${ik}$, n-l+i ), 1_${ik}$, csq, snq )
call stdlib${ii}$_${ci}$rot( l, b( 1_${ik}$, n-l+j ), 1_${ik}$, b( 1_${ik}$, n-l+i ), 1_${ik}$, csq,snq )
if( upper ) then
if( k+i<=m )a( k+i, n-l+j ) = czero
b( i, n-l+j ) = czero
else
if( k+j<=m )a( k+j, n-l+i ) = czero
b( j, n-l+i ) = czero
end if
! ensure that the diagonal elements of a and b are real.
if( k+i<=m )a( k+i, n-l+i ) = real( a( k+i, n-l+i ),KIND=${ck}$)
if( k+j<=m )a( k+j, n-l+j ) = real( a( k+j, n-l+j ),KIND=${ck}$)
b( i, n-l+i ) = real( b( i, n-l+i ),KIND=${ck}$)
b( j, n-l+j ) = real( b( j, n-l+j ),KIND=${ck}$)
! update unitary matrices u, v, q, if desired.
if( wantu .and. k+j<=m )call stdlib${ii}$_${ci}$rot( m, u( 1_${ik}$, k+j ), 1_${ik}$, u( 1_${ik}$, k+i ), 1_${ik}$, &
csu,snu )
if( wantv )call stdlib${ii}$_${ci}$rot( p, v( 1_${ik}$, j ), 1_${ik}$, v( 1_${ik}$, i ), 1_${ik}$, csv, snv )
if( wantq )call stdlib${ii}$_${ci}$rot( n, q( 1_${ik}$, n-l+j ), 1_${ik}$, q( 1_${ik}$, n-l+i ), 1_${ik}$, csq,snq )
end do loop_10
end do loop_20
if( .not.upper ) then
! the matrices a13 and b13 were lower triangular at the start
! of the cycle, and are now upper triangular.
! convergence test: test the parallelism of the corresponding
! rows of a and b.
error = zero
do i = 1, min( l, m-k )
call stdlib${ii}$_${ci}$copy( l-i+1, a( k+i, n-l+i ), lda, work, 1_${ik}$ )
call stdlib${ii}$_${ci}$copy( l-i+1, b( i, n-l+i ), ldb, work( l+1 ), 1_${ik}$ )
call stdlib${ii}$_${ci}$lapll( l-i+1, work, 1_${ik}$, work( l+1 ), 1_${ik}$, ssmin )
error = max( error, ssmin )
end do
if( abs( error )<=min( tola, tolb ) )go to 50
end if
! end of cycle loop
end do loop_40
! the algorithm has not converged after maxit cycles.
info = 1_${ik}$
go to 100
50 continue
! if error <= min(tola,tolb), then the algorithm has converged.
! compute the generalized singular value pairs (alpha, beta), and
! set the triangular matrix r to array a.
do i = 1, k
alpha( i ) = one
beta( i ) = zero
end do
do i = 1, min( l, m-k )
a1 = real( a( k+i, n-l+i ),KIND=${ck}$)
b1 = real( b( i, n-l+i ),KIND=${ck}$)
gamma = b1 / a1
if( (gamma<=hugenum).and.(gamma>=-hugenum) ) then
if( gamma<zero ) then
call stdlib${ii}$_${ci}$dscal( l-i+1, -one, b( i, n-l+i ), ldb )
if( wantv )call stdlib${ii}$_${ci}$dscal( p, -one, v( 1_${ik}$, i ), 1_${ik}$ )
end if
call stdlib${ii}$_${c2ri(ci)}$lartg( abs( gamma ), one, beta( k+i ), alpha( k+i ),rwk )
if( alpha( k+i )>=beta( k+i ) ) then
call stdlib${ii}$_${ci}$dscal( l-i+1, one / alpha( k+i ), a( k+i, n-l+i ),lda )
else
call stdlib${ii}$_${ci}$dscal( l-i+1, one / beta( k+i ), b( i, n-l+i ),ldb )
call stdlib${ii}$_${ci}$copy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),lda )
end if
else
alpha( k+i ) = zero
beta( k+i ) = one
call stdlib${ii}$_${ci}$copy( l-i+1, b( i, n-l+i ), ldb, a( k+i, n-l+i ),lda )
end if
end do
! post-assignment
do i = m + 1, k + l
alpha( i ) = zero
beta( i ) = one
end do
if( k+l<n ) then
do i = k + l + 1, n
alpha( i ) = zero
beta( i ) = zero
end do
end if
100 continue
ncycle = kcycle
return
end subroutine stdlib${ii}$_${ci}$tgsja
#:endif
#:endfor
pure module subroutine stdlib${ii}$_cungbr( vect, m, n, k, a, lda, tau, work, lwork, info )
!! CUNGBR generates one of the complex unitary matrices Q or P**H
!! determined by CGEBRD when reducing a complex matrix A to bidiagonal
!! form: A = Q * B * P**H. Q and P**H are defined as products of
!! elementary reflectors H(i) or G(i) respectively.
!! If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q
!! is of order M:
!! if m >= k, Q = H(1) H(2) . . . H(k) and CUNGBR returns the first n
!! columns of Q, where m >= n >= k;
!! if m < k, Q = H(1) H(2) . . . H(m-1) and CUNGBR returns Q as an
!! M-by-M matrix.
!! If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**H
!! is of order N:
!! if k < n, P**H = G(k) . . . G(2) G(1) and CUNGBR returns the first m
!! rows of P**H, where n >= m >= k;
!! if k >= n, P**H = G(n-1) . . . G(2) G(1) and CUNGBR returns P**H as
!! an N-by-N matrix.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: k, lda, lwork, m, n
! Array Arguments
complex(sp), intent(inout) :: a(lda,*)
complex(sp), intent(in) :: tau(*)
complex(sp), intent(out) :: work(*)
! =====================================================================
! Local Scalars
logical(lk) :: lquery, wantq
integer(${ik}$) :: i, iinfo, j, lwkopt, mn
! Intrinsic Functions
! Executable Statements
! test the input arguments
info = 0_${ik}$
wantq = stdlib_lsame( vect, 'Q' )
mn = min( m, n )
lquery = ( lwork==-1_${ik}$ )
if( .not.wantq .and. .not.stdlib_lsame( vect, 'P' ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( n<0_${ik}$ .or. ( wantq .and. ( n>m .or. n<min( m,k ) ) ) .or. ( .not.wantq .and. ( &
m>n .or. m<min( n, k ) ) ) ) then
info = -3_${ik}$
else if( k<0_${ik}$ ) then
info = -4_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -6_${ik}$
else if( lwork<max( 1_${ik}$, mn ) .and. .not.lquery ) then
info = -9_${ik}$
end if
if( info==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
if( wantq ) then
if( m>=k ) then
call stdlib${ii}$_cungqr( m, n, k, a, lda, tau, work, -1_${ik}$, iinfo )
else
if( m>1_${ik}$ ) then
call stdlib${ii}$_cungqr( m-1, m-1, m-1, a, lda, tau, work, -1_${ik}$,iinfo )
end if
end if
else
if( k<n ) then
call stdlib${ii}$_cunglq( m, n, k, a, lda, tau, work, -1_${ik}$, iinfo )
else
if( n>1_${ik}$ ) then
call stdlib${ii}$_cunglq( n-1, n-1, n-1, a, lda, tau, work, -1_${ik}$,iinfo )
end if
end if
end if
lwkopt = real( work( 1_${ik}$ ),KIND=sp)
lwkopt = max (lwkopt, mn)
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'CUNGBR', -info )
return
else if( lquery ) then
work( 1_${ik}$ ) = lwkopt
return
end if
! quick return if possible
if( m==0_${ik}$ .or. n==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
return
end if
if( wantq ) then
! form q, determined by a call to stdlib${ii}$_cgebrd to reduce an m-by-k
! matrix
if( m>=k ) then
! if m >= k, assume m >= n >= k
call stdlib${ii}$_cungqr( m, n, k, a, lda, tau, work, lwork, iinfo )
else
! if m < k, assume m = n
! shift the vectors which define the elementary reflectors cone
! column to the right, and set the first row and column of q
! to those of the unit matrix
do j = m, 2, -1
a( 1_${ik}$, j ) = czero
do i = j + 1, m
a( i, j ) = a( i, j-1 )
end do
end do
a( 1_${ik}$, 1_${ik}$ ) = cone
do i = 2, m
a( i, 1_${ik}$ ) = czero
end do
if( m>1_${ik}$ ) then
! form q(2:m,2:m)
call stdlib${ii}$_cungqr( m-1, m-1, m-1, a( 2_${ik}$, 2_${ik}$ ), lda, tau, work,lwork, iinfo )
end if
end if
else
! form p**h, determined by a call to stdlib${ii}$_cgebrd to reduce a k-by-n
! matrix
if( k<n ) then
! if k < n, assume k <= m <= n
call stdlib${ii}$_cunglq( m, n, k, a, lda, tau, work, lwork, iinfo )
else
! if k >= n, assume m = n
! shift the vectors which define the elementary reflectors cone
! row downward, and set the first row and column of p**h to
! those of the unit matrix
a( 1_${ik}$, 1_${ik}$ ) = cone
do i = 2, n
a( i, 1_${ik}$ ) = czero
end do
do j = 2, n
do i = j - 1, 2, -1
a( i, j ) = a( i-1, j )
end do
a( 1_${ik}$, j ) = czero
end do
if( n>1_${ik}$ ) then
! form p**h(2:n,2:n)
call stdlib${ii}$_cunglq( n-1, n-1, n-1, a( 2_${ik}$, 2_${ik}$ ), lda, tau, work,lwork, iinfo )
end if
end if
end if
work( 1_${ik}$ ) = lwkopt
return
end subroutine stdlib${ii}$_cungbr
pure module subroutine stdlib${ii}$_zungbr( vect, m, n, k, a, lda, tau, work, lwork, info )
!! ZUNGBR generates one of the complex unitary matrices Q or P**H
!! determined by ZGEBRD when reducing a complex matrix A to bidiagonal
!! form: A = Q * B * P**H. Q and P**H are defined as products of
!! elementary reflectors H(i) or G(i) respectively.
