#:include "common.fypp" module stdlib_linalg_blas_d use stdlib_linalg_constants use stdlib_linalg_blas_aux use stdlib_linalg_blas_s use stdlib_linalg_blas_c implicit none(type,external) private public :: sp,dp,qp,lk,ilp public :: stdlib_dasum public :: stdlib_daxpy public :: stdlib_dcopy public :: stdlib_ddot public :: stdlib_dgbmv public :: stdlib_dgemm public :: stdlib_dgemv public :: stdlib_dger public :: stdlib_dnrm2 public :: stdlib_drot public :: stdlib_drotg public :: stdlib_drotm public :: stdlib_drotmg public :: stdlib_dsbmv public :: stdlib_dscal public :: stdlib_dsdot public :: stdlib_dspmv public :: stdlib_dspr public :: stdlib_dspr2 public :: stdlib_dswap public :: stdlib_dsymm public :: stdlib_dsymv public :: stdlib_dsyr public :: stdlib_dsyr2 public :: stdlib_dsyr2k public :: stdlib_dsyrk public :: stdlib_dtbmv public :: stdlib_dtbsv public :: stdlib_dtpmv public :: stdlib_dtpsv public :: stdlib_dtrmm public :: stdlib_dtrmv public :: stdlib_dtrsm public :: stdlib_dtrsv public :: stdlib_dzasum public :: stdlib_dznrm2 ! 64-bit real constants real(dp), parameter, private :: negone = -1.00_dp real(dp), parameter, private :: zero = 0.00_dp real(dp), parameter, private :: half = 0.50_dp real(dp), parameter, private :: one = 1.00_dp real(dp), parameter, private :: two = 2.00_dp real(dp), parameter, private :: three = 3.00_dp real(dp), parameter, private :: four = 4.00_dp real(dp), parameter, private :: eight = 8.00_dp real(dp), parameter, private :: ten = 10.00_dp ! 64-bit complex constants complex(dp), parameter, private :: czero = ( 0.0_dp,0.0_dp) complex(dp), parameter, private :: chalf = ( 0.5_dp,0.0_dp) complex(dp), parameter, private :: cone = ( 1.0_dp,0.0_dp) complex(dp), parameter, private :: cnegone = (-1.0_dp,0.0_dp) ! 64-bit scaling constants integer, parameter, private :: maxexp = maxexponent(zero) integer, parameter, private :: minexp = minexponent(zero) real(dp), parameter, private :: rradix = real(radix(zero),dp) real(dp), parameter, private :: ulp = epsilon(zero) real(dp), parameter, private :: eps = ulp*half real(dp), parameter, private :: safmin = rradix**max(minexp-1,1-maxexp) real(dp), parameter, private :: safmax = one/safmin real(dp), parameter, private :: smlnum = safmin/ulp real(dp), parameter, private :: bignum = safmax*ulp real(dp), parameter, private :: rtmin = sqrt(smlnum) real(dp), parameter, private :: rtmax = sqrt(bignum) ! 64-bit Blue's scaling constants ! ssml>=1/s and sbig==1/S with s,S as defined in https://doi.org/10.1145/355769.355771 real(dp), parameter, private :: tsml = rradix**ceiling((minexp-1)*half) real(dp), parameter, private :: tbig = rradix**floor((maxexp-digits(zero)+1)*half) real(dp), parameter, private :: ssml = rradix**(-floor((minexp-digits(zero))*half)) real(dp), parameter, private :: sbig = rradix**(-ceiling((maxexp+digits(zero)-1)*half)) contains pure real(dp) function stdlib_dasum(n,dx,incx) !! DASUM takes the sum of the absolute values. ! -- reference blas level1 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments integer(ilp), intent(in) :: incx, n ! Array Arguments real(dp), intent(in) :: dx(*) ! ===================================================================== ! Local Scalars real(dp) :: dtemp integer(ilp) :: i, m, mp1, nincx ! Intrinsic Functions intrinsic :: abs,mod stdlib_dasum = zero dtemp = zero if (n<=0 .or. incx<=0) return if (incx==1) then ! code for increment equal to 1 ! clean-up loop m = mod(n,6) if (m/=0) then do i = 1,m dtemp = dtemp + abs(dx(i)) end do if (n<6) then stdlib_dasum = dtemp return end if end if mp1 = m + 1 do i = mp1,n,6 dtemp = dtemp + abs(dx(i)) + abs(dx(i+1)) +abs(dx(i+2)) + abs(dx(i+3)) +abs(dx(i+& 4)) + abs(dx(i+5)) end do else ! code for increment not equal to 1 nincx = n*incx do i = 1,nincx,incx dtemp = dtemp + abs(dx(i)) end do end if stdlib_dasum = dtemp return end function stdlib_dasum pure subroutine stdlib_daxpy(n,da,dx,incx,dy,incy) !! DAXPY constant times a vector plus a vector. !! uses unrolled loops for increments equal to one. ! -- reference blas level1 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: da integer(ilp), intent(in) :: incx, incy, n ! Array Arguments real(dp), intent(in) :: dx(*) real(dp), intent(inout) :: dy(*) ! ===================================================================== ! Local Scalars integer(ilp) :: i, ix, iy, m, mp1 ! Intrinsic Functions intrinsic :: mod if (n<=0) return if (da==0.0_dp) return if (incx==1 .and. incy==1) then ! code for both increments equal to 1 ! clean-up loop m = mod(n,4) if (m/=0) then do i = 1,m dy(i) = dy(i) + da*dx(i) end do end if if (n<4) return mp1 = m + 1 do i = mp1,n,4 dy(i) = dy(i) + da*dx(i) dy(i+1) = dy(i+1) + da*dx(i+1) dy(i+2) = dy(i+2) + da*dx(i+2) dy(i+3) = dy(i+3) + da*dx(i+3) end do else ! code for unequal increments or equal increments ! not equal to 1 ix = 1 iy = 1 if (incx<0) ix = (-n+1)*incx + 1 if (incy<0) iy = (-n+1)*incy + 1 do i = 1,n dy(iy) = dy(iy) + da*dx(ix) ix = ix + incx iy = iy + incy end do end if return end subroutine stdlib_daxpy pure subroutine stdlib_dcopy(n,dx,incx,dy,incy) !! DCOPY copies a vector, x, to a vector, y. !! uses unrolled loops for increments equal to 1. ! -- reference blas level1 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments integer(ilp), intent(in) :: incx, incy, n ! Array Arguments real(dp), intent(in) :: dx(*) real(dp), intent(out) :: dy(*) ! ===================================================================== ! Local Scalars integer(ilp) :: i, ix, iy, m, mp1 ! Intrinsic Functions intrinsic :: mod if (n<=0) return if (incx==1 .and. incy==1) then ! code for both increments equal to 1 ! clean-up loop m = mod(n,7) if (m/=0) then do i = 1,m dy(i) = dx(i) end do if (n<7) return end if mp1 = m + 1 do i = mp1,n,7 dy(i) = dx(i) dy(i+1) = dx(i+1) dy(i+2) = dx(i+2) dy(i+3) = dx(i+3) dy(i+4) = dx(i+4) dy(i+5) = dx(i+5) dy(i+6) = dx(i+6) end do else ! code for unequal increments or equal increments ! not equal to 1 ix = 1 iy = 1 if (incx<0) ix = (-n+1)*incx + 1 if (incy<0) iy = (-n+1)*incy + 1 do i = 1,n dy(iy) = dx(ix) ix = ix + incx iy = iy + incy end do end if return end subroutine stdlib_dcopy pure real(dp) function stdlib_ddot(n,dx,incx,dy,incy) !! DDOT forms the dot product of two vectors. !! uses unrolled loops for increments equal to one. ! -- reference blas level1 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments integer(ilp), intent(in) :: incx, incy, n ! Array Arguments real(dp), intent(in) :: dx(*), dy(*) ! ===================================================================== ! Local Scalars real(dp) :: dtemp integer(ilp) :: i, ix, iy, m, mp1 ! Intrinsic Functions intrinsic :: mod stdlib_ddot = zero dtemp = zero if (n<=0) return if (incx==1 .and. incy==1) then ! code for both increments equal to 1 ! clean-up loop m = mod(n,5) if (m/=0) then do i = 1,m dtemp = dtemp + dx(i)*dy(i) end do if (n<5) then stdlib_ddot=dtemp return end if end if mp1 = m + 1 do i = mp1,n,5 dtemp = dtemp + dx(i)*dy(i) + dx(i+1)*dy(i+1) +dx(i+2)*dy(i+2) + dx(i+3)*dy(i+3) + & dx(i+4)*dy(i+4) end do else ! code for unequal increments or equal increments ! not equal to 1 ix = 1 iy = 1 if (incx<0) ix = (-n+1)*incx + 1 if (incy<0) iy = (-n+1)*incy + 1 do i = 1,n dtemp = dtemp + dx(ix)*dy(iy) ix = ix + incx iy = iy + incy end do end if stdlib_ddot = dtemp return end function stdlib_ddot pure subroutine stdlib_dgbmv(trans,m,n,kl,ku,alpha,a,lda,x,incx,beta,y,incy) !! DGBMV performs one of the matrix-vector operations !! y := alpha*A*x + beta*y, or y := alpha*A**T*x + beta*y, !! where alpha and beta are scalars, x and y are vectors and A is an !! m by n band matrix, with kl sub-diagonals and ku super-diagonals. ! -- reference blas level2 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha, beta integer(ilp), intent(in) :: incx, incy, kl, ku, lda, m, n character, intent(in) :: trans ! Array Arguments real(dp), intent(in) :: a(lda,*), x(*) real(dp), intent(inout) :: y(*) ! ===================================================================== ! Local Scalars real(dp) :: temp integer(ilp) :: i, info, ix, iy, j, jx, jy, k, kup1, kx, ky, lenx, leny ! Intrinsic Functions intrinsic :: max,min ! test the input parameters. info = 0 if (.not.stdlib_lsame(trans,'N') .and. .not.stdlib_lsame(trans,'T') & .and..not.stdlib_lsame(trans,'C')) then info = 1 else if (m<0) then info = 2 else if (n<0) then info = 3 else if (kl<0) then info = 4 else if (ku<0) then info = 5 else if (lda< (kl+ku+1)) then info = 8 else if (incx==0) then info = 10 else if (incy==0) then info = 13 end if if (info/=0) then call stdlib_xerbla('DGBMV ',info) return end if ! quick return if possible. if ((m==0) .or. (n==0) .or.((alpha==zero).and. (beta==one))) return ! set lenx and leny, the lengths of the vectors x and y, and set ! up the start points in x and y. if (stdlib_lsame(trans,'N')) then lenx = n leny = m else lenx = m leny = n end if if (incx>0) then kx = 1 else kx = 1 - (lenx-1)*incx end if if (incy>0) then ky = 1 else ky = 1 - (leny-1)*incy end if ! start the operations. in this version the elements of a are ! accessed sequentially with one pass through the band part of a. ! first form y := beta*y. if (beta/=one) then if (incy==1) then if (beta==zero) then do i = 1,leny y(i) = zero end do else do i = 1,leny y(i) = beta*y(i) end do end if else iy = ky if (beta==zero) then do i = 1,leny y(iy) = zero iy = iy + incy end do else do i = 1,leny y(iy) = beta*y(iy) iy = iy + incy end do end if end if end if if (alpha==zero) return kup1 = ku + 1 if (stdlib_lsame(trans,'N')) then ! form y := alpha*a*x + y. jx = kx if (incy==1) then do j = 1,n temp = alpha*x(jx) k = kup1 - j do i = max(1,j-ku),min(m,j+kl) y(i) = y(i) + temp*a(k+i,j) end do jx = jx + incx end do else do j = 1,n temp = alpha*x(jx) iy = ky k = kup1 - j do i = max(1,j-ku),min(m,j+kl) y(iy) = y(iy) + temp*a(k+i,j) iy = iy + incy end do jx = jx + incx if (j>ku) ky = ky + incy end do end if else ! form y := alpha*a**t*x + y. jy = ky if (incx==1) then do j = 1,n temp = zero k = kup1 - j do i = max(1,j-ku),min(m,j+kl) temp = temp + a(k+i,j)*x(i) end do y(jy) = y(jy) + alpha*temp jy = jy + incy end do else do j = 1,n temp = zero ix = kx k = kup1 - j do i = max(1,j-ku),min(m,j+kl) temp = temp + a(k+i,j)*x(ix) ix = ix + incx end do y(jy) = y(jy) + alpha*temp jy = jy + incy if (j>ku) kx = kx + incx end do end if end if return end subroutine stdlib_dgbmv pure subroutine stdlib_dgemm(transa,transb,m,n,k,alpha,a,lda,b,ldb,beta,c,ldc) !! DGEMM performs one of the matrix-matrix operations !! C := alpha*op( A )*op( B ) + beta*C, !! where op( X ) is one of !! op( X ) = X or op( X ) = X**T, !! alpha and beta are scalars, and A, B and C are matrices, with op( A ) !! an m by k matrix, op( B ) a k by n matrix and C an m by n matrix. ! -- reference blas level3 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha, beta integer(ilp), intent(in) :: k, lda, ldb, ldc, m, n character, intent(in) :: transa, transb ! Array Arguments real(dp), intent(in) :: a(lda,*), b(ldb,*) real(dp), intent(inout) :: c(ldc,*) ! ===================================================================== ! Intrinsic Functions intrinsic :: max ! Local Scalars real(dp) :: temp integer(ilp) :: i, info, j, l, nrowa, nrowb logical(lk) :: nota, notb ! set nota and notb as true if a and b respectively are not ! transposed and set nrowa and nrowb as the number of rows of a ! and b respectively. nota = stdlib_lsame(transa,'N') notb = stdlib_lsame(transb,'N') if (nota) then nrowa = m else nrowa = k end if if (notb) then nrowb = k else nrowb = n end if ! test the input parameters. info = 0 if ((.not.nota) .and. (.not.stdlib_lsame(transa,'C')) .and.(.not.stdlib_lsame(transa,& 'T'))) then info = 1 else if ((.not.notb) .and. (.not.stdlib_lsame(transb,'C')) .and.(.not.stdlib_lsame(& transb,'T'))) then info = 2 else if (m<0) then info = 3 else if (n<0) then info = 4 else if (k<0) then info = 5 else if (lda<max(1,nrowa)) then info = 8 else if (ldb<max(1,nrowb)) then info = 10 else if (ldc<max(1,m)) then info = 13 end if if (info/=0) then call stdlib_xerbla('DGEMM ',info) return end if ! quick return if possible. if ((m==0) .or. (n==0) .or.(((alpha==zero).or. (k==0)).and. (beta==one))) & return ! and if alpha.eq.zero. if (alpha==zero) then if (beta==zero) then do j = 1,n do i = 1,m c(i,j) = zero end do end do else do j = 1,n do i = 1,m c(i,j) = beta*c(i,j) end do end do end if return end if ! start the operations. if (notb) then if (nota) then ! form c := alpha*a*b + beta*c. do j = 1,n if (beta==zero) then do i = 1,m c(i,j) = zero end do else if (beta/=one) then do i = 1,m c(i,j) = beta*c(i,j) end do end if do l = 1,k temp = alpha*b(l,j) do i = 1,m c(i,j) = c(i,j) + temp*a(i,l) end do end do end do else ! form c := alpha*a**t*b + beta*c do j = 1,n do i = 1,m temp = zero do l = 1,k temp = temp + a(l,i)*b(l,j) end do if (beta==zero) then c(i,j) = alpha*temp else c(i,j) = alpha*temp + beta*c(i,j) end if end do end do end if else if (nota) then ! form c := alpha*a*b**t + beta*c do j = 1,n if (beta==zero) then do i = 1,m c(i,j) = zero end do else if (beta/=one) then do i = 1,m c(i,j) = beta*c(i,j) end do end if do l = 1,k temp = alpha*b(j,l) do i = 1,m c(i,j) = c(i,j) + temp*a(i,l) end do end do end do else ! form c := alpha*a**t*b**t + beta*c do j = 1,n do i = 1,m temp = zero do l = 1,k temp = temp + a(l,i)*b(j,l) end do if (beta==zero) then c(i,j) = alpha*temp else c(i,j) = alpha*temp + beta*c(i,j) end if end do end do end if end if return end subroutine stdlib_dgemm pure subroutine stdlib_dgemv(trans,m,n,alpha,a,lda,x,incx,beta,y,incy) !! DGEMV performs one of the matrix-vector operations !! y := alpha*A*x + beta*y, or y := alpha*A**T*x + beta*y, !! where alpha and beta are scalars, x and y are vectors and A is an !! m by n matrix. ! -- reference blas level2 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha, beta integer(ilp), intent(in) :: incx, incy, lda, m, n character, intent(in) :: trans ! Array Arguments real(dp), intent(in) :: a(lda,*), x(*) real(dp), intent(inout) :: y(*) ! ===================================================================== ! Local Scalars real(dp) :: temp integer(ilp) :: i, info, ix, iy, j, jx, jy, kx, ky, lenx, leny ! Intrinsic Functions intrinsic :: max ! test the input parameters. info = 0 if (.not.stdlib_lsame(trans,'N') .and. .not.stdlib_lsame(trans,'T') & .and..not.stdlib_lsame(trans,'C')) then info = 1 else if (m<0) then info = 2 else if (n<0) then info = 3 else if (lda<max(1,m)) then info = 6 else if (incx==0) then info = 8 else if (incy==0) then info = 11 end if if (info/=0) then call stdlib_xerbla('DGEMV ',info) return end if ! quick return if possible. if ((m==0) .or. (n==0) .or.((alpha==zero).and. (beta==one))) return ! set lenx and leny, the lengths of the vectors x and y, and set ! up the start points in x and y. if (stdlib_lsame(trans,'N')) then lenx = n leny = m else lenx = m leny = n end if if (incx>0) then kx = 1 else kx = 1 - (lenx-1)*incx end if if (incy>0) then ky = 1 else ky = 1 - (leny-1)*incy end if ! start the operations. in this version the elements of a are ! accessed sequentially with one pass through a. ! first form y := beta*y. if (beta/=one) then if (incy==1) then if (beta==zero) then do i = 1,leny y(i) = zero end do else do i = 1,leny y(i) = beta*y(i) end do end if else iy = ky if (beta==zero) then do i = 1,leny y(iy) = zero iy = iy + incy end do else do i = 1,leny y(iy) = beta*y(iy) iy = iy + incy end do end if end if end if if (alpha==zero) return if (stdlib_lsame(trans,'N')) then ! form y := alpha*a*x + y. jx = kx if (incy==1) then do j = 1,n temp = alpha*x(jx) do i = 1,m y(i) = y(i) + temp*a(i,j) end do jx = jx + incx end do else do j = 1,n temp = alpha*x(jx) iy = ky do i = 1,m y(iy) = y(iy) + temp*a(i,j) iy = iy + incy end do jx = jx + incx end do end if else ! form y := alpha*a**t*x + y. jy = ky if (incx==1) then do j = 1,n temp = zero do i = 1,m temp = temp + a(i,j)*x(i) end do y(jy) = y(jy) + alpha*temp jy = jy + incy end do else do j = 1,n temp = zero ix = kx do i = 1,m temp = temp + a(i,j)*x(ix) ix = ix + incx end do y(jy) = y(jy) + alpha*temp jy = jy + incy end do end if end if return end subroutine stdlib_dgemv pure subroutine stdlib_dger(m,n,alpha,x,incx,y,incy,a,lda) !! DGER performs the rank 1 operation !! A := alpha*x*y**T + A, !! where alpha is a scalar, x is an m element vector, y is an n element !! vector and A is an m by n matrix. ! -- reference blas level2 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha integer(ilp), intent(in) :: incx, incy, lda, m, n ! Array Arguments real(dp), intent(inout) :: a(lda,*) real(dp), intent(in) :: x(*), y(*) ! ===================================================================== ! Local Scalars real(dp) :: temp integer(ilp) :: i, info, ix, j, jy, kx ! Intrinsic Functions intrinsic :: max ! test the input parameters. info = 0 if (m<0) then info = 1 else if (n<0) then info = 2 else if (incx==0) then info = 5 else if (incy==0) then info = 7 else if (lda<max(1,m)) then info = 9 end if if (info/=0) then call stdlib_xerbla('DGER ',info) return end if ! quick return if possible. if ((m==0) .or. (n==0) .or. (alpha==zero)) return ! start the operations. in this version the elements of a are ! accessed sequentially with one pass through a. if (incy>0) then jy = 1 else jy = 1 - (n-1)*incy end if if (incx==1) then do j = 1,n if (y(jy)/=zero) then temp = alpha*y(jy) do i = 1,m a(i,j) = a(i,j) + x(i)*temp end do end if jy = jy + incy end do else if (incx>0) then kx = 1 else kx = 1 - (m-1)*incx end if do j = 1,n if (y(jy)/=zero) then temp = alpha*y(jy) ix = kx do i = 1,m a(i,j) = a(i,j) + x(ix)*temp ix = ix + incx end do end if jy = jy + incy end do end if return end subroutine stdlib_dger pure function stdlib_dnrm2( n, x, incx ) !! DNRM2 returns the euclidean norm of a vector via the function !! name, so that !! DNRM2 := sqrt( x'*x ) real(dp) :: stdlib_dnrm2 ! -- reference blas level1 routine (version 