!! If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q
!! is of order M:
!! if m >= k, Q = H(1) H(2) . . . H(k) and ZUNGBR returns the first n
!! columns of Q, where m >= n >= k;
!! if m < k, Q = H(1) H(2) . . . H(m-1) and ZUNGBR returns Q as an
!! M-by-M matrix.
!! If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**H
!! is of order N:
!! if k < n, P**H = G(k) . . . G(2) G(1) and ZUNGBR returns the first m
!! rows of P**H, where n >= m >= k;
!! if k >= n, P**H = G(n-1) . . . G(2) G(1) and ZUNGBR returns P**H as
!! an N-by-N matrix.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: k, lda, lwork, m, n
! Array Arguments
complex(dp), intent(inout) :: a(lda,*)
complex(dp), intent(in) :: tau(*)
complex(dp), intent(out) :: work(*)
! =====================================================================
! Local Scalars
logical(lk) :: lquery, wantq
integer(${ik}$) :: i, iinfo, j, lwkopt, mn
! Intrinsic Functions
! Executable Statements
! test the input arguments
info = 0_${ik}$
wantq = stdlib_lsame( vect, 'Q' )
mn = min( m, n )
lquery = ( lwork==-1_${ik}$ )
if( .not.wantq .and. .not.stdlib_lsame( vect, 'P' ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( n<0_${ik}$ .or. ( wantq .and. ( n>m .or. n<min( m,k ) ) ) .or. ( .not.wantq .and. ( &
m>n .or. m<min( n, k ) ) ) ) then
info = -3_${ik}$
else if( k<0_${ik}$ ) then
info = -4_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -6_${ik}$
else if( lwork<max( 1_${ik}$, mn ) .and. .not.lquery ) then
info = -9_${ik}$
end if
if( info==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
if( wantq ) then
if( m>=k ) then
call stdlib${ii}$_zungqr( m, n, k, a, lda, tau, work, -1_${ik}$, iinfo )
else
if( m>1_${ik}$ ) then
call stdlib${ii}$_zungqr( m-1, m-1, m-1, a, lda, tau, work, -1_${ik}$,iinfo )
end if
end if
else
if( k<n ) then
call stdlib${ii}$_zunglq( m, n, k, a, lda, tau, work, -1_${ik}$, iinfo )
else
if( n>1_${ik}$ ) then
call stdlib${ii}$_zunglq( n-1, n-1, n-1, a, lda, tau, work, -1_${ik}$,iinfo )
end if
end if
end if
lwkopt = real( work( 1_${ik}$ ),KIND=dp)
lwkopt = max (lwkopt, mn)
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZUNGBR', -info )
return
else if( lquery ) then
work( 1_${ik}$ ) = lwkopt
return
end if
! quick return if possible
if( m==0_${ik}$ .or. n==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
return
end if
if( wantq ) then
! form q, determined by a call to stdlib${ii}$_zgebrd to reduce an m-by-k
! matrix
if( m>=k ) then
! if m >= k, assume m >= n >= k
call stdlib${ii}$_zungqr( m, n, k, a, lda, tau, work, lwork, iinfo )
else
! if m < k, assume m = n
! shift the vectors which define the elementary reflectors cone
! column to the right, and set the first row and column of q
! to those of the unit matrix
do j = m, 2, -1
a( 1_${ik}$, j ) = czero
do i = j + 1, m
a( i, j ) = a( i, j-1 )
end do
end do
a( 1_${ik}$, 1_${ik}$ ) = cone
do i = 2, m
a( i, 1_${ik}$ ) = czero
end do
if( m>1_${ik}$ ) then
! form q(2:m,2:m)
call stdlib${ii}$_zungqr( m-1, m-1, m-1, a( 2_${ik}$, 2_${ik}$ ), lda, tau, work,lwork, iinfo )
end if
end if
else
! form p**h, determined by a call to stdlib${ii}$_zgebrd to reduce a k-by-n
! matrix
if( k<n ) then
! if k < n, assume k <= m <= n
call stdlib${ii}$_zunglq( m, n, k, a, lda, tau, work, lwork, iinfo )
else
! if k >= n, assume m = n
! shift the vectors which define the elementary reflectors cone
! row downward, and set the first row and column of p**h to
! those of the unit matrix
a( 1_${ik}$, 1_${ik}$ ) = cone
do i = 2, n
a( i, 1_${ik}$ ) = czero
end do
do j = 2, n
do i = j - 1, 2, -1
a( i, j ) = a( i-1, j )
end do
a( 1_${ik}$, j ) = czero
end do
if( n>1_${ik}$ ) then
! form p**h(2:n,2:n)
call stdlib${ii}$_zunglq( n-1, n-1, n-1, a( 2_${ik}$, 2_${ik}$ ), lda, tau, work,lwork, iinfo )
end if
end if
end if
work( 1_${ik}$ ) = lwkopt
return
end subroutine stdlib${ii}$_zungbr
#:for ck,ct,ci in CMPLX_KINDS_TYPES
#:if not ck in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ci}$ungbr( vect, m, n, k, a, lda, tau, work, lwork, info )
!! ZUNGBR: generates one of the complex unitary matrices Q or P**H
!! determined by ZGEBRD when reducing a complex matrix A to bidiagonal
!! form: A = Q * B * P**H. Q and P**H are defined as products of
!! elementary reflectors H(i) or G(i) respectively.
!! If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q
!! is of order M:
!! if m >= k, Q = H(1) H(2) . . . H(k) and ZUNGBR returns the first n
!! columns of Q, where m >= n >= k;
!! if m < k, Q = H(1) H(2) . . . H(m-1) and ZUNGBR returns Q as an
!! M-by-M matrix.