3.9.1_dp) -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! march 2021 ! Constants integer, parameter :: wp = kind(1._dp) real(dp), parameter :: maxn = huge(0.0_dp) ! .. blue's scaling constants .. ! Scalar Arguments integer(ilp), intent(in) :: incx, n ! Array Arguments real(dp), intent(in) :: x(*) ! Local Scalars integer(ilp) :: i, ix logical(lk) :: notbig real(dp) :: abig, amed, asml, ax, scl, sumsq, ymax, ymin ! quick return if possible stdlib_dnrm2 = zero if( n <= 0 ) return scl = one sumsq = zero ! compute the sum of squares in 3 accumulators: ! abig -- sums of squares scaled down to avoid overflow ! asml -- sums of squares scaled up to avoid underflow ! amed -- sums of squares that do not require scaling ! the thresholds and multipliers are ! tbig -- values bigger than this are scaled down by sbig ! tsml -- values smaller than this are scaled up by ssml notbig = .true. asml = zero amed = zero abig = zero ix = 1 if( incx < 0 ) ix = 1 - (n-1)*incx do i = 1, n ax = abs(x(ix)) if (ax > tbig) then abig = abig + (ax*sbig)**2 notbig = .false. else if (ax < tsml) then if (notbig) asml = asml + (ax*ssml)**2 else amed = amed + ax**2 end if ix = ix + incx end do ! combine abig and amed or amed and asml if more than one ! accumulator was used. if (abig > zero) then ! combine abig and amed if abig > 0. if ( (amed > zero) .or. (amed > maxn) .or. (amed /= amed) ) then abig = abig + (amed*sbig)*sbig end if scl = one / sbig sumsq = abig else if (asml > zero) then ! combine amed and asml if asml > 0. if ( (amed > zero) .or. (amed > maxn) .or. (amed /= amed) ) then amed = sqrt(amed) asml = sqrt(asml) / ssml if (asml > amed) then ymin = amed ymax = asml else ymin = asml ymax = amed end if scl = one sumsq = ymax**2*( one + (ymin/ymax)**2 ) else scl = one / ssml sumsq = asml end if else ! otherwise all values are mid-range scl = one sumsq = amed end if stdlib_dnrm2 = scl*sqrt( sumsq ) return end function stdlib_dnrm2 pure subroutine stdlib_drot(n,dx,incx,dy,incy,c,s) !! DROT applies a plane rotation. ! -- reference blas level1 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: c, s integer(ilp), intent(in) :: incx, incy, n ! Array Arguments real(dp), intent(inout) :: dx(*), dy(*) ! ===================================================================== ! Local Scalars real(dp) :: dtemp integer(ilp) :: i, ix, iy if (n<=0) return if (incx==1 .and. incy==1) then ! code for both increments equal to 1 do i = 1,n dtemp = c*dx(i) + s*dy(i) dy(i) = c*dy(i) - s*dx(i) dx(i) = dtemp end do else ! code for unequal increments or equal increments not equal ! to 1 ix = 1 iy = 1 if (incx<0) ix = (-n+1)*incx + 1 if (incy<0) iy = (-n+1)*incy + 1 do i = 1,n dtemp = c*dx(ix) + s*dy(iy) dy(iy) = c*dy(iy) - s*dx(ix) dx(ix) = dtemp ix = ix + incx iy = iy + incy end do end if return end subroutine stdlib_drot pure subroutine stdlib_drotg( a, b, c, s ) !! The computation uses the formulas !! sigma = sgn(a) if |a| > |b| !! = sgn(b) if |b| >= |a| !! r = sigma*sqrt( a**2 + b**2 ) !! c = 1; s = 0 if r = 0 !! c = a/r; s = b/r if r != 0 !! The subroutine also computes !! z = s if |a| > |b|, !! = 1/c if |b| >= |a| and c != 0 !! = 1 if c = 0 !! This allows c and s to be reconstructed from z as follows: !! If z = 1, set c = 0, s = 1. !! If |z| < 1, set c = sqrt(1 - z**2) and s = z. !! If |z| > 1, set c = 1/z and s = sqrt( 1 - c**2). ! -- reference blas level1 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Constants integer, parameter :: wp = kind(1._dp) ! Scaling Constants ! Scalar Arguments real(dp), intent(inout) :: a, b real(dp), intent(out) :: c, s ! Local Scalars real(dp) :: anorm, bnorm, scl, sigma, r, z anorm = abs(a) bnorm = abs(b) if( bnorm == zero ) then c = one s = zero b = zero else if( anorm == zero ) then c = zero s = one a = b b = one else scl = min( safmax, max( safmin, anorm, bnorm ) ) if( anorm > bnorm ) then sigma = sign(one,a) else sigma = sign(one,b) end if r = sigma*( scl*sqrt((a/scl)**2 + (b/scl)**2) ) c = a/r s = b/r if( anorm > bnorm ) then z = s else if( c /= zero ) then z = one/c else z = one end if a = r b = z end if return end subroutine stdlib_drotg pure subroutine stdlib_drotm(n,dx,incx,dy,incy,dparam) !! DROTM applies the modified Givens transformation, \(H\), to the 2-by-N matrix !! $$ \left[ \begin{array}{c}DX^T\\DY^T\\ \end{array} \right], $$ !! where \(^T\) indicates transpose. The elements of \(DX\) are in !! DX(LX+I*INCX), I = 0:N-1, where LX = 1 if INCX >= 0, else LX = (-INCX)*N, !! and similarly for DY using LY and INCY. !! With DPARAM(1)=DFLAG, \(H\) has one of the following forms: !! $$ H=\underbrace{\begin{bmatrix}DH_{11} & DH_{12}\\DH_{21} & DH_{22}\end{bmatrix}}_{DFLAG=-1}, !! \underbrace{\begin{bmatrix}1 & DH_{12}\\DH_{21} & 1\end{bmatrix}}_{DFLAG=0}, !! \underbrace{\begin{bmatrix}DH_{11} & 1\\-1 & DH_{22}\end{bmatrix}}_{DFLAG=1}, !! \underbrace{\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}}_{DFLAG=-2}. $$ !! See DROTMG for a description of data storage in DPARAM. ! -- reference blas level1 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments integer(ilp), intent(in) :: incx, incy, n ! Array Arguments real(dp), intent(in) :: dparam(5) real(dp), intent(inout) :: dx(*), dy(*) ! ===================================================================== ! Local Scalars real(dp) :: dflag, dh11, dh12, dh21, dh22, two, w, z, zero integer(ilp) :: i, kx, ky, nsteps ! Data Statements zero = 0.0_dp two = 2.0_dp dflag = dparam(1) if (n<=0 .or. (dflag+two==zero)) return if (incx==incy.and.incx>0) then nsteps = n*incx if (dflag<zero) then dh11 = dparam(2) dh12 = dparam(4) dh21 = dparam(3) dh22 = dparam(5) do i = 1,nsteps,incx w = dx(i) z = dy(i) dx(i) = w*dh11 + z*dh12 dy(i) = w*dh21 + z*dh22 end do else if (dflag==zero) then dh12 = dparam(4) dh21 = dparam(3) do i = 1,nsteps,incx w = dx(i) z = dy(i) dx(i) = w + z*dh12 dy(i) = w*dh21 + z end do else dh11 = dparam(2) dh22 = dparam(5) do i = 1,nsteps,incx w = dx(i) z = dy(i) dx(i) = w*dh11 + z dy(i) = -w + dh22*z end do end if else kx = 1 ky = 1 if (incx<0) kx = 1 + (1-n)*incx if (incy<0) ky = 1 + (1-n)*incy if (dflag<zero) then dh11 = dparam(2) dh12 = dparam(4) dh21 = dparam(3) dh22 = dparam(5) do i = 1,n w = dx(kx) z = dy(ky) dx(kx) = w*dh11 + z*dh12 dy(ky) = w*dh21 + z*dh22 kx = kx + incx ky = ky + incy end do else if (dflag==zero) then dh12 = dparam(4) dh21 = dparam(3) do i = 1,n w = dx(kx) z = dy(ky) dx(kx) = w + z*dh12 dy(ky) = w*dh21 + z kx = kx + incx ky = ky + incy end do else dh11 = dparam(2) dh22 = dparam(5) do i = 1,n w = dx(kx) z = dy(ky) dx(kx) = w*dh11 + z dy(ky) = -w + dh22*z kx = kx + incx ky = ky + incy end do end if end if return end subroutine stdlib_drotm pure subroutine stdlib_drotmg(dd1,dd2,dx1,dy1,dparam) !! DROTMG Constructs the modified Givens transformation matrix \(H\) which zeros the !! second component of the 2-vector !! $$ \left[ {\sqrt{DD_1}\cdot DX_1,\sqrt{DD_2}\cdot DY_2} \right]^T. $$ !! With DPARAM(1)=DFLAG, \(H\) has one of the following forms: !! $$ H=\underbrace{\begin{bmatrix}DH_{11} & DH_{12}\\DH_{21} & DH_{22}\end{bmatrix}}_{DFLAG=-1}, !! \underbrace{\begin{bmatrix}1 & DH_{12}\\DH_{21} & 1\end{bmatrix}}_{DFLAG=0}, !! \underbrace{\begin{bmatrix}DH_{11} & 1\\-1 & DH_{22}\end{bmatrix}}_{DFLAG=1}, !! \underbrace{\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}}_{DFLAG=-2}. $$ !! Locations 2-4 of DPARAM contain DH11, DH21, DH12 and DH22 respectively. !! (Values of 1.0, -1.0, or 0.0 implied by the value of DPARAM(1) are not stored in DPARAM.) !! The values of parameters GAMSQ and RGAMSQ may be inexact. This is OK as they are only !! used for testing the size of DD1 and DD2. All actual scaling of data is done using GAM. ! -- reference blas level1 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(inout) :: dd1, dd2, dx1 real(dp), intent(in) :: dy1 ! Array Arguments real(dp), intent(out) :: dparam(5) ! ===================================================================== ! Local Scalars real(dp) :: dflag, dh11, dh12, dh21, dh22, dp1, dp2, dq1, dq2, dtemp, du, gam, gamsq, & one, rgamsq, two, zero ! Intrinsic Functions intrinsic :: abs ! Data Statements zero = 0.0_dp one = 1.0_dp two = 2.0_dp gam = 4096.0_dp gamsq = 16777216.0_dp rgamsq = 5.9604645e-8_dp if (dd1<zero) then ! go zero-h-d-and-dx1.. dflag = -one dh11 = zero dh12 = zero dh21 = zero dh22 = zero dd1 = zero dd2 = zero dx1 = zero else ! case-dd1-nonnegative dp2 = dd2*dy1 if (dp2==zero) then dflag = -two dparam(1) = dflag return end if ! regular-case.. dp1 = dd1*dx1 dq2 = dp2*dy1 dq1 = dp1*dx1 if (abs(dq1)>abs(dq2)) then dh21 = -dy1/dx1 dh12 = dp2/dp1 du = one - dh12*dh21 if (du>zero) then dflag = zero dd1 = dd1/du dd2 = dd2/du dx1 = dx1*du else ! this code path if here for safety. we do not expect this ! condition to ever hold except in edge cases with rounding ! errors. see doi: 10.1145/355841.355847 dflag = -one dh11 = zero dh12 = zero dh21 = zero dh22 = zero dd1 = zero dd2 = zero dx1 = zero end if else if (dq2<zero) then ! go zero-h-d-and-dx1.. dflag = -one dh11 = zero dh12 = zero dh21 = zero dh22 = zero dd1 = zero dd2 = zero dx1 = zero else dflag = one dh11 = dp1/dp2 dh22 = dx1/dy1 du = one + dh11*dh22 dtemp = dd2/du dd2 = dd1/du dd1 = dtemp dx1 = dy1*du end if end if ! procedure..scale-check if (dd1/=zero) then do while ((dd1<=rgamsq) .or. (dd1>=gamsq)) if (dflag==zero) then dh11 = one dh22 = one dflag = -one else dh21 = -one dh12 = one dflag = -one end if if (dd1<=rgamsq) then dd1 = dd1*gam**2 dx1 = dx1/gam dh11 = dh11/gam dh12 = dh12/gam else dd1 = dd1/gam**2 dx1 = dx1*gam dh11 = dh11*gam dh12 = dh12*gam end if enddo end if if (dd2/=zero) then do while ( (abs(dd2)<=rgamsq) .or. (abs(dd2)>=gamsq) ) if (dflag==zero) then dh11 = one dh22 = one dflag = -one else dh21 = -one dh12 = one dflag = -one end if if (abs(dd2)<=rgamsq) then dd2 = dd2*gam**2 dh21 = dh21/gam dh22 = dh22/gam else dd2 = dd2/gam**2 dh21 = dh21*gam dh22 = dh22*gam end if end do end if end if if (dflag<zero) then dparam(2) = dh11 dparam(3) = dh21 dparam(4) = dh12 dparam(5) = dh22 else if (dflag==zero) then dparam(3) = dh21 dparam(4) = dh12 else dparam(2) = dh11 dparam(5) = dh22 end if dparam(1) = dflag return end subroutine stdlib_drotmg pure subroutine stdlib_dsbmv(uplo,n,k,alpha,a,lda,x,incx,beta,y,incy) !! DSBMV performs the matrix-vector operation !! y := alpha*A*x + beta*y, !! where alpha and beta are scalars, x and y are n element vectors and !! A is an n by n symmetric band matrix, with k super-diagonals. ! -- reference blas level2 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha, beta integer(ilp), intent(in) :: incx, incy, k, lda, n character, intent(in) :: uplo ! Array Arguments real(dp), intent(in) :: a(lda,*), x(*) real(dp), intent(inout) :: y(*) ! ===================================================================== ! Local Scalars real(dp) :: temp1, temp2 integer(ilp) :: i, info, ix, iy, j, jx, jy, kplus1, kx, ky, l ! Intrinsic Functions intrinsic :: max,min ! test the input parameters. info = 0 if (.not.stdlib_lsame(uplo,'U') .and. .not.stdlib_lsame(uplo,'L')) then info = 1 else if (n<0) then info = 2 else if (k<0) then info = 3 else if (lda< (k+1)) then info = 6 else if (incx==0) then info = 8 else if (incy==0) then info = 11 end if if (info/=0) then call stdlib_xerbla('DSBMV ',info) return end if ! quick return if possible. if ((n==0) .or. ((alpha==zero).and. (beta==one))) return ! set up the start points in x and y. if (incx>0) then kx = 1 else kx = 1 - (n-1)*incx end if if (incy>0) then ky = 1 else ky = 1 - (n-1)*incy end if ! start the operations. in this version the elements of the array a ! are accessed sequentially with one pass through a. ! first form y := beta*y. if (beta/=one) then if (incy==1) then if (beta==zero) then do i = 1,n y(i) = zero end do else do i = 1,n y(i) = beta*y(i) end do end if else iy = ky if (beta==zero) then do i = 1,n y(iy) = zero iy = iy + incy end do else do i = 1,n y(iy) = beta*y(iy) iy = iy + incy end do end if end if end if if (alpha==zero) return if (stdlib_lsame(uplo,'U')) then ! form y when upper triangle of a is stored. kplus1 = k + 1 if ((incx==1) .and. (incy==1)) then do j = 1,n temp1 = alpha*x(j) temp2 = zero l = kplus1 - j do i = max(1,j-k),j - 1 y(i) = y(i) + temp1*a(l+i,j) temp2 = temp2 + a(l+i,j)*x(i) end do y(j) = y(j) + temp1*a(kplus1,j) + alpha*temp2 end do else jx = kx jy = ky do j = 1,n temp1 = alpha*x(jx) temp2 = zero ix = kx iy = ky l = kplus1 - j do i = max(1,j-k),j - 1 y(iy) = y(iy) + temp1*a(l+i,j) temp2 = temp2 + a(l+i,j)*x(ix) ix = ix + incx iy = iy + incy end do y(jy) = y(jy) + temp1*a(kplus1,j) + alpha*temp2 jx = jx + incx jy = jy + incy if (j>k) then kx = kx + incx ky = ky + incy end if end do end if else ! form y when lower triangle of a is stored. if ((incx==1) .and. (incy==1)) then do j = 1,n temp1 = alpha*x(j) temp2 = zero y(j) = y(j) + temp1*a(1,j) l = 1 - j do i = j + 1,min(n,j+k) y(i) = y(i) + temp1*a(l+i,j) temp2 = temp2 + a(l+i,j)*x(i) end do y(j) = y(j) + alpha*temp2 end do else jx = kx jy = ky do j = 1,n temp1 = alpha*x(jx) temp2 = zero y(jy) = y(jy) + temp1*a(1,j) l = 1 - j ix = jx iy = jy do i = j + 1,min(n,j+k) ix = ix + incx iy = iy + incy y(iy) = y(iy) + temp1*a(l+i,j) temp2 = temp2 + a(l+i,j)*x(ix) end do y(jy) = y(jy) + alpha*temp2 jx = jx + incx jy = jy + incy end do end if end if return end subroutine stdlib_dsbmv pure subroutine stdlib_dscal(n,da,dx,incx) !! DSCAL scales a vector by a constant. !! uses unrolled loops for increment equal to 1. ! -- reference blas level1 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: da integer(ilp), intent(in) :: incx, n ! Array Arguments real(dp), intent(inout) :: dx(*) ! ===================================================================== ! Local Scalars integer(ilp) :: i, m, mp1, nincx ! Intrinsic Functions intrinsic :: mod if (n<=0 .or. incx<=0) return if (incx==1) then ! code for increment equal to 1 ! clean-up loop m = mod(n,5) if (m/=0) then do i = 1,m dx(i) = da*dx(i) end do if (n<5) return end if mp1 = m + 1 do i = mp1,n,5 dx(i) = da*dx(i) dx(i+1) = da*dx(i+1) dx(i+2) = da*dx(i+2) dx(i+3) = da*dx(i+3) dx(i+4) = da*dx(i+4) end do else ! code for increment not equal to 1 nincx = n*incx do i = 1,nincx,incx dx(i) = da*dx(i) end do end if return end subroutine stdlib_dscal pure real(dp) function stdlib_dsdot(n,sx,incx,sy,incy) !! Compute the inner product of two vectors with extended !! precision accumulation and result. !! Returns D.P. dot product accumulated in D.P., for S.P. SX and SY !! DSDOT = sum for I = 0 to N-1 of SX(LX+I*INCX) * SY(LY+I*INCY), !! where LX = 1 if INCX >= 0, else LX = 1+(1-N)*INCX, and LY is !! defined in a similar way using INCY. ! -- reference blas level1 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments integer(ilp), intent(in) :: incx, incy, n ! Array Arguments real(sp), intent(in) :: sx(*), sy(*) ! authors: ! ======== ! lawson, c. l., (jpl), hanson, r. j., (snla), ! kincaid, d. r., (u. of texas), krogh, f. t., (jpl) ! ===================================================================== ! Local Scalars integer(ilp) :: i, kx, ky, ns ! Intrinsic Functions intrinsic :: real stdlib_dsdot = zero if (n<=0) return if (incx==incy .and. incx>0) then ! code for equal, positive, non-unit increments. ns = n*incx do i = 1,ns,incx stdlib_dsdot = stdlib_dsdot + real(sx(i),KIND=dp)*real(sy(i),KIND=dp) end do else ! code for unequal or nonpositive increments. kx = 1 ky = 1 if (incx<0) kx = 1 + (1-n)*incx if (incy<0) ky = 1 + (1-n)*incy do i = 1,n stdlib_dsdot = stdlib_dsdot + real(sx(kx),KIND=dp)*real(sy(ky),KIND=dp) kx = kx + incx ky = ky + incy end do end if return end function stdlib_dsdot pure subroutine stdlib_dspmv(uplo,n,alpha,ap,x,incx,beta,y,incy) !! DSPMV performs the matrix-vector operation !! y := alpha*A*x + beta*y, !! where alpha and beta are scalars, x and y are n element vectors and !! A is an n by n symmetric matrix, supplied in packed form. ! -- reference blas level2 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha, beta integer(ilp), intent(in) :: incx, incy, n character, intent(in) :: uplo ! Array Arguments real(dp), intent(in) :: ap(*), x(*) real(dp), intent(inout) :: y(*) ! ===================================================================== ! Local Scalars real(dp) :: temp1, temp2 integer(ilp) :: i, info, ix, iy, j, jx, jy, k, kk, kx, ky ! test the input parameters. info = 0 if (.not.stdlib_lsame(uplo,'U') .and. .not.stdlib_lsame(uplo,'L')) then info = 1 else if (n<0) then info = 2 else if (incx==0) then info = 6 else if (incy==0) then info = 9 end if if (info/=0) then call stdlib_xerbla('DSPMV ',info) return end if ! quick return if possible. if ((n==0) .or. ((alpha==zero).and. (beta==one))) return ! set up the start points in x and y. if (incx>0) then kx = 1 else kx = 1 - (n-1)*incx end if if (incy>0) then ky = 1 else ky = 1 - (n-1)*incy end if ! start the operations. in this version the elements of the array ap ! are accessed sequentially with one pass through ap. ! first form y := beta*y. if (beta/=one) then if (incy==1) then if (beta==zero) then do i = 1,n y(i) = zero end do else do i = 1,n y(i) = beta*y(i) end do end if else iy = ky if (beta==zero) then do i = 1,n y(iy) = zero iy = iy + incy end do else do i = 1,n y(iy) = beta*y(iy) iy = iy + incy end do end if end if end if if (alpha==zero) return kk = 1 if (stdlib_lsame(uplo,'U')) then ! form y when ap contains the upper triangle. if ((incx==1) .and. (incy==1)) then do j = 1,n temp1 = alpha*x(j) temp2 = zero k = kk do i = 1,j - 1 y(i) = y(i) + temp1*ap(k) temp2 = temp2 + ap(k)*x(i) k = k + 1 end do y(j) = y(j) + temp1*ap(kk+j-1) + alpha*temp2 kk = kk + j end do else jx = kx jy = ky do j = 1,n temp1 = alpha*x(jx) temp2 = zero ix = kx iy = ky do k = kk,kk + j - 2 y(iy) = y(iy) + temp1*ap(k) temp2 = temp2 + ap(k)*x(ix) ix = ix + incx iy = iy + incy end do y(jy) = y(jy) + temp1*ap(kk+j-1) + alpha*temp2 jx = jx + incx jy = jy + incy kk = kk + j end do end if else ! form y when ap contains the lower triangle. if ((incx==1) .and. (incy==1)) then