!! If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**H
!! is of order N:
!! if k < n, P**H = G(k) . . . G(2) G(1) and ZUNGBR returns the first m
!! rows of P**H, where n >= m >= k;
!! if k >= n, P**H = G(n-1) . . . G(2) G(1) and ZUNGBR returns P**H as
!! an N-by-N matrix.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${ck}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: k, lda, lwork, m, n
! Array Arguments
complex(${ck}$), intent(inout) :: a(lda,*)
complex(${ck}$), intent(in) :: tau(*)
complex(${ck}$), intent(out) :: work(*)
! =====================================================================
! Local Scalars
logical(lk) :: lquery, wantq
integer(${ik}$) :: i, iinfo, j, lwkopt, mn
! Intrinsic Functions
! Executable Statements
! test the input arguments
info = 0_${ik}$
wantq = stdlib_lsame( vect, 'Q' )
mn = min( m, n )
lquery = ( lwork==-1_${ik}$ )
if( .not.wantq .and. .not.stdlib_lsame( vect, 'P' ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( n<0_${ik}$ .or. ( wantq .and. ( n>m .or. n<min( m,k ) ) ) .or. ( .not.wantq .and. ( &
m>n .or. m<min( n, k ) ) ) ) then
info = -3_${ik}$
else if( k<0_${ik}$ ) then
info = -4_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -6_${ik}$
else if( lwork<max( 1_${ik}$, mn ) .and. .not.lquery ) then
info = -9_${ik}$
end if
if( info==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
if( wantq ) then
if( m>=k ) then
call stdlib${ii}$_${ci}$ungqr( m, n, k, a, lda, tau, work, -1_${ik}$, iinfo )
else
if( m>1_${ik}$ ) then
call stdlib${ii}$_${ci}$ungqr( m-1, m-1, m-1, a, lda, tau, work, -1_${ik}$,iinfo )
end if
end if
else
if( k<n ) then
call stdlib${ii}$_${ci}$unglq( m, n, k, a, lda, tau, work, -1_${ik}$, iinfo )
else
if( n>1_${ik}$ ) then
call stdlib${ii}$_${ci}$unglq( n-1, n-1, n-1, a, lda, tau, work, -1_${ik}$,iinfo )
end if
end if
end if
lwkopt = real( work( 1_${ik}$ ),KIND=${ck}$)
lwkopt = max (lwkopt, mn)
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZUNGBR', -info )
return
else if( lquery ) then
work( 1_${ik}$ ) = lwkopt
return
end if
! quick return if possible
if( m==0_${ik}$ .or. n==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
return
end if
if( wantq ) then
! form q, determined by a call to stdlib${ii}$_${ci}$gebrd to reduce an m-by-k
! matrix
if( m>=k ) then
! if m >= k, assume m >= n >= k
call stdlib${ii}$_${ci}$ungqr( m, n, k, a, lda, tau, work, lwork, iinfo )
else
! if m < k, assume m = n
! shift the vectors which define the elementary reflectors cone
! column to the right, and set the first row and column of q
! to those of the unit matrix
do j = m, 2, -1
a( 1_${ik}$, j ) = czero
do i = j + 1, m
a( i, j ) = a( i, j-1 )
end do
end do
a( 1_${ik}$, 1_${ik}$ ) = cone
do i = 2, m
a( i, 1_${ik}$ ) = czero
end do
if( m>1_${ik}$ ) then
! form q(2:m,2:m)
call stdlib${ii}$_${ci}$ungqr( m-1, m-1, m-1, a( 2_${ik}$, 2_${ik}$ ), lda, tau, work,lwork, iinfo )
end if
end if
else
! form p**h, determined by a call to stdlib${ii}$_${ci}$gebrd to reduce a k-by-n
! matrix
if( k<n ) then
! if k < n, assume k <= m <= n
call stdlib${ii}$_${ci}$unglq( m, n, k, a, lda, tau, work, lwork, iinfo )
else
! if k >= n, assume m = n
! shift the vectors which define the elementary reflectors cone
! row downward, and set the first row and column of p**h to
! those of the unit matrix
a( 1_${ik}$, 1_${ik}$ ) = cone
do i = 2, n
a( i, 1_${ik}$ ) = czero
end do
do j = 2, n
do i = j - 1, 2, -1
a( i, j ) = a( i-1, j )
end do
a( 1_${ik}$, j ) = czero
end do
if( n>1_${ik}$ ) then
! form p**h(2:n,2:n)
call stdlib${ii}$_${ci}$unglq( n-1, n-1, n-1, a( 2_${ik}$, 2_${ik}$ ), lda, tau, work,lwork, iinfo )
end if
end if
end if
work( 1_${ik}$ ) = lwkopt
return
end subroutine stdlib${ii}$_${ci}$ungbr
#:endif
#:endfor
pure module subroutine stdlib${ii}$_sorgbr( vect, m, n, k, a, lda, tau, work, lwork, info )
!! SORGBR generates one of the real orthogonal matrices Q or P**T
!! determined by SGEBRD when reducing a real matrix A to bidiagonal
!! form: A = Q * B * P**T. Q and P**T are defined as products of
!! elementary reflectors H(i) or G(i) respectively.
!! If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q
!! is of order M:
!! if m >= k, Q = H(1) H(2) . . . H(k) and SORGBR returns the first n
!! columns of Q, where m >= n >= k;
!! if m < k, Q = H(1) H(2) . . . H(m-1) and SORGBR returns Q as an
!! M-by-M matrix.
!! If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**T
!! is of order N:
!! if k < n, P**T = G(k) . . . G(2) G(1) and SORGBR returns the first m
!! rows of P**T, where n >= m >= k;
!! if k >= n, P**T = G(n-1) . . . G(2) G(1) and SORGBR returns P**T as
!! an N-by-N matrix.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: k, lda, lwork, m, n
! Array Arguments
real(sp), intent(inout) :: a(lda,*)
real(sp), intent(in) :: tau(*)
real(sp), intent(out) :: work(*)
! =====================================================================
! Local Scalars
logical(lk) :: lquery, wantq
integer(${ik}$) :: i, iinfo, j, lwkopt, mn
! Intrinsic Functions
! Executable Statements
! test the input arguments
info = 0_${ik}$
wantq = stdlib_lsame( vect, 'Q' )
mn = min( m, n )
lquery = ( lwork==-1_${ik}$ )
if( .not.wantq .and. .not.stdlib_lsame( vect, 'P' ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( n<0_${ik}$ .or. ( wantq .and. ( n>m .or. n<min( m,k ) ) ) .or. ( .not.wantq .and. ( &
m>n .or. m<min( n, k ) ) ) ) then
info = -3_${ik}$
else if( k<0_${ik}$ ) then
info = -4_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -6_${ik}$
else if( lwork<max( 1_${ik}$, mn ) .and. .not.lquery ) then
info = -9_${ik}$
end if
if( info==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
if( wantq ) then
if( m>=k ) then
call stdlib${ii}$_sorgqr( m, n, k, a, lda, tau, work, -1_${ik}$, iinfo )
else
if( m>1_${ik}$ ) then
call stdlib${ii}$_sorgqr( m-1, m-1, m-1, a, lda, tau, work, -1_${ik}$,iinfo )
end if
end if
else
if( k<n ) then
call stdlib${ii}$_sorglq( m, n, k, a, lda, tau, work, -1_${ik}$, iinfo )
else
if( n>1_${ik}$ ) then
call stdlib${ii}$_sorglq( n-1, n-1, n-1, a, lda, tau, work, -1_${ik}$,iinfo )
end if
end if
end if
lwkopt = work( 1_${ik}$ )
lwkopt = max (lwkopt, mn)
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'SORGBR', -info )
return
else if( lquery ) then
work( 1_${ik}$ ) = lwkopt
return
end if
! quick return if possible
if( m==0_${ik}$ .or. n==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
return
end if
if( wantq ) then
! form q, determined by a call to stdlib${ii}$_sgebrd to reduce an m-by-k
! matrix
if( m>=k ) then
! if m >= k, assume m >= n >= k
call stdlib${ii}$_sorgqr( m, n, k, a, lda, tau, work, lwork, iinfo )
else
! if m < k, assume m = n
! shift the vectors which define the elementary reflectors one
! column to the right, and set the first row and column of q
! to those of the unit matrix
do j = m, 2, -1
a( 1_${ik}$, j ) = zero
do i = j + 1, m
a( i, j ) = a( i, j-1 )
end do
end do
a( 1_${ik}$, 1_${ik}$ ) = one
do i = 2, m
a( i, 1_${ik}$ ) = zero
end do
if( m>1_${ik}$ ) then
! form q(2:m,2:m)
call stdlib${ii}$_sorgqr( m-1, m-1, m-1, a( 2_${ik}$, 2_${ik}$ ), lda, tau, work,lwork, iinfo )
end if
end if
else
! form p**t, determined by a call to stdlib${ii}$_sgebrd to reduce a k-by-n
! matrix
if( k<n ) then
! if k < n, assume k <= m <= n
call stdlib${ii}$_sorglq( m, n, k, a, lda, tau, work, lwork, iinfo )
else
! if k >= n, assume m = n
! shift the vectors which define the elementary reflectors one
! row downward, and set the first row and column of p**t to
! those of the unit matrix
a( 1_${ik}$, 1_${ik}$ ) = one
do i = 2, n
a( i, 1_${ik}$ ) = zero
end do
do j = 2, n
do i = j - 1, 2, -1
a( i, j ) = a( i-1, j )
end do
a( 1_${ik}$, j ) = zero
end do
if( n>1_${ik}$ ) then
! form p**t(2:n,2:n)
call stdlib${ii}$_sorglq( n-1, n-1, n-1, a( 2_${ik}$, 2_${ik}$ ), lda, tau, work,lwork, iinfo )
end if
end if
end if
work( 1_${ik}$ ) = lwkopt
return
end subroutine stdlib${ii}$_sorgbr
pure module subroutine stdlib${ii}$_dorgbr( vect, m, n, k, a, lda, tau, work, lwork, info )
!! DORGBR generates one of the real orthogonal matrices Q or P**T
!! determined by DGEBRD when reducing a real matrix A to bidiagonal
!! form: A = Q * B * P**T. Q and P**T are defined as products of
!! elementary reflectors H(i) or G(i) respectively.
!! If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q
!! is of order M:
!! if m >= k, Q = H(1) H(2) . . . H(k) and DORGBR returns the first n
!! columns of Q, where m >= n >= k;
!! if m < k, Q = H(1) H(2) . . . H(m-1) and DORGBR returns Q as an
!! M-by-M matrix.