do j = 1,n temp1 = alpha*x(j) temp2 = zero y(j) = y(j) + temp1*ap(kk) k = kk + 1 do i = j + 1,n y(i) = y(i) + temp1*ap(k) temp2 = temp2 + ap(k)*x(i) k = k + 1 end do y(j) = y(j) + alpha*temp2 kk = kk + (n-j+1) end do else jx = kx jy = ky do j = 1,n temp1 = alpha*x(jx) temp2 = zero y(jy) = y(jy) + temp1*ap(kk) ix = jx iy = jy do k = kk + 1,kk + n - j ix = ix + incx iy = iy + incy y(iy) = y(iy) + temp1*ap(k) temp2 = temp2 + ap(k)*x(ix) end do y(jy) = y(jy) + alpha*temp2 jx = jx + incx jy = jy + incy kk = kk + (n-j+1) end do end if end if return end subroutine stdlib_dspmv pure subroutine stdlib_dspr(uplo,n,alpha,x,incx,ap) !! DSPR performs the symmetric rank 1 operation !! A := alpha*x*x**T + A, !! where alpha is a real scalar, x is an n element vector and A is an !! n by n symmetric matrix, supplied in packed form. ! -- reference blas level2 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha integer(ilp), intent(in) :: incx, n character, intent(in) :: uplo ! Array Arguments real(dp), intent(inout) :: ap(*) real(dp), intent(in) :: x(*) ! ===================================================================== ! Local Scalars real(dp) :: temp integer(ilp) :: i, info, ix, j, jx, k, kk, kx ! test the input parameters. info = 0 if (.not.stdlib_lsame(uplo,'U') .and. .not.stdlib_lsame(uplo,'L')) then info = 1 else if (n<0) then info = 2 else if (incx==0) then info = 5 end if if (info/=0) then call stdlib_xerbla('DSPR ',info) return end if ! quick return if possible. if ((n==0) .or. (alpha==zero)) return ! set the start point in x if the increment is not unity. if (incx<=0) then kx = 1 - (n-1)*incx else if (incx/=1) then kx = 1 end if ! start the operations. in this version the elements of the array ap ! are accessed sequentially with one pass through ap. kk = 1 if (stdlib_lsame(uplo,'U')) then ! form a when upper triangle is stored in ap. if (incx==1) then do j = 1,n if (x(j)/=zero) then temp = alpha*x(j) k = kk do i = 1,j ap(k) = ap(k) + x(i)*temp k = k + 1 end do end if kk = kk + j end do else jx = kx do j = 1,n if (x(jx)/=zero) then temp = alpha*x(jx) ix = kx do k = kk,kk + j - 1 ap(k) = ap(k) + x(ix)*temp ix = ix + incx end do end if jx = jx + incx kk = kk + j end do end if else ! form a when lower triangle is stored in ap. if (incx==1) then do j = 1,n if (x(j)/=zero) then temp = alpha*x(j) k = kk do i = j,n ap(k) = ap(k) + x(i)*temp k = k + 1 end do end if kk = kk + n - j + 1 end do else jx = kx do j = 1,n if (x(jx)/=zero) then temp = alpha*x(jx) ix = jx do k = kk,kk + n - j ap(k) = ap(k) + x(ix)*temp ix = ix + incx end do end if jx = jx + incx kk = kk + n - j + 1 end do end if end if return end subroutine stdlib_dspr pure subroutine stdlib_dspr2(uplo,n,alpha,x,incx,y,incy,ap) !! DSPR2 performs the symmetric rank 2 operation !! A := alpha*x*y**T + alpha*y*x**T + A, !! where alpha is a scalar, x and y are n element vectors and A is an !! n by n symmetric matrix, supplied in packed form. ! -- reference blas level2 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha integer(ilp), intent(in) :: incx, incy, n character, intent(in) :: uplo ! Array Arguments real(dp), intent(inout) :: ap(*) real(dp), intent(in) :: x(*), y(*) ! ===================================================================== ! Local Scalars real(dp) :: temp1, temp2 integer(ilp) :: i, info, ix, iy, j, jx, jy, k, kk, kx, ky ! test the input parameters. info = 0 if (.not.stdlib_lsame(uplo,'U') .and. .not.stdlib_lsame(uplo,'L')) then info = 1 else if (n<0) then info = 2 else if (incx==0) then info = 5 else if (incy==0) then info = 7 end if if (info/=0) then call stdlib_xerbla('DSPR2 ',info) return end if ! quick return if possible. if ((n==0) .or. (alpha==zero)) return ! set up the start points in x and y if the increments are not both ! unity. if ((incx/=1) .or. (incy/=1)) then if (incx>0) then kx = 1 else kx = 1 - (n-1)*incx end if if (incy>0) then ky = 1 else ky = 1 - (n-1)*incy end if jx = kx jy = ky end if ! start the operations. in this version the elements of the array ap ! are accessed sequentially with one pass through ap. kk = 1 if (stdlib_lsame(uplo,'U')) then ! form a when upper triangle is stored in ap. if ((incx==1) .and. (incy==1)) then do j = 1,n if ((x(j)/=zero) .or. (y(j)/=zero)) then temp1 = alpha*y(j) temp2 = alpha*x(j) k = kk do i = 1,j ap(k) = ap(k) + x(i)*temp1 + y(i)*temp2 k = k + 1 end do end if kk = kk + j end do else do j = 1,n if ((x(jx)/=zero) .or. (y(jy)/=zero)) then temp1 = alpha*y(jy) temp2 = alpha*x(jx) ix = kx iy = ky do k = kk,kk + j - 1 ap(k) = ap(k) + x(ix)*temp1 + y(iy)*temp2 ix = ix + incx iy = iy + incy end do end if jx = jx + incx jy = jy + incy kk = kk + j end do end if else ! form a when lower triangle is stored in ap. if ((incx==1) .and. (incy==1)) then do j = 1,n if ((x(j)/=zero) .or. (y(j)/=zero)) then temp1 = alpha*y(j) temp2 = alpha*x(j) k = kk do i = j,n ap(k) = ap(k) + x(i)*temp1 + y(i)*temp2 k = k + 1 end do end if kk = kk + n - j + 1 end do else do j = 1,n if ((x(jx)/=zero) .or. (y(jy)/=zero)) then temp1 = alpha*y(jy) temp2 = alpha*x(jx) ix = jx iy = jy do k = kk,kk + n - j ap(k) = ap(k) + x(ix)*temp1 + y(iy)*temp2 ix = ix + incx iy = iy + incy end do end if jx = jx + incx jy = jy + incy kk = kk + n - j + 1 end do end if end if return end subroutine stdlib_dspr2 pure subroutine stdlib_dswap(n,dx,incx,dy,incy) !! DSWAP interchanges two vectors. !! uses unrolled loops for increments equal to 1. ! -- reference blas level1 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments integer(ilp), intent(in) :: incx, incy, n ! Array Arguments real(dp), intent(inout) :: dx(*), dy(*) ! ===================================================================== ! Local Scalars real(dp) :: dtemp integer(ilp) :: i, ix, iy, m, mp1 ! Intrinsic Functions intrinsic :: mod if (n<=0) return if (incx==1 .and. incy==1) then ! code for both increments equal to 1 ! clean-up loop m = mod(n,3) if (m/=0) then do i = 1,m dtemp = dx(i) dx(i) = dy(i) dy(i) = dtemp end do if (n<3) return end if mp1 = m + 1 do i = mp1,n,3 dtemp = dx(i) dx(i) = dy(i) dy(i) = dtemp dtemp = dx(i+1) dx(i+1) = dy(i+1) dy(i+1) = dtemp dtemp = dx(i+2) dx(i+2) = dy(i+2) dy(i+2) = dtemp end do else ! code for unequal increments or equal increments not equal ! to 1 ix = 1 iy = 1 if (incx<0) ix = (-n+1)*incx + 1 if (incy<0) iy = (-n+1)*incy + 1 do i = 1,n dtemp = dx(ix) dx(ix) = dy(iy) dy(iy) = dtemp ix = ix + incx iy = iy + incy end do end if return end subroutine stdlib_dswap pure subroutine stdlib_dsymm(side,uplo,m,n,alpha,a,lda,b,ldb,beta,c,ldc) !! DSYMM performs one of the matrix-matrix operations !! C := alpha*A*B + beta*C, !! or !! C := alpha*B*A + beta*C, !! where alpha and beta are scalars, A is a symmetric matrix and B and !! C are m by n matrices. ! -- reference blas level3 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha, beta integer(ilp), intent(in) :: lda, ldb, ldc, m, n character, intent(in) :: side, uplo ! Array Arguments real(dp), intent(in) :: a(lda,*), b(ldb,*) real(dp), intent(inout) :: c(ldc,*) ! ===================================================================== ! Intrinsic Functions intrinsic :: max ! Local Scalars real(dp) :: temp1, temp2 integer(ilp) :: i, info, j, k, nrowa logical(lk) :: upper ! set nrowa as the number of rows of a. if (stdlib_lsame(side,'L')) then nrowa = m else nrowa = n end if upper = stdlib_lsame(uplo,'U') ! test the input parameters. info = 0 if ((.not.stdlib_lsame(side,'L')) .and. (.not.stdlib_lsame(side,'R'))) then info = 1 else if ((.not.upper) .and. (.not.stdlib_lsame(uplo,'L'))) then info = 2 else if (m<0) then info = 3 else if (n<0) then info = 4 else if (lda<max(1,nrowa)) then info = 7 else if (ldb<max(1,m)) then info = 9 else if (ldc<max(1,m)) then info = 12 end if if (info/=0) then call stdlib_xerbla('DSYMM ',info) return end if ! quick return if possible. if ((m==0) .or. (n==0) .or.((alpha==zero).and. (beta==one))) return ! and when alpha.eq.zero. if (alpha==zero) then if (beta==zero) then do j = 1,n do i = 1,m c(i,j) = zero end do end do else do j = 1,n do i = 1,m c(i,j) = beta*c(i,j) end do end do end if return end if ! start the operations. if (stdlib_lsame(side,'L')) then ! form c := alpha*a*b + beta*c. if (upper) then do j = 1,n do i = 1,m temp1 = alpha*b(i,j) temp2 = zero do k = 1,i - 1 c(k,j) = c(k,j) + temp1*a(k,i) temp2 = temp2 + b(k,j)*a(k,i) end do if (beta==zero) then c(i,j) = temp1*a(i,i) + alpha*temp2 else c(i,j) = beta*c(i,j) + temp1*a(i,i) +alpha*temp2 end if end do end do else do j = 1,n do i = m,1,-1 temp1 = alpha*b(i,j) temp2 = zero do k = i + 1,m c(k,j) = c(k,j) + temp1*a(k,i) temp2 = temp2 + b(k,j)*a(k,i) end do if (beta==zero) then c(i,j) = temp1*a(i,i) + alpha*temp2 else c(i,j) = beta*c(i,j) + temp1*a(i,i) +alpha*temp2 end if end do end do end if else ! form c := alpha*b*a + beta*c. loop_170: do j = 1,n temp1 = alpha*a(j,j) if (beta==zero) then do i = 1,m c(i,j) = temp1*b(i,j) end do else do i = 1,m c(i,j) = beta*c(i,j) + temp1*b(i,j) end do end if do k = 1,j - 1 if (upper) then temp1 = alpha*a(k,j) else temp1 = alpha*a(j,k) end if do i = 1,m c(i,j) = c(i,j) + temp1*b(i,k) end do end do do k = j + 1,n if (upper) then temp1 = alpha*a(j,k) else temp1 = alpha*a(k,j) end if do i = 1,m c(i,j) = c(i,j) + temp1*b(i,k) end do end do end do loop_170 end if return end subroutine stdlib_dsymm pure subroutine stdlib_dsymv(uplo,n,alpha,a,lda,x,incx,beta,y,incy) !! DSYMV performs the matrix-vector operation !! y := alpha*A*x + beta*y, !! where alpha and beta are scalars, x and y are n element vectors and !! A is an n by n symmetric matrix. ! -- reference blas level2 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha, beta integer(ilp), intent(in) :: incx, incy, lda, n character, intent(in) :: uplo ! Array Arguments real(dp), intent(in) :: a(lda,*), x(*) real(dp), intent(inout) :: y(*) ! ===================================================================== ! Local Scalars real(dp) :: temp1, temp2 integer(ilp) :: i, info, ix, iy, j, jx, jy, kx, ky ! Intrinsic Functions intrinsic :: max ! test the input parameters. info = 0 if (.not.stdlib_lsame(uplo,'U') .and. .not.stdlib_lsame(uplo,'L')) then info = 1 else if (n<0) then info = 2 else if (lda<max(1,n)) then info = 5 else if (incx==0) then info = 7 else if (incy==0) then info = 10 end if if (info/=0) then call stdlib_xerbla('DSYMV ',info) return end if ! quick return if possible. if ((n==0) .or. ((alpha==zero).and. (beta==one))) return ! set up the start points in x and y. if (incx>0) then kx = 1 else kx = 1 - (n-1)*incx end if if (incy>0) then ky = 1 else ky = 1 - (n-1)*incy end if ! start the operations. in this version the elements of a are ! accessed sequentially with one pass through the triangular part ! of a. ! first form y := beta*y. if (beta/=one) then if (incy==1) then if (beta==zero) then do i = 1,n y(i) = zero end do else do i = 1,n y(i) = beta*y(i) end do end if else iy = ky if (beta==zero) then do i = 1,n y(iy) = zero iy = iy + incy end do else do i = 1,n y(iy) = beta*y(iy) iy = iy + incy end do end if end if end if if (alpha==zero) return if (stdlib_lsame(uplo,'U')) then ! form y when a is stored in upper triangle. if ((incx==1) .and. (incy==1)) then do j = 1,n temp1 = alpha*x(j) temp2 = zero do i = 1,j - 1 y(i) = y(i) + temp1*a(i,j) temp2 = temp2 + a(i,j)*x(i) end do y(j) = y(j) + temp1*a(j,j) + alpha*temp2 end do else jx = kx jy = ky do j = 1,n temp1 = alpha*x(jx) temp2 = zero ix = kx iy = ky do i = 1,j - 1 y(iy) = y(iy) + temp1*a(i,j) temp2 = temp2 + a(i,j)*x(ix) ix = ix + incx iy = iy + incy end do y(jy) = y(jy) + temp1*a(j,j) + alpha*temp2 jx = jx + incx jy = jy + incy end do end if else ! form y when a is stored in lower triangle. if ((incx==1) .and. (incy==1)) then do j = 1,n temp1 = alpha*x(j) temp2 = zero y(j) = y(j) + temp1*a(j,j) do i = j + 1,n y(i) = y(i) + temp1*a(i,j) temp2 = temp2 + a(i,j)*x(i) end do y(j) = y(j) + alpha*temp2 end do else jx = kx jy = ky do j = 1,n temp1 = alpha*x(jx) temp2 = zero y(jy) = y(jy) + temp1*a(j,j) ix = jx iy = jy do i = j + 1,n ix = ix + incx iy = iy + incy y(iy) = y(iy) + temp1*a(i,j) temp2 = temp2 + a(i,j)*x(ix) end do y(jy) = y(jy) + alpha*temp2 jx = jx + incx jy = jy + incy end do end if end if return end subroutine stdlib_dsymv pure subroutine stdlib_dsyr(uplo,n,alpha,x,incx,a,lda) !! DSYR performs the symmetric rank 1 operation !! A := alpha*x*x**T + A, !! where alpha is a real scalar, x is an n element vector and A is an !! n by n symmetric matrix. ! -- reference blas level2 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha integer(ilp), intent(in) :: incx, lda, n character, intent(in) :: uplo ! Array Arguments real(dp), intent(inout) :: a(lda,*) real(dp), intent(in) :: x(*) ! ===================================================================== ! Local Scalars real(dp) :: temp integer(ilp) :: i, info, ix, j, jx, kx ! Intrinsic Functions intrinsic :: max ! test the input parameters. info = 0 if (.not.stdlib_lsame(uplo,'U') .and. .not.stdlib_lsame(uplo,'L')) then info = 1 else if (n<0) then info = 2 else if (incx==0) then info = 5 else if (lda<max(1,n)) then info = 7 end if if (info/=0) then call stdlib_xerbla('DSYR ',info) return end if ! quick return if possible. if ((n==0) .or. (alpha==zero)) return ! set the start point in x if the increment is not unity. if (incx<=0) then kx = 1 - (n-1)*incx else if (incx/=1) then kx = 1 end if ! start the operations. in this version the elements of a are ! accessed sequentially with one pass through the triangular part ! of a. if (stdlib_lsame(uplo,'U')) then ! form a when a is stored in upper triangle. if (incx==1) then do j = 1,n if (x(j)/=zero) then temp = alpha*x(j) do i = 1,j a(i,j) = a(i,j) + x(i)*temp end do end if end do else jx = kx do j = 1,n if (x(jx)/=zero) then temp = alpha*x(jx) ix = kx do i = 1,j a(i,j) = a(i,j) + x(ix)*temp ix = ix + incx end do end if jx = jx + incx end do end if else ! form a when a is stored in lower triangle. if (incx==1) then do j = 1,n if (x(j)/=zero) then temp = alpha*x(j) do i = j,n a(i,j) = a(i,j) + x(i)*temp end do end if end do else jx = kx do j = 1,n if (x(jx)/=zero) then temp = alpha*x(jx) ix = jx do i = j,n a(i,j) = a(i,j) + x(ix)*temp ix = ix + incx end do end if jx = jx + incx end do end if end if return end subroutine stdlib_dsyr pure subroutine stdlib_dsyr2(uplo,n,alpha,x,incx,y,incy,a,lda) !! DSYR2 performs the symmetric rank 2 operation !! A := alpha*x*y**T + alpha*y*x**T + A, !! where alpha is a scalar, x and y are n element vectors and A is an n !! by n symmetric matrix. ! -- reference blas level2 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha integer(ilp), intent(in) :: incx, incy, lda, n character, intent(in) :: uplo ! Array Arguments real(dp), intent(inout) :: a(lda,*) real(dp), intent(in) :: x(*), y(*) ! ===================================================================== ! Local Scalars real(dp) :: temp1, temp2 integer(ilp) :: i, info, ix, iy, j, jx, jy, kx, ky ! Intrinsic Functions intrinsic :: max ! test the input parameters. info = 0 if (.not.stdlib_lsame(uplo,'U') .and. .not.stdlib_lsame(uplo,'L')) then info = 1 else if (n<0) then info = 2 else if (incx==0) then info = 5 else if (incy==0) then info = 7 else if (lda<max(1,n)) then info = 9 end if if (info/=0) then call stdlib_xerbla('DSYR2 ',info) return end if ! quick return if possible. if ((n==0) .or. (alpha==zero)) return ! set up the start points in x and y if the increments are not both ! unity. if ((incx/=1) .or. (incy/=1)) then if (incx>0) then kx = 1 else kx = 1 - (n-1)*incx end if if (incy>0) then ky = 1 else ky = 1 - (n-1)*incy end if jx = kx jy = ky end if ! start the operations. in this version the elements of a are ! accessed sequentially with one pass through the triangular part ! of a. if (stdlib_lsame(uplo,'U')) then ! form a when a is stored in the upper triangle. if ((incx==1) .and. (incy==1)) then do j = 1,n if ((x(j)/=zero) .or. (y(j)/=zero)) then temp1 = alpha*y(j) temp2 = alpha*x(j) do i = 1,j a(i,j) = a(i,j) + x(i)*temp1 + y(i)*temp2 end do end if end do else do j = 1,n if ((x(jx)/=zero) .or. (y(jy)/=zero)) then temp1 = alpha*y(jy) temp2 = alpha*x(jx) ix = kx iy = ky do i = 1,j a(i,j) = a(i,j) + x(ix)*temp1 + y(iy)*temp2 ix = ix + incx iy = iy + incy end do end if jx = jx + incx jy = jy + incy end do end if else ! form a when a is stored in the lower triangle. if ((incx==1) .and. (incy==1)) then do j = 1,n if ((x(j)/=zero) .or. (y(j)/=zero)) then temp1 = alpha*y(j) temp2 = alpha*x(j) do i = j,n a(i,j) = a(i,j) + x(i)*temp1 + y(i)*temp2 end do end if end do else do j = 1,n if ((x(jx)/=zero) .or. (y(jy)/=zero)) then temp1 = alpha*y(jy) temp2 = alpha*x(jx) ix = jx iy = jy do i = j,n a(i,j) = a(i,j) + x(ix)*temp1 + y(iy)*temp2 ix = ix + incx iy = iy + incy end do end if jx = jx + incx jy = jy + incy end do end if end if return end subroutine stdlib_dsyr2 pure subroutine stdlib_dsyr2k(uplo,trans,n,k,alpha,a,lda,b,ldb,beta,c,ldc) !! DSYR2K performs one of the symmetric rank 2k operations !! C := alpha*A*B**T + alpha*B*A**T + beta*C, !! or !! C := alpha*A**T*B + alpha*B**T*A + beta*C, !! where alpha and beta are scalars, C is an n by n symmetric matrix !! and A and B are n by k matrices in the first case and k by n !! matrices in the second case. ! -- reference blas level3 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha, beta integer(ilp), intent(in) :: k, lda, ldb, ldc, n character, intent(in) :: trans, uplo ! Array Arguments real(dp), intent(in) :: a(lda,*), b(ldb,*) real(dp), intent(inout) :: c(ldc,*) ! ===================================================================== ! Intrinsic Functions intrinsic :: max ! Local Scalars real(dp) :: temp1, temp2 integer(ilp) :: i, info, j, l, nrowa logical(lk) :: upper ! test the input parameters. if (stdlib_lsame(trans,'N')) then nrowa = n else nrowa = k end if upper = stdlib_lsame(uplo,'U') info = 0 if ((.not.upper) .and. (.not.stdlib_lsame(uplo,'L'))) then info = 1 else if ((.not.stdlib_lsame(trans,'N')) .and.