!! If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**T
!! is of order N:
!! if k < n, P**T = G(k) . . . G(2) G(1) and DORGBR returns the first m
!! rows of P**T, where n >= m >= k;
!! if k >= n, P**T = G(n-1) . . . G(2) G(1) and DORGBR returns P**T as
!! an N-by-N matrix.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: k, lda, lwork, m, n
! Array Arguments
real(dp), intent(inout) :: a(lda,*)
real(dp), intent(in) :: tau(*)
real(dp), intent(out) :: work(*)
! =====================================================================
! Local Scalars
logical(lk) :: lquery, wantq
integer(${ik}$) :: i, iinfo, j, lwkopt, mn
! Intrinsic Functions
! Executable Statements
! test the input arguments
info = 0_${ik}$
wantq = stdlib_lsame( vect, 'Q' )
mn = min( m, n )
lquery = ( lwork==-1_${ik}$ )
if( .not.wantq .and. .not.stdlib_lsame( vect, 'P' ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( n<0_${ik}$ .or. ( wantq .and. ( n>m .or. n<min( m,k ) ) ) .or. ( .not.wantq .and. ( &
m>n .or. m<min( n, k ) ) ) ) then
info = -3_${ik}$
else if( k<0_${ik}$ ) then
info = -4_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -6_${ik}$
else if( lwork<max( 1_${ik}$, mn ) .and. .not.lquery ) then
info = -9_${ik}$
end if
if( info==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
if( wantq ) then
if( m>=k ) then
call stdlib${ii}$_dorgqr( m, n, k, a, lda, tau, work, -1_${ik}$, iinfo )
else
if( m>1_${ik}$ ) then
call stdlib${ii}$_dorgqr( m-1, m-1, m-1, a, lda, tau, work, -1_${ik}$,iinfo )
end if
end if
else
if( k<n ) then
call stdlib${ii}$_dorglq( m, n, k, a, lda, tau, work, -1_${ik}$, iinfo )
else
if( n>1_${ik}$ ) then
call stdlib${ii}$_dorglq( n-1, n-1, n-1, a, lda, tau, work, -1_${ik}$,iinfo )
end if
end if
end if
lwkopt = work( 1_${ik}$ )
lwkopt = max (lwkopt, mn)
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DORGBR', -info )
return
else if( lquery ) then
work( 1_${ik}$ ) = lwkopt
return
end if
! quick return if possible
if( m==0_${ik}$ .or. n==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
return
end if
if( wantq ) then
! form q, determined by a call to stdlib${ii}$_dgebrd to reduce an m-by-k
! matrix
if( m>=k ) then
! if m >= k, assume m >= n >= k
call stdlib${ii}$_dorgqr( m, n, k, a, lda, tau, work, lwork, iinfo )
else
! if m < k, assume m = n
! shift the vectors which define the elementary reflectors one
! column to the right, and set the first row and column of q
! to those of the unit matrix
do j = m, 2, -1
a( 1_${ik}$, j ) = zero
do i = j + 1, m
a( i, j ) = a( i, j-1 )
end do
end do
a( 1_${ik}$, 1_${ik}$ ) = one
do i = 2, m
a( i, 1_${ik}$ ) = zero
end do
if( m>1_${ik}$ ) then
! form q(2:m,2:m)
call stdlib${ii}$_dorgqr( m-1, m-1, m-1, a( 2_${ik}$, 2_${ik}$ ), lda, tau, work,lwork, iinfo )
end if
end if
else
! form p**t, determined by a call to stdlib${ii}$_dgebrd to reduce a k-by-n
! matrix
if( k<n ) then
! if k < n, assume k <= m <= n
call stdlib${ii}$_dorglq( m, n, k, a, lda, tau, work, lwork, iinfo )
else
! if k >= n, assume m = n
! shift the vectors which define the elementary reflectors one
! row downward, and set the first row and column of p**t to
! those of the unit matrix
a( 1_${ik}$, 1_${ik}$ ) = one
do i = 2, n
a( i, 1_${ik}$ ) = zero
end do
do j = 2, n
do i = j - 1, 2, -1
a( i, j ) = a( i-1, j )
end do
a( 1_${ik}$, j ) = zero
end do
if( n>1_${ik}$ ) then
! form p**t(2:n,2:n)
call stdlib${ii}$_dorglq( n-1, n-1, n-1, a( 2_${ik}$, 2_${ik}$ ), lda, tau, work,lwork, iinfo )
end if
end if
end if
work( 1_${ik}$ ) = lwkopt
return
end subroutine stdlib${ii}$_dorgbr
#:for rk,rt,ri in REAL_KINDS_TYPES
#:if not rk in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ri}$orgbr( vect, m, n, k, a, lda, tau, work, lwork, info )
!! DORGBR: generates one of the real orthogonal matrices Q or P**T
!! determined by DGEBRD when reducing a real matrix A to bidiagonal
!! form: A = Q * B * P**T. Q and P**T are defined as products of
!! elementary reflectors H(i) or G(i) respectively.
!! If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q
!! is of order M:
!! if m >= k, Q = H(1) H(2) . . . H(k) and DORGBR returns the first n
!! columns of Q, where m >= n >= k;
!! if m < k, Q = H(1) H(2) . . . H(m-1) and DORGBR returns Q as an
!! M-by-M matrix.
!! If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**T
!! is of order N:
!! if k < n, P**T = G(k) . . . G(2) G(1) and DORGBR returns the first m
!! rows of P**T, where n >= m >= k;
!! if k >= n, P**T = G(n-1) . . . G(2) G(1) and DORGBR returns P**T as
!! an N-by-N matrix.
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${rk}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: k, lda, lwork, m, n
! Array Arguments
real(${rk}$), intent(inout) :: a(lda,*)
real(${rk}$), intent(in) :: tau(*)
real(${rk}$), intent(out) :: work(*)
! =====================================================================
! Local Scalars
logical(lk) :: lquery, wantq
integer(${ik}$) :: i, iinfo, j, lwkopt, mn
! Intrinsic Functions
! Executable Statements
! test the input arguments
info = 0_${ik}$
wantq = stdlib_lsame( vect, 'Q' )
mn = min( m, n )
lquery = ( lwork==-1_${ik}$ )
if( .not.wantq .and. .not.stdlib_lsame( vect, 'P' ) ) then
info = -1_${ik}$
else if( m<0_${ik}$ ) then
info = -2_${ik}$
else if( n<0_${ik}$ .or. ( wantq .and. ( n>m .or. n<min( m,k ) ) ) .or. ( .not.wantq .and. ( &
m>n .or. m<min( n, k ) ) ) ) then
info = -3_${ik}$
else if( k<0_${ik}$ ) then
info = -4_${ik}$
else if( lda<max( 1_${ik}$, m ) ) then
info = -6_${ik}$
else if( lwork<max( 1_${ik}$, mn ) .and. .not.lquery ) then
info = -9_${ik}$
end if
if( info==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
if( wantq ) then
if( m>=k ) then
call stdlib${ii}$_${ri}$orgqr( m, n, k, a, lda, tau, work, -1_${ik}$, iinfo )
else
if( m>1_${ik}$ ) then
call stdlib${ii}$_${ri}$orgqr( m-1, m-1, m-1, a, lda, tau, work, -1_${ik}$,iinfo )
end if
end if
else
if( k<n ) then
call stdlib${ii}$_${ri}$orglq( m, n, k, a, lda, tau, work, -1_${ik}$, iinfo )
else
if( n>1_${ik}$ ) then
call stdlib${ii}$_${ri}$orglq( n-1, n-1, n-1, a, lda, tau, work, -1_${ik}$,iinfo )
end if
end if
end if
lwkopt = work( 1_${ik}$ )
lwkopt = max (lwkopt, mn)
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DORGBR', -info )
return
else if( lquery ) then
work( 1_${ik}$ ) = lwkopt
return
end if
! quick return if possible
if( m==0_${ik}$ .or. n==0_${ik}$ ) then
work( 1_${ik}$ ) = 1_${ik}$
return
end if
if( wantq ) then
! form q, determined by a call to stdlib${ii}$_${ri}$gebrd to reduce an m-by-k
! matrix
if( m>=k ) then
! if m >= k, assume m >= n >= k
call stdlib${ii}$_${ri}$orgqr( m, n, k, a, lda, tau, work, lwork, iinfo )
else
! if m < k, assume m = n
! shift the vectors which define the elementary reflectors one
! column to the right, and set the first row and column of q
! to those of the unit matrix
do j = m, 2, -1
a( 1_${ik}$, j ) = zero
do i = j + 1, m
a( i, j ) = a( i, j-1 )
end do
end do
a( 1_${ik}$, 1_${ik}$ ) = one
do i = 2, m
a( i, 1_${ik}$ ) = zero
end do
if( m>1_${ik}$ ) then
! form q(2:m,2:m)
call stdlib${ii}$_${ri}$orgqr( m-1, m-1, m-1, a( 2_${ik}$, 2_${ik}$ ), lda, tau, work,lwork, iinfo )
end if
end if
else
! form p**t, determined by a call to stdlib${ii}$_${ri}$gebrd to reduce a k-by-n
! matrix
if( k<n ) then
! if k < n, assume k <= m <= n
call stdlib${ii}$_${ri}$orglq( m, n, k, a, lda, tau, work, lwork, iinfo )
else
! if k >= n, assume m = n
! shift the vectors which define the elementary reflectors one
! row downward, and set the first row and column of p**t to
! those of the unit matrix
a( 1_${ik}$, 1_${ik}$ ) = one
do i = 2, n
a( i, 1_${ik}$ ) = zero
end do
do j = 2, n
do i = j - 1, 2, -1
a( i, j ) = a( i-1, j )
end do
a( 1_${ik}$, j ) = zero
end do
if( n>1_${ik}$ ) then
! form p**t(2:n,2:n)
call stdlib${ii}$_${ri}$orglq( n-1, n-1, n-1, a( 2_${ik}$, 2_${ik}$ ), lda, tau, work,lwork, iinfo )
end if
end if
end if
work( 1_${ik}$ ) = lwkopt
return
end subroutine stdlib${ii}$_${ri}$orgbr
#:endif
#:endfor
pure module subroutine stdlib${ii}$_cunmbr( vect, side, trans, m, n, k, a, lda, tau, c,ldc, work, lwork, &
!! If VECT = 'Q', CUNMBR: overwrites the general complex M-by-N matrix C
!! with
!! SIDE = 'L' SIDE = 'R'
!! TRANS = 'N': Q * C C * Q
!! TRANS = 'C': Q**H * C C * Q**H
!! If VECT = 'P', CUNMBR overwrites the general complex M-by-N matrix C
!! with
!! SIDE = 'L' SIDE = 'R'
!! TRANS = 'N': P * C C * P
!! TRANS = 'C': P**H * C C * P**H
!! Here Q and P**H are the unitary matrices determined by CGEBRD when
!! reducing a complex matrix A to bidiagonal form: A = Q * B * P**H. Q
!! and P**H are defined as products of elementary reflectors H(i) and
!! G(i) respectively.
!! Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
!! order of the unitary matrix Q or P**H that is applied.
!! If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
!! if nq >= k, Q = H(1) H(2) . . . H(k);
!! if nq < k, Q = H(1) H(2) . . . H(nq-1).
!! If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
!! if k < nq, P = G(1) G(2) . . . G(k);
!! if k >= nq, P = G(1) G(2) . . . G(nq-1).
info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: side, trans, vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: k, lda, ldc, lwork, m, n
! Array Arguments
complex(sp), intent(inout) :: a(lda,*), c(ldc,*)
complex(sp), intent(in) :: tau(*)
complex(sp), intent(out) :: work(*)
! =====================================================================
! Local Scalars
logical(lk) :: applyq, left, lquery, notran
character :: transt
integer(${ik}$) :: i1, i2, iinfo, lwkopt, mi, nb, ni, nq, nw
! Intrinsic Functions
! Executable Statements
! test the input arguments
info = 0_${ik}$
applyq = stdlib_lsame( vect, 'Q' )
left = stdlib_lsame( side, 'L' )
notran = stdlib_lsame( trans, 'N' )
lquery = ( lwork==-1_${ik}$ )
! nq is the order of q or p and nw is the minimum dimension of work
if( left ) then
nq = m
nw = max( 1_${ik}$, n )
else
nq = n
nw = max( 1_${ik}$, m )
end if
if( .not.applyq .and. .not.stdlib_lsame( vect, 'P' ) ) then
info = -1_${ik}$
else if( .not.left .and. .not.stdlib_lsame( side, 'R' ) ) then
info = -2_${ik}$
else if( .not.notran .and. .not.stdlib_lsame( trans, 'C' ) ) then
info = -3_${ik}$
else if( m<0_${ik}$ ) then
info = -4_${ik}$
else if( n<0_${ik}$ ) then
info = -5_${ik}$
else if( k<0_${ik}$ ) then
info = -6_${ik}$
else if( ( applyq .and. lda<max( 1_${ik}$, nq ) ) .or.( .not.applyq .and. lda<max( 1_${ik}$, min( nq,&
k ) ) ) )then
info = -8_${ik}$
else if( ldc<max( 1_${ik}$, m ) ) then
info = -11_${ik}$
else if( lwork<nw .and. .not.lquery ) then
info = -13_${ik}$
end if
if( info==0_${ik}$ ) then
if( m>0_${ik}$ .and. n>0_${ik}$ ) then
if( applyq ) then
if( left ) then
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'CUNMQR', side // trans, m-1, n, m-1,-1_${ik}$ )
else
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'CUNMQR', side // trans, m, n-1, n-1,-1_${ik}$ )
end if
else
if( left ) then
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'CUNMLQ', side // trans, m-1, n, m-1,-1_${ik}$ )
else
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'CUNMLQ', side // trans, m, n-1, n-1,-1_${ik}$ )
end if
end if
lwkopt = nw*nb
else
lwkopt = 1_${ik}$
end if
work( 1_${ik}$ ) = lwkopt
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'CUNMBR', -info )
return
else if( lquery ) then
return
end if
! quick return if possible
if( m==0 .or. n==0 )return
if( applyq ) then
! apply q
if( nq>=k ) then
! q was determined by a call to stdlib${ii}$_cgebrd with nq >= k
call stdlib${ii}$_cunmqr( side, trans, m, n, k, a, lda, tau, c, ldc,work, lwork, iinfo &
)
else if( nq>1_${ik}$ ) then
! q was determined by a call to stdlib${ii}$_cgebrd with nq < k
if( left ) then
mi = m - 1_${ik}$
ni = n
i1 = 2_${ik}$
i2 = 1_${ik}$
else
mi = m
ni = n - 1_${ik}$
i1 = 1_${ik}$
i2 = 2_${ik}$
end if
call stdlib${ii}$_cunmqr( side, trans, mi, ni, nq-1, a( 2_${ik}$, 1_${ik}$ ), lda, tau,c( i1, i2 ), &
ldc, work, lwork, iinfo )
end if
else
! apply p
if( notran ) then
transt = 'C'
else
transt = 'N'
end if
if( nq>k ) then
! p was determined by a call to stdlib${ii}$_cgebrd with nq > k
call stdlib${ii}$_cunmlq( side, transt, m, n, k, a, lda, tau, c, ldc,work, lwork, &
iinfo )
else if( nq>1_${ik}$ ) then
! p was determined by a call to stdlib${ii}$_cgebrd with nq <= k
if( left ) then
mi = m - 1_${ik}$
ni = n
i1 = 2_${ik}$
i2 = 1_${ik}$
else
mi = m
ni = n - 1_${ik}$
i1 = 1_${ik}$
i2 = 2_${ik}$
end if
call stdlib${ii}$_cunmlq( side, transt, mi, ni, nq-1, a( 1_${ik}$, 2_${ik}$ ), lda,tau, c( i1, i2 ), &
ldc, work, lwork, iinfo )
end if
end if
work( 1_${ik}$ ) = lwkopt
return
end subroutine stdlib${ii}$_cunmbr
pure module subroutine stdlib${ii}$_zunmbr( vect, side, trans, m, n, k, a, lda, tau, c,ldc, work, lwork, &
!! If VECT = 'Q', ZUNMBR: overwrites the general complex M-by-N matrix C
!! with
!! SIDE = 'L' SIDE = 'R'
!! TRANS = 'N': Q * C C * Q
!! TRANS = 'C': Q**H * C C * Q**H
!! If VECT = 'P', ZUNMBR overwrites the general complex M-by-N matrix C
!! with
!! SIDE = 'L' SIDE = 'R'
!! TRANS = 'N': P * C C * P
!! TRANS = 'C': P**H * C C * P**H
!! Here Q and P**H are the unitary matrices determined by ZGEBRD when
!! reducing a complex matrix A to bidiagonal form: A = Q * B * P**H. Q
!! and P**H are defined as products of elementary reflectors H(i) and
!! G(i) respectively.
!! Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
!! order of the unitary matrix Q or P**H that is applied.
!! If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
!! if nq >= k, Q = H(1) H(2) . . . H(k);
!! if nq < k, Q = H(1) H(2) . . . H(nq-1).
!! If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
!! if k < nq, P = G(1) G(2) . . . G(k);
!! if k >= nq, P = G(1) G(2) . . . G(nq-1).