(.not.stdlib_lsame(trans,'T')) .and.(& .not.stdlib_lsame(trans,'C'))) then info = 2 else if (n<0) then info = 3 else if (k<0) then info = 4 else if (lda<max(1,nrowa)) then info = 7 else if (ldb<max(1,nrowa)) then info = 9 else if (ldc<max(1,n)) then info = 12 end if if (info/=0) then call stdlib_xerbla('DSYR2K',info) return end if ! quick return if possible. if ((n==0) .or. (((alpha==zero).or.(k==0)).and. (beta==one))) return ! and when alpha.eq.zero. if (alpha==zero) then if (upper) then if (beta==zero) then do j = 1,n do i = 1,j c(i,j) = zero end do end do else do j = 1,n do i = 1,j c(i,j) = beta*c(i,j) end do end do end if else if (beta==zero) then do j = 1,n do i = j,n c(i,j) = zero end do end do else do j = 1,n do i = j,n c(i,j) = beta*c(i,j) end do end do end if end if return end if ! start the operations. if (stdlib_lsame(trans,'N')) then ! form c := alpha*a*b**t + alpha*b*a**t + c. if (upper) then do j = 1,n if (beta==zero) then do i = 1,j c(i,j) = zero end do else if (beta/=one) then do i = 1,j c(i,j) = beta*c(i,j) end do end if do l = 1,k if ((a(j,l)/=zero) .or. (b(j,l)/=zero)) then temp1 = alpha*b(j,l) temp2 = alpha*a(j,l) do i = 1,j c(i,j) = c(i,j) + a(i,l)*temp1 +b(i,l)*temp2 end do end if end do end do else do j = 1,n if (beta==zero) then do i = j,n c(i,j) = zero end do else if (beta/=one) then do i = j,n c(i,j) = beta*c(i,j) end do end if do l = 1,k if ((a(j,l)/=zero) .or. (b(j,l)/=zero)) then temp1 = alpha*b(j,l) temp2 = alpha*a(j,l) do i = j,n c(i,j) = c(i,j) + a(i,l)*temp1 +b(i,l)*temp2 end do end if end do end do end if else ! form c := alpha*a**t*b + alpha*b**t*a + c. if (upper) then do j = 1,n do i = 1,j temp1 = zero temp2 = zero do l = 1,k temp1 = temp1 + a(l,i)*b(l,j) temp2 = temp2 + b(l,i)*a(l,j) end do if (beta==zero) then c(i,j) = alpha*temp1 + alpha*temp2 else c(i,j) = beta*c(i,j) + alpha*temp1 +alpha*temp2 end if end do end do else do j = 1,n do i = j,n temp1 = zero temp2 = zero do l = 1,k temp1 = temp1 + a(l,i)*b(l,j) temp2 = temp2 + b(l,i)*a(l,j) end do if (beta==zero) then c(i,j) = alpha*temp1 + alpha*temp2 else c(i,j) = beta*c(i,j) + alpha*temp1 +alpha*temp2 end if end do end do end if end if return end subroutine stdlib_dsyr2k pure subroutine stdlib_dsyrk(uplo,trans,n,k,alpha,a,lda,beta,c,ldc) !! DSYRK performs one of the symmetric rank k operations !! C := alpha*A*A**T + beta*C, !! or !! C := alpha*A**T*A + beta*C, !! where alpha and beta are scalars, C is an n by n symmetric matrix !! and A is an n by k matrix in the first case and a k by n matrix !! in the second case. ! -- reference blas level3 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha, beta integer(ilp), intent(in) :: k, lda, ldc, n character, intent(in) :: trans, uplo ! Array Arguments real(dp), intent(in) :: a(lda,*) real(dp), intent(inout) :: c(ldc,*) ! ===================================================================== ! Intrinsic Functions intrinsic :: max ! Local Scalars real(dp) :: temp integer(ilp) :: i, info, j, l, nrowa logical(lk) :: upper ! test the input parameters. if (stdlib_lsame(trans,'N')) then nrowa = n else nrowa = k end if upper = stdlib_lsame(uplo,'U') info = 0 if ((.not.upper) .and. (.not.stdlib_lsame(uplo,'L'))) then info = 1 else if ((.not.stdlib_lsame(trans,'N')) .and.(.not.stdlib_lsame(trans,'T')) .and.(& .not.stdlib_lsame(trans,'C'))) then info = 2 else if (n<0) then info = 3 else if (k<0) then info = 4 else if (lda<max(1,nrowa)) then info = 7 else if (ldc<max(1,n)) then info = 10 end if if (info/=0) then call stdlib_xerbla('DSYRK ',info) return end if ! quick return if possible. if ((n==0) .or. (((alpha==zero).or.(k==0)).and. (beta==one))) return ! and when alpha.eq.zero. if (alpha==zero) then if (upper) then if (beta==zero) then do j = 1,n do i = 1,j c(i,j) = zero end do end do else do j = 1,n do i = 1,j c(i,j) = beta*c(i,j) end do end do end if else if (beta==zero) then do j = 1,n do i = j,n c(i,j) = zero end do end do else do j = 1,n do i = j,n c(i,j) = beta*c(i,j) end do end do end if end if return end if ! start the operations. if (stdlib_lsame(trans,'N')) then ! form c := alpha*a*a**t + beta*c. if (upper) then do j = 1,n if (beta==zero) then do i = 1,j c(i,j) = zero end do else if (beta/=one) then do i = 1,j c(i,j) = beta*c(i,j) end do end if do l = 1,k if (a(j,l)/=zero) then temp = alpha*a(j,l) do i = 1,j c(i,j) = c(i,j) + temp*a(i,l) end do end if end do end do else do j = 1,n if (beta==zero) then do i = j,n c(i,j) = zero end do else if (beta/=one) then do i = j,n c(i,j) = beta*c(i,j) end do end if do l = 1,k if (a(j,l)/=zero) then temp = alpha*a(j,l) do i = j,n c(i,j) = c(i,j) + temp*a(i,l) end do end if end do end do end if else ! form c := alpha*a**t*a + beta*c. if (upper) then do j = 1,n do i = 1,j temp = zero do l = 1,k temp = temp + a(l,i)*a(l,j) end do if (beta==zero) then c(i,j) = alpha*temp else c(i,j) = alpha*temp + beta*c(i,j) end if end do end do else do j = 1,n do i = j,n temp = zero do l = 1,k temp = temp + a(l,i)*a(l,j) end do if (beta==zero) then c(i,j) = alpha*temp else c(i,j) = alpha*temp + beta*c(i,j) end if end do end do end if end if return end subroutine stdlib_dsyrk pure subroutine stdlib_dtbmv(uplo,trans,diag,n,k,a,lda,x,incx) !! DTBMV performs one of the matrix-vector operations !! x := A*x, or x := A**T*x, !! where x is an n element vector and A is an n by n unit, or non-unit, !! upper or lower triangular band matrix, with ( k + 1 ) diagonals. ! -- reference blas level2 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments integer(ilp), intent(in) :: incx, k, lda, n character, intent(in) :: diag, trans, uplo ! Array Arguments real(dp), intent(in) :: a(lda,*) real(dp), intent(inout) :: x(*) ! ===================================================================== ! Local Scalars real(dp) :: temp integer(ilp) :: i, info, ix, j, jx, kplus1, kx, l logical(lk) :: nounit ! Intrinsic Functions intrinsic :: max,min ! test the input parameters. info = 0 if (.not.stdlib_lsame(uplo,'U') .and. .not.stdlib_lsame(uplo,'L')) then info = 1 else if (.not.stdlib_lsame(trans,'N') .and. .not.stdlib_lsame(trans,'T') & .and..not.stdlib_lsame(trans,'C')) then info = 2 else if (.not.stdlib_lsame(diag,'U') .and. .not.stdlib_lsame(diag,'N')) then info = 3 else if (n<0) then info = 4 else if (k<0) then info = 5 else if (lda< (k+1)) then info = 7 else if (incx==0) then info = 9 end if if (info/=0) then call stdlib_xerbla('DTBMV ',info) return end if ! quick return if possible. if (n==0) return nounit = stdlib_lsame(diag,'N') ! set up the start point in x if the increment is not unity. this ! will be ( n - 1 )*incx too small for descending loops. if (incx<=0) then kx = 1 - (n-1)*incx else if (incx/=1) then kx = 1 end if ! start the operations. in this version the elements of a are ! accessed sequentially with one pass through a. if (stdlib_lsame(trans,'N')) then ! form x := a*x. if (stdlib_lsame(uplo,'U')) then kplus1 = k + 1 if (incx==1) then do j = 1,n if (x(j)/=zero) then temp = x(j) l = kplus1 - j do i = max(1,j-k),j - 1 x(i) = x(i) + temp*a(l+i,j) end do if (nounit) x(j) = x(j)*a(kplus1,j) end if end do else jx = kx do j = 1,n if (x(jx)/=zero) then temp = x(jx) ix = kx l = kplus1 - j do i = max(1,j-k),j - 1 x(ix) = x(ix) + temp*a(l+i,j) ix = ix + incx end do if (nounit) x(jx) = x(jx)*a(kplus1,j) end if jx = jx + incx if (j>k) kx = kx + incx end do end if else if (incx==1) then do j = n,1,-1 if (x(j)/=zero) then temp = x(j) l = 1 - j do i = min(n,j+k),j + 1,-1 x(i) = x(i) + temp*a(l+i,j) end do if (nounit) x(j) = x(j)*a(1,j) end if end do else kx = kx + (n-1)*incx jx = kx do j = n,1,-1 if (x(jx)/=zero) then temp = x(jx) ix = kx l = 1 - j do i = min(n,j+k),j + 1,-1 x(ix) = x(ix) + temp*a(l+i,j) ix = ix - incx end do if (nounit) x(jx) = x(jx)*a(1,j) end if jx = jx - incx if ((n-j)>=k) kx = kx - incx end do end if end if else ! form x := a**t*x. if (stdlib_lsame(uplo,'U')) then kplus1 = k + 1 if (incx==1) then do j = n,1,-1 temp = x(j) l = kplus1 - j if (nounit) temp = temp*a(kplus1,j) do i = j - 1,max(1,j-k),-1 temp = temp + a(l+i,j)*x(i) end do x(j) = temp end do else kx = kx + (n-1)*incx jx = kx do j = n,1,-1 temp = x(jx) kx = kx - incx ix = kx l = kplus1 - j if (nounit) temp = temp*a(kplus1,j) do i = j - 1,max(1,j-k),-1 temp = temp + a(l+i,j)*x(ix) ix = ix - incx end do x(jx) = temp jx = jx - incx end do end if else if (incx==1) then do j = 1,n temp = x(j) l = 1 - j if (nounit) temp = temp*a(1,j) do i = j + 1,min(n,j+k) temp = temp + a(l+i,j)*x(i) end do x(j) = temp end do else jx = kx do j = 1,n temp = x(jx) kx = kx + incx ix = kx l = 1 - j if (nounit) temp = temp*a(1,j) do i = j + 1,min(n,j+k) temp = temp + a(l+i,j)*x(ix) ix = ix + incx end do x(jx) = temp jx = jx + incx end do end if end if end if return end subroutine stdlib_dtbmv pure subroutine stdlib_dtbsv(uplo,trans,diag,n,k,a,lda,x,incx) !! DTBSV solves one of the systems of equations !! A*x = b, or A**T*x = b, !! where b and x are n element vectors and A is an n by n unit, or !! non-unit, upper or lower triangular