info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: side, trans, vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: k, lda, ldc, lwork, m, n
! Array Arguments
complex(dp), intent(inout) :: a(lda,*), c(ldc,*)
complex(dp), intent(in) :: tau(*)
complex(dp), intent(out) :: work(*)
! =====================================================================
! Local Scalars
logical(lk) :: applyq, left, lquery, notran
character :: transt
integer(${ik}$) :: i1, i2, iinfo, lwkopt, mi, nb, ni, nq, nw
! Intrinsic Functions
! Executable Statements
! test the input arguments
info = 0_${ik}$
applyq = stdlib_lsame( vect, 'Q' )
left = stdlib_lsame( side, 'L' )
notran = stdlib_lsame( trans, 'N' )
lquery = ( lwork==-1_${ik}$ )
! nq is the order of q or p and nw is the minimum dimension of work
if( left ) then
nq = m
nw = max( 1_${ik}$, n )
else
nq = n
nw = max( 1_${ik}$, m )
end if
if( .not.applyq .and. .not.stdlib_lsame( vect, 'P' ) ) then
info = -1_${ik}$
else if( .not.left .and. .not.stdlib_lsame( side, 'R' ) ) then
info = -2_${ik}$
else if( .not.notran .and. .not.stdlib_lsame( trans, 'C' ) ) then
info = -3_${ik}$
else if( m<0_${ik}$ ) then
info = -4_${ik}$
else if( n<0_${ik}$ ) then
info = -5_${ik}$
else if( k<0_${ik}$ ) then
info = -6_${ik}$
else if( ( applyq .and. lda<max( 1_${ik}$, nq ) ) .or.( .not.applyq .and. lda<max( 1_${ik}$, min( nq,&
k ) ) ) )then
info = -8_${ik}$
else if( ldc<max( 1_${ik}$, m ) ) then
info = -11_${ik}$
else if( lwork<nw .and. .not.lquery ) then
info = -13_${ik}$
end if
if( info==0_${ik}$ ) then
if( m>0_${ik}$ .and. n>0_${ik}$ ) then
if( applyq ) then
if( left ) then
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'ZUNMQR', side // trans, m-1, n, m-1,-1_${ik}$ )
else
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'ZUNMQR', side // trans, m, n-1, n-1,-1_${ik}$ )
end if
else
if( left ) then
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'ZUNMLQ', side // trans, m-1, n, m-1,-1_${ik}$ )
else
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'ZUNMLQ', side // trans, m, n-1, n-1,-1_${ik}$ )
end if
end if
lwkopt = nw*nb
else
lwkopt = 1_${ik}$
end if
work( 1_${ik}$ ) = lwkopt
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZUNMBR', -info )
return
else if( lquery ) then
return
end if
! quick return if possible
if( m==0 .or. n==0 )return
if( applyq ) then
! apply q
if( nq>=k ) then
! q was determined by a call to stdlib${ii}$_zgebrd with nq >= k
call stdlib${ii}$_zunmqr( side, trans, m, n, k, a, lda, tau, c, ldc,work, lwork, iinfo &
)
else if( nq>1_${ik}$ ) then
! q was determined by a call to stdlib${ii}$_zgebrd with nq < k
if( left ) then
mi = m - 1_${ik}$
ni = n
i1 = 2_${ik}$
i2 = 1_${ik}$
else
mi = m
ni = n - 1_${ik}$
i1 = 1_${ik}$
i2 = 2_${ik}$
end if
call stdlib${ii}$_zunmqr( side, trans, mi, ni, nq-1, a( 2_${ik}$, 1_${ik}$ ), lda, tau,c( i1, i2 ), &
ldc, work, lwork, iinfo )
end if
else
! apply p
if( notran ) then
transt = 'C'
else
transt = 'N'
end if
if( nq>k ) then
! p was determined by a call to stdlib${ii}$_zgebrd with nq > k
call stdlib${ii}$_zunmlq( side, transt, m, n, k, a, lda, tau, c, ldc,work, lwork, &
iinfo )
else if( nq>1_${ik}$ ) then
! p was determined by a call to stdlib${ii}$_zgebrd with nq <= k
if( left ) then
mi = m - 1_${ik}$
ni = n
i1 = 2_${ik}$
i2 = 1_${ik}$
else
mi = m
ni = n - 1_${ik}$
i1 = 1_${ik}$
i2 = 2_${ik}$
end if
call stdlib${ii}$_zunmlq( side, transt, mi, ni, nq-1, a( 1_${ik}$, 2_${ik}$ ), lda,tau, c( i1, i2 ), &
ldc, work, lwork, iinfo )
end if
end if
work( 1_${ik}$ ) = lwkopt
return
end subroutine stdlib${ii}$_zunmbr
#:for ck,ct,ci in CMPLX_KINDS_TYPES
#:if not ck in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ci}$unmbr( vect, side, trans, m, n, k, a, lda, tau, c,ldc, work, lwork, &
!! If VECT = 'Q', ZUNMBR: overwrites the general complex M-by-N matrix C
!! with
!! SIDE = 'L' SIDE = 'R'
!! TRANS = 'N': Q * C C * Q
!! TRANS = 'C': Q**H * C C * Q**H
!! If VECT = 'P', ZUNMBR overwrites the general complex M-by-N matrix C
!! with
!! SIDE = 'L' SIDE = 'R'
!! TRANS = 'N': P * C C * P
!! TRANS = 'C': P**H * C C * P**H
!! Here Q and P**H are the unitary matrices determined by ZGEBRD when
!! reducing a complex matrix A to bidiagonal form: A = Q * B * P**H. Q
!! and P**H are defined as products of elementary reflectors H(i) and
!! G(i) respectively.
!! Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
!! order of the unitary matrix Q or P**H that is applied.
!! If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
!! if nq >= k, Q = H(1) H(2) . . . H(k);
!! if nq < k, Q = H(1) H(2) . . . H(nq-1).
!! If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
!! if k < nq, P = G(1) G(2) . . . G(k);
!! if k >= nq, P = G(1) G(2) . . . G(nq-1).
info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${ck}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: side, trans, vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: k, lda, ldc, lwork, m, n
! Array Arguments
complex(${ck}$), intent(inout) :: a(lda,*), c(ldc,*)
complex(${ck}$), intent(in) :: tau(*)
complex(${ck}$), intent(out) :: work(*)
! =====================================================================
! Local Scalars
logical(lk) :: applyq, left, lquery, notran
character :: transt
integer(${ik}$) :: i1, i2, iinfo, lwkopt, mi, nb, ni, nq, nw
! Intrinsic Functions
! Executable Statements
! test the input arguments
info = 0_${ik}$
applyq = stdlib_lsame( vect, 'Q' )
left = stdlib_lsame( side, 'L' )
notran = stdlib_lsame( trans, 'N' )
lquery = ( lwork==-1_${ik}$ )
! nq is the order of q or p and nw is the minimum dimension of work
if( left ) then
nq = m
nw = max( 1_${ik}$, n )
else
nq = n
nw = max( 1_${ik}$, m )
end if
if( .not.applyq .and. .not.stdlib_lsame( vect, 'P' ) ) then
info = -1_${ik}$
else if( .not.left .and. .not.stdlib_lsame( side, 'R' ) ) then
info = -2_${ik}$
else if( .not.notran .and. .not.stdlib_lsame( trans, 'C' ) ) then
info = -3_${ik}$
else if( m<0_${ik}$ ) then
info = -4_${ik}$
else if( n<0_${ik}$ ) then
info = -5_${ik}$
else if( k<0_${ik}$ ) then
info = -6_${ik}$
else if( ( applyq .and. lda<max( 1_${ik}$, nq ) ) .or.( .not.applyq .and. lda<max( 1_${ik}$, min( nq,&
k ) ) ) )then
info = -8_${ik}$
else if( ldc<max( 1_${ik}$, m ) ) then
info = -11_${ik}$
else if( lwork<nw .and. .not.lquery ) then
info = -13_${ik}$
end if
if( info==0_${ik}$ ) then
if( m>0_${ik}$ .and. n>0_${ik}$ ) then
if( applyq ) then
if( left ) then
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'ZUNMQR', side // trans, m-1, n, m-1,-1_${ik}$ )
else
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'ZUNMQR', side // trans, m, n-1, n-1,-1_${ik}$ )
end if
else
if( left ) then
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'ZUNMLQ', side // trans, m-1, n, m-1,-1_${ik}$ )
else
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'ZUNMLQ', side // trans, m, n-1, n-1,-1_${ik}$ )
end if
end if
lwkopt = nw*nb
else
lwkopt = 1_${ik}$
end if
work( 1_${ik}$ ) = lwkopt
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'ZUNMBR', -info )
return
else if( lquery ) then
return
end if
! quick return if possible
if( m==0 .or. n==0 )return
if( applyq ) then
! apply q
if( nq>=k ) then
! q was determined by a call to stdlib${ii}$_${ci}$gebrd with nq >= k
call stdlib${ii}$_${ci}$unmqr( side, trans, m, n, k, a, lda, tau, c, ldc,work, lwork, iinfo &
)
else if( nq>1_${ik}$ ) then
! q was determined by a call to stdlib${ii}$_${ci}$gebrd with nq < k
if( left ) then
mi = m - 1_${ik}$
ni = n
i1 = 2_${ik}$
i2 = 1_${ik}$
else
mi = m
ni = n - 1_${ik}$
i1 = 1_${ik}$
i2 = 2_${ik}$
end if
call stdlib${ii}$_${ci}$unmqr( side, trans, mi, ni, nq-1, a( 2_${ik}$, 1_${ik}$ ), lda, tau,c( i1, i2 ), &
ldc, work, lwork, iinfo )
end if
else
! apply p
if( notran ) then
transt = 'C'
else
transt = 'N'
end if
if( nq>k ) then
! p was determined by a call to stdlib${ii}$_${ci}$gebrd with nq > k
call stdlib${ii}$_${ci}$unmlq( side, transt, m, n, k, a, lda, tau, c, ldc,work, lwork, &
iinfo )
else if( nq>1_${ik}$ ) then
! p was determined by a call to stdlib${ii}$_${ci}$gebrd with nq <= k
if( left ) then
mi = m - 1_${ik}$
ni = n
i1 = 2_${ik}$
i2 = 1_${ik}$
else
mi = m
ni = n - 1_${ik}$
i1 = 1_${ik}$
i2 = 2_${ik}$
end if
call stdlib${ii}$_${ci}$unmlq( side, transt, mi, ni, nq-1, a( 1_${ik}$, 2_${ik}$ ), lda,tau, c( i1, i2 ), &
ldc, work, lwork, iinfo )
end if
end if
work( 1_${ik}$ ) = lwkopt
return
end subroutine stdlib${ii}$_${ci}$unmbr
#:endif
#:endfor
pure module subroutine stdlib${ii}$_sormbr( vect, side, trans, m, n, k, a, lda, tau, c,ldc, work, lwork, &
!! If VECT = 'Q', SORMBR: overwrites the general real M-by-N matrix C
!! with
!! SIDE = 'L' SIDE = 'R'
!! TRANS = 'N': Q * C C * Q
!! TRANS = 'T': Q**T * C C * Q**T
!! If VECT = 'P', SORMBR overwrites the general real M-by-N matrix C
!! with
!! SIDE = 'L' SIDE = 'R'
!! TRANS = 'N': P * C C * P
!! TRANS = 'T': P**T * C C * P**T
!! Here Q and P**T are the orthogonal matrices determined by SGEBRD when
!! reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and
!! P**T are defined as products of elementary reflectors H(i) and G(i)
!! respectively.
!! Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
!! order of the orthogonal matrix Q or P**T that is applied.
!! If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
!! if nq >= k, Q = H(1) H(2) . . . H(k);
!! if nq < k, Q = H(1) H(2) . . . H(nq-1).
!! If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
!! if k < nq, P = G(1) G(2) . . . G(k);
!! if k >= nq, P = G(1) G(2) . . . G(nq-1).
info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_sp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: side, trans, vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: k, lda, ldc, lwork, m, n
! Array Arguments
real(sp), intent(inout) :: a(lda,*), c(ldc,*)
real(sp), intent(in) :: tau(*)
real(sp), intent(out) :: work(*)
! =====================================================================
! Local Scalars
logical(lk) :: applyq, left, lquery, notran
character :: transt
integer(${ik}$) :: i1, i2, iinfo, lwkopt, mi, nb, ni, nq, nw
! Intrinsic Functions
! Executable Statements
! test the input arguments
info = 0_${ik}$
applyq = stdlib_lsame( vect, 'Q' )
left = stdlib_lsame( side, 'L' )
notran = stdlib_lsame( trans, 'N' )
lquery = ( lwork==-1_${ik}$ )
! nq is the order of q or p and nw is the minimum dimension of work
if( left ) then
nq = m
nw = max( 1_${ik}$, n )
else
nq = n
nw = max( 1_${ik}$, m )
end if
if( .not.applyq .and. .not.stdlib_lsame( vect, 'P' ) ) then
info = -1_${ik}$
else if( .not.left .and. .not.stdlib_lsame( side, 'R' ) ) then
info = -2_${ik}$
else if( .not.notran .and. .not.stdlib_lsame( trans, 'T' ) ) then
info = -3_${ik}$
else if( m<0_${ik}$ ) then
info = -4_${ik}$
else if( n<0_${ik}$ ) then
info = -5_${ik}$
else if( k<0_${ik}$ ) then
info = -6_${ik}$
else if( ( applyq .and. lda<max( 1_${ik}$, nq ) ) .or.( .not.applyq .and. lda<max( 1_${ik}$, min( nq,&
k ) ) ) )then
info = -8_${ik}$
else if( ldc<max( 1_${ik}$, m ) ) then
info = -11_${ik}$
else if( lwork<nw .and. .not.lquery ) then
info = -13_${ik}$
end if
if( info==0_${ik}$ ) then
if( applyq ) then
if( left ) then
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'SORMQR', side // trans, m-1, n, m-1,-1_${ik}$ )
else
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'SORMQR', side // trans, m, n-1, n-1,-1_${ik}$ )
end if
else
if( left ) then
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'SORMLQ', side // trans, m-1, n, m-1,-1_${ik}$ )
else
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'SORMLQ', side // trans, m, n-1, n-1,-1_${ik}$ )
end if
end if
lwkopt = nw*nb
work( 1_${ik}$ ) = lwkopt
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'SORMBR', -info )
return
else if( lquery ) then
return
end if
! quick return if possible
work( 1_${ik}$ ) = 1_${ik}$
if( m==0 .or. n==0 )return
if( applyq ) then
! apply q
if( nq>=k ) then
! q was determined by a call to stdlib${ii}$_sgebrd with nq >= k
call stdlib${ii}$_sormqr( side, trans, m, n, k, a, lda, tau, c, ldc,work, lwork, iinfo &
)
else if( nq>1_${ik}$ ) then
! q was determined by a call to stdlib${ii}$_sgebrd with nq < k
if( left ) then
mi = m - 1_${ik}$
ni = n
i1 = 2_${ik}$
i2 = 1_${ik}$
else
mi = m
ni = n - 1_${ik}$
i1 = 1_${ik}$
i2 = 2_${ik}$
end if
call stdlib${ii}$_sormqr( side, trans, mi, ni, nq-1, a( 2_${ik}$, 1_${ik}$ ), lda, tau,c( i1, i2 ), &
ldc, work, lwork, iinfo )
end if
else
! apply p
if( notran ) then
transt = 'T'
else
transt = 'N'
end if
if( nq>k ) then
! p was determined by a call to stdlib${ii}$_sgebrd with nq > k
call stdlib${ii}$_sormlq( side, transt, m, n, k, a, lda, tau, c, ldc,work, lwork, &
iinfo )
else if( nq>1_${ik}$ ) then
! p was determined by a call to stdlib${ii}$_sgebrd with nq <= k
if( left ) then
mi = m - 1_${ik}$
ni = n
i1 = 2_${ik}$
i2 = 1_${ik}$
else
mi = m
ni = n - 1_${ik}$
i1 = 1_${ik}$
i2 = 2_${ik}$
end if
call stdlib${ii}$_sormlq( side, transt, mi, ni, nq-1, a( 1_${ik}$, 2_${ik}$ ), lda,tau, c( i1, i2 ), &
ldc, work, lwork, iinfo )
end if
end if
work( 1_${ik}$ ) = lwkopt
return
end subroutine stdlib${ii}$_sormbr
pure module subroutine stdlib${ii}$_dormbr( vect, side, trans, m, n, k, a, lda, tau, c,ldc, work, lwork, &
!! If VECT = 'Q', DORMBR: overwrites the general real M-by-N matrix C
!! with
!! SIDE = 'L' SIDE = 'R'
!! TRANS = 'N': Q * C C * Q
!! TRANS = 'T': Q**T * C C * Q**T
!! If VECT = 'P', DORMBR overwrites the general real M-by-N matrix C
!! with
!! SIDE = 'L' SIDE = 'R'
!! TRANS = 'N': P * C C * P
!! TRANS = 'T': P**T * C C * P**T
!! Here Q and P**T are the orthogonal matrices determined by DGEBRD when
!! reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and
!! P**T are defined as products of elementary reflectors H(i) and G(i)
!! respectively.
!! Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
!! order of the orthogonal matrix Q or P**T that is applied.
!! If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
!! if nq >= k, Q = H(1) H(2) . . . H(k);
!! if nq < k, Q = H(1) H(2) . . . H(nq-1).
!! If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
!! if k < nq, P = G(1) G(2) . . . G(k);
!! if k >= nq, P = G(1) G(2) . . . G(nq-1).
info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_dp, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: side, trans, vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: k, lda, ldc, lwork, m, n
! Array Arguments
real(dp), intent(inout) :: a(lda,*), c(ldc,*)
real(dp), intent(in) :: tau(*)
real(dp), intent(out) :: work(*)
! =====================================================================
! Local Scalars
logical(lk) :: applyq, left, lquery, notran
character :: transt
integer(${ik}$) :: i1, i2, iinfo, lwkopt, mi, nb, ni, nq, nw
! Intrinsic Functions
! Executable Statements
! test the input arguments
info = 0_${ik}$
applyq = stdlib_lsame( vect, 'Q' )
left = stdlib_lsame( side, 'L' )
notran = stdlib_lsame( trans, 'N' )
lquery = ( lwork==-1_${ik}$ )
! nq is the order of q or p and nw is the minimum dimension of work
if( left ) then
nq = m
nw = max( 1_${ik}$, n )
else
nq = n
nw = max( 1_${ik}$, m )
end if
if( .not.applyq .and. .not.stdlib_lsame( vect, 'P' ) ) then
info = -1_${ik}$
else if( .not.left .and. .not.stdlib_lsame( side, 'R' ) ) then
info = -2_${ik}$
else if( .not.notran .and. .not.stdlib_lsame( trans, 'T' ) ) then
info = -3_${ik}$
else if( m<0_${ik}$ ) then
info = -4_${ik}$
else if( n<0_${ik}$ ) then
info = -5_${ik}$
else if( k<0_${ik}$ ) then
info = -6_${ik}$
else if( ( applyq .and. lda<max( 1_${ik}$, nq ) ) .or.( .not.applyq .and. lda<max( 1_${ik}$, min( nq,&
k ) ) ) )then
info = -8_${ik}$
else if( ldc<max( 1_${ik}$, m ) ) then
info = -11_${ik}$
else if( lwork<nw .and. .not.lquery ) then
info = -13_${ik}$
end if
if( info==0_${ik}$ ) then
if( applyq ) then
if( left ) then
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'DORMQR', side // trans, m-1, n, m-1,-1_${ik}$ )
else
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'DORMQR', side // trans, m, n-1, n-1,-1_${ik}$ )
end if
else
if( left ) then
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'DORMLQ', side // trans, m-1, n, m-1,-1_${ik}$ )
else
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'DORMLQ', side // trans, m, n-1, n-1,-1_${ik}$ )
end if
end if
lwkopt = nw*nb
work( 1_${ik}$ ) = lwkopt
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DORMBR', -info )
return
else if( lquery ) then
return
end if
! quick return if possible
work( 1_${ik}$ ) = 1_${ik}$
if( m==0 .or. n==0 )return
if( applyq ) then
! apply q
if( nq>=k ) then
! q was determined by a call to stdlib${ii}$_dgebrd with nq >= k
call stdlib${ii}$_dormqr( side, trans, m, n, k, a, lda, tau, c, ldc,work, lwork, iinfo &
)
else if( nq>1_${ik}$ ) then
! q was determined by a call to stdlib${ii}$_dgebrd with nq < k
if( left ) then
mi = m - 1_${ik}$
ni = n
i1 = 2_${ik}$
i2 = 1_${ik}$
else
mi = m
ni = n - 1_${ik}$
i1 = 1_${ik}$
i2 = 2_${ik}$
end if
call stdlib${ii}$_dormqr( side, trans, mi, ni, nq-1, a( 2_${ik}$, 1_${ik}$ ), lda, tau,c( i1, i2 ), &
ldc, work, lwork, iinfo )
end if
else
! apply p
if( notran ) then
transt = 'T'
else
transt = 'N'
end if
if( nq>k ) then
! p was determined by a call to stdlib${ii}$_dgebrd with nq > k
call stdlib${ii}$_dormlq( side, transt, m, n, k, a, lda, tau, c, ldc,work, lwork, &
iinfo )
else if( nq>1_${ik}$ ) then
! p was determined by a call to stdlib${ii}$_dgebrd with nq <= k
if( left ) then
mi = m - 1_${ik}$
ni = n
i1 = 2_${ik}$
i2 = 1_${ik}$
else
mi = m
ni = n - 1_${ik}$
i1 = 1_${ik}$
i2 = 2_${ik}$
end if
call stdlib${ii}$_dormlq( side, transt, mi, ni, nq-1, a( 1_${ik}$, 2_${ik}$ ), lda,tau, c( i1, i2 ), &
ldc, work, lwork, iinfo )
end if
end if
work( 1_${ik}$ ) = lwkopt
return
end subroutine stdlib${ii}$_dormbr
#:for rk,rt,ri in REAL_KINDS_TYPES
#:if not rk in ["sp","dp"]
pure module subroutine stdlib${ii}$_${ri}$ormbr( vect, side, trans, m, n, k, a, lda, tau, c,ldc, work, lwork, &
!! If VECT = 'Q', DORMBR: overwrites the general real M-by-N matrix C
!! with
!! SIDE = 'L' SIDE = 'R'
!! TRANS = 'N': Q * C C * Q
!! TRANS = 'T': Q**T * C C * Q**T
!! If VECT = 'P', DORMBR overwrites the general real M-by-N matrix C
!! with
!! SIDE = 'L' SIDE = 'R'
!! TRANS = 'N': P * C C * P
!! TRANS = 'T': P**T * C C * P**T
!! Here Q and P**T are the orthogonal matrices determined by DGEBRD when
!! reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and
!! P**T are defined as products of elementary reflectors H(i) and G(i)
!! respectively.
!! Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
!! order of the orthogonal matrix Q or P**T that is applied.
!! If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
!! if nq >= k, Q = H(1) H(2) . . . H(k);
!! if nq < k, Q = H(1) H(2) . . . H(nq-1).
!! If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
!! if k < nq, P = G(1) G(2) . . . G(k);
!! if k >= nq, P = G(1) G(2) . . . G(nq-1).
info )
! -- lapack computational routine --
! -- lapack is a software package provided by univ. of tennessee, --
! -- univ. of california berkeley, univ. of colorado denver and nag ltd..--
use stdlib_blas_constants_${rk}$, only: negone, zero, half, one, two, three, four, eight, ten, czero, chalf, cone, cnegone
! Scalar Arguments
character, intent(in) :: side, trans, vect
integer(${ik}$), intent(out) :: info
integer(${ik}$), intent(in) :: k, lda, ldc, lwork, m, n
! Array Arguments
real(${rk}$), intent(inout) :: a(lda,*), c(ldc,*)
real(${rk}$), intent(in) :: tau(*)
real(${rk}$), intent(out) :: work(*)
! =====================================================================
! Local Scalars
logical(lk) :: applyq, left, lquery, notran
character :: transt
integer(${ik}$) :: i1, i2, iinfo, lwkopt, mi, nb, ni, nq, nw
! Intrinsic Functions
! Executable Statements
! test the input arguments
info = 0_${ik}$
applyq = stdlib_lsame( vect, 'Q' )
left = stdlib_lsame( side, 'L' )
notran = stdlib_lsame( trans, 'N' )
lquery = ( lwork==-1_${ik}$ )
! nq is the order of q or p and nw is the minimum dimension of work
if( left ) then
nq = m
nw = max( 1_${ik}$, n )
else
nq = n
nw = max( 1_${ik}$, m )
end if
if( .not.applyq .and. .not.stdlib_lsame( vect, 'P' ) ) then
info = -1_${ik}$
else if( .not.left .and. .not.stdlib_lsame( side, 'R' ) ) then
info = -2_${ik}$
else if( .not.notran .and. .not.stdlib_lsame( trans, 'T' ) ) then
info = -3_${ik}$
else if( m<0_${ik}$ ) then
info = -4_${ik}$
else if( n<0_${ik}$ ) then
info = -5_${ik}$
else if( k<0_${ik}$ ) then
info = -6_${ik}$
else if( ( applyq .and. lda<max( 1_${ik}$, nq ) ) .or.( .not.applyq .and. lda<max( 1_${ik}$, min( nq,&
k ) ) ) )then
info = -8_${ik}$
else if( ldc<max( 1_${ik}$, m ) ) then
info = -11_${ik}$
else if( lwork<nw .and. .not.lquery ) then
info = -13_${ik}$
end if
if( info==0_${ik}$ ) then
if( applyq ) then
if( left ) then
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'DORMQR', side // trans, m-1, n, m-1,-1_${ik}$ )
else
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'DORMQR', side // trans, m, n-1, n-1,-1_${ik}$ )
end if
else
if( left ) then
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'DORMLQ', side // trans, m-1, n, m-1,-1_${ik}$ )
else
nb = stdlib${ii}$_ilaenv( 1_${ik}$, 'DORMLQ', side // trans, m, n-1, n-1,-1_${ik}$ )
end if
end if
lwkopt = nw*nb
work( 1_${ik}$ ) = lwkopt
end if
if( info/=0_${ik}$ ) then
call stdlib${ii}$_xerbla( 'DORMBR', -info )
return
else if( lquery ) then
return
end if
! quick return if possible
work( 1_${ik}$ ) = 1_${ik}$
if( m==0 .or. n==0 )return
if( applyq ) then
! apply q
if( nq>=k ) then
! q was determined by a call to stdlib${ii}$_${ri}$gebrd with nq >= k
call stdlib${ii}$_${ri}$ormqr( side, trans, m, n, k, a, lda, tau, c, ldc,work, lwork, iinfo &
)
else if( nq>1_${ik}$ ) then
! q was determined by a call to stdlib${ii}$_${ri}$gebrd with nq < k
if( left ) then
mi = m - 1_${ik}$
ni = n
i1 = 2_${ik}$
i2 = 1_${ik}$
else
mi = m
ni = n - 1_${ik}$
i1 = 1_${ik}$
i2 = 2_${ik}$
end if
call stdlib${ii}$_${ri}$ormqr( side, trans, mi, ni, nq-1, a( 2_${ik}$, 1_${ik}$ ), lda, tau,c( i1, i2 ), &
ldc, work, lwork, iinfo )
end if
else
! apply p
if( notran ) then
transt = 'T'
else
transt = 'N'
end if
if( nq>k ) then
! p was determined by a call to stdlib${ii}$_${ri}$gebrd with nq > k
call stdlib${ii}$_${ri}$ormlq( side, transt, m, n, k, a, lda, tau, c, ldc,work, lwork, &
iinfo )
else if( nq>1_${ik}$ ) then
! p was determined by a call to stdlib${ii}$_${ri}$gebrd with nq <= k
if( left ) then
mi = m - 1_${ik}$
ni = n
i1 = 2_${ik}$
i2 = 1_${ik}$
else
mi = m
ni = n - 1_${ik}$
i1 = 1_${ik}$
i2 = 2_${ik}$
end if
call stdlib${ii}$_${ri}$ormlq( side, transt, mi, ni, nq-1, a( 1_${ik}$, 2_${ik}$ ), lda,tau, c( i1, i2 ), &
ldc, work, lwork, iinfo )
end if
end if
work( 1_${ik}$ ) = lwkopt
return
end subroutine stdlib${ii}$_${ri}$ormbr
#:endif
#:endfor
#:endfor
end submodule stdlib_lapack_svd_comp