band matrix, with ( k + 1 ) !! diagonals. !! No test for singularity or near-singularity is included in this !! routine. Such tests must be performed before calling this routine. ! -- reference blas level2 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments integer(ilp), intent(in) :: incx, k, lda, n character, intent(in) :: diag, trans, uplo ! Array Arguments real(dp), intent(in) :: a(lda,*) real(dp), intent(inout) :: x(*) ! ===================================================================== ! Local Scalars real(dp) :: temp integer(ilp) :: i, info, ix, j, jx, kplus1, kx, l logical(lk) :: nounit ! Intrinsic Functions intrinsic :: max,min ! test the input parameters. info = 0 if (.not.stdlib_lsame(uplo,'U') .and. .not.stdlib_lsame(uplo,'L')) then info = 1 else if (.not.stdlib_lsame(trans,'N') .and. .not.stdlib_lsame(trans,'T') & .and..not.stdlib_lsame(trans,'C')) then info = 2 else if (.not.stdlib_lsame(diag,'U') .and. .not.stdlib_lsame(diag,'N')) then info = 3 else if (n<0) then info = 4 else if (k<0) then info = 5 else if (lda< (k+1)) then info = 7 else if (incx==0) then info = 9 end if if (info/=0) then call stdlib_xerbla('DTBSV ',info) return end if ! quick return if possible. if (n==0) return nounit = stdlib_lsame(diag,'N') ! set up the start point in x if the increment is not unity. this ! will be ( n - 1 )*incx too small for descending loops. if (incx<=0) then kx = 1 - (n-1)*incx else if (incx/=1) then kx = 1 end if ! start the operations. in this version the elements of a are ! accessed by sequentially with one pass through a. if (stdlib_lsame(trans,'N')) then ! form x := inv( a )*x. if (stdlib_lsame(uplo,'U')) then kplus1 = k + 1 if (incx==1) then do j = n,1,-1 if (x(j)/=zero) then l = kplus1 - j if (nounit) x(j) = x(j)/a(kplus1,j) temp = x(j) do i = j - 1,max(1,j-k),-1 x(i) = x(i) - temp*a(l+i,j) end do end if end do else kx = kx + (n-1)*incx jx = kx do j = n,1,-1 kx = kx - incx if (x(jx)/=zero) then ix = kx l = kplus1 - j if (nounit) x(jx) = x(jx)/a(kplus1,j) temp = x(jx) do i = j - 1,max(1,j-k),-1 x(ix) = x(ix) - temp*a(l+i,j) ix = ix - incx end do end if jx = jx - incx end do end if else if (incx==1) then do j = 1,n if (x(j)/=zero) then l = 1 - j if (nounit) x(j) = x(j)/a(1,j) temp = x(j) do i = j + 1,min(n,j+k) x(i) = x(i) - temp*a(l+i,j) end do end if end do else jx = kx do j = 1,n kx = kx + incx if (x(jx)/=zero) then ix = kx l = 1 - j if (nounit) x(jx) = x(jx)/a(1,j) temp = x(jx) do i = j + 1,min(n,j+k) x(ix) = x(ix) - temp*a(l+i,j) ix = ix + incx end do end if jx = jx + incx end do end if end if else ! form x := inv( a**t)*x. if (stdlib_lsame(uplo,'U')) then kplus1 = k + 1 if (incx==1) then do j = 1,n temp = x(j) l = kplus1 - j do i = max(1,j-k),j - 1 temp = temp - a(l+i,j)*x(i) end do if (nounit) temp = temp/a(kplus1,j) x(j) = temp end do else jx = kx do j = 1,n temp = x(jx) ix = kx l = kplus1 - j do i = max(1,j-k),j - 1 temp = temp - a(l+i,j)*x(ix) ix = ix + incx end do if (nounit) temp = temp/a(kplus1,j) x(jx) = temp jx = jx + incx if (j>k) kx = kx + incx end do end if else if (incx==1) then do j = n,1,-1 temp = x(j) l = 1 - j do i = min(n,j+k),j + 1,-1 temp = temp - a(l+i,j)*x(i) end do if (nounit) temp = temp/a(1,j) x(j) = temp end do else kx = kx + (n-1)*incx jx = kx do j = n,1,-1 temp = x(jx) ix = kx l = 1 - j do i = min(n,j+k),j + 1,-1 temp = temp - a(l+i,j)*x(ix) ix = ix - incx end do if (nounit) temp = temp/a(1,j) x(jx) = temp jx = jx - incx if ((n-j)>=k) kx = kx - incx end do end if end if end if return end subroutine stdlib_dtbsv pure subroutine stdlib_dtpmv(uplo,trans,diag,n,ap,x,incx) !! DTPMV performs one of the matrix-vector operations !! x := A*x, or x := A**T*x, !! where x is an n element vector and A is an n by n unit, or non-unit, !! upper or lower triangular matrix, supplied in packed form. ! -- reference blas level2 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments integer(ilp), intent(in) :: incx, n character, intent(in) :: diag, trans, uplo ! Array Arguments real(dp), intent(in) :: ap(*) real(dp), intent(inout) :: x(*) ! ===================================================================== ! Local Scalars real(dp) :: temp integer(ilp) :: i, info, ix, j, jx, k, kk, kx logical(lk) :: nounit ! test the input parameters. info = 0 if (.not.stdlib_lsame(uplo,'U') .and. .not.stdlib_lsame(uplo,'L')) then info = 1 else if (.not.stdlib_lsame(trans,'N') .and. .not.stdlib_lsame(trans,'T') & .and..not.stdlib_lsame(trans,'C')) then info = 2 else if (.not.stdlib_lsame(diag,'U') .and. .not.stdlib_lsame(diag,'N')) then info = 3 else if (n<0) then info = 4 else if (incx==0) then info = 7 end if if (info/=0) then call stdlib_xerbla('DTPMV ',info) return end if ! quick return if possible. if (n==0) return nounit = stdlib_lsame(diag,'N') ! set up the start point in x if the increment is not unity. this ! will be ( n - 1 )*incx too small for descending loops. if (incx<=0) then kx = 1 - (n-1)*incx else if (incx/=1) then kx = 1 end if ! start the operations. in this version the elements of ap are ! accessed sequentially with one pass through ap. if (stdlib_lsame(trans,'N')) then ! form x:= a*x. if (stdlib_lsame(uplo,'U')) then kk = 1 if (incx==1) then do j = 1,n if (x(j)/=zero) then temp = x(j) k = kk do i = 1,j - 1 x(i) = x(i) + temp*ap(k) k = k + 1 end do if (nounit) x(j) = x(j)*ap(kk+j-1) end if kk = kk + j end do else jx = kx do j = 1,n if (x(jx)/=zero) then temp = x(jx) ix = kx do k = kk,kk + j - 2 x(ix) = x(ix) + temp*ap(k) ix = ix + incx end do if (nounit) x(jx) = x(jx)*ap(kk+j-1) end if jx = jx + incx kk = kk + j end do end if else kk = (n* (n+1))/2 if (incx==1) then do j = n,1,-1 if (x(j)/=zero) then temp = x(j) k = kk do i = n,j + 1,-1 x(i) = x(i) + temp*ap(k) k = k - 1 end do if (nounit) x(j) = x(j)*ap(kk-n+j) end if kk = kk - (n-j+1) end do else kx = kx + (n-1)*incx jx = kx do j = n,1,-1 if (x(jx)/=zero) then temp = x(jx) ix = kx do k = kk,kk - (n- (j+1)),-1 x(ix) = x(ix) + temp*ap(k) ix = ix - incx end do if (nounit) x(jx) = x(jx)*ap(kk-n+j) end if jx = jx - incx kk = kk - (n-j+1) end do end if end if else ! form x := a**t*x. if (stdlib_lsame(uplo,'U')) then kk = (n* (n+1))/2 if (incx==1) then do j = n,1,-1 temp = x(j) if (nounit) temp = temp*ap(kk) k = kk - 1 do i = j - 1,1,-1 temp = temp + ap(k)*x(i) k = k - 1 end do x(j) = temp kk = kk - j end do else jx = kx + (n-1)*incx do j = n,1,-1 temp = x(jx) ix = jx if (nounit) temp = temp*ap(kk) do k = kk - 1,kk - j + 1,-1 ix = ix - incx temp = temp + ap(k)*x(ix) end do x(jx) = temp jx = jx - incx kk = kk - j end do end if else kk = 1 if (incx==1) then do j = 1,n temp = x(j) if (nounit) temp = temp*ap(kk) k = kk + 1 do i = j + 1,n temp = temp + ap(k)*x(i) k = k + 1 end do x(j) = temp kk = kk + (n-j+1) end do else jx = kx do j = 1,n temp = x(jx) ix = jx if (nounit) temp = temp*ap(kk) do k = kk + 1,kk + n - j ix = ix + incx temp = temp + ap(k)*x(ix) end do x(jx) = temp jx = jx + incx kk = kk + (n-j+1) end do end if end if end if return end subroutine stdlib_dtpmv pure subroutine stdlib_dtpsv(uplo,trans,diag,n,ap,x,incx) !! DTPSV solves one of the systems of equations !! A*x = b, or A**T*x = b, !! where b and x are n element vectors and A is an n by n unit, or !! non-unit, upper or lower triangular matrix, supplied in packed form. !! No test for singularity or near-singularity is included in this !! routine. Such tests must be performed before calling this routine. ! -- reference blas level2 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments integer(ilp), intent(in) :: incx, n character, intent(in) :: diag, trans, uplo ! Array Arguments real(dp), intent(in) :: ap(*) real(dp), intent(inout) :: x(*) ! ===================================================================== ! Local Scalars real(dp) :: temp integer(ilp) :: i, info, ix, j, jx, k, kk, kx logical(lk) :: nounit ! test the input parameters. info = 0 if (.not.stdlib_lsame(uplo,'U') .and. .not.stdlib_lsame(uplo,'L')) then info = 1 else if (.not.stdlib_lsame(trans,'N') .and. .not.stdlib_lsame(trans,'T') & .and..not.stdlib_lsame(trans,'C')) then info = 2 else if (.not.stdlib_lsame(diag,'U') .and. .not.stdlib_lsame(diag,'N')) then info = 3 else if (n<0) then info = 4 else if (incx==0) then info = 7 end if if (info/=0) then call stdlib_xerbla('DTPSV ',info) return end if ! quick return if possible. if (n==0) return nounit = stdlib_lsame(diag,'N') ! set up the start point in x if the increment is not unity. this ! will be ( n - 1 )*incx too small for descending loops. if (incx<=0) then kx = 1 - (n-1)*incx else if (incx/=1) then kx = 1 end if ! start the operations. in this version the elements of ap are ! accessed sequentially with one pass through ap. if (stdlib_lsame(trans,'N')) then ! form x := inv( a )*x. if (stdlib_lsame(uplo,'U')) then kk = (n* (n+1))/2 if (incx==1) then do j = n,1,-1 if (x(j)/=zero) then if (nounit) x(j) = x(j)/ap(kk) temp = x(j) k = kk - 1 do i = j - 1,1,-1 x(i) = x(i) - temp*ap(k) k = k - 1 end do end if kk = kk - j end do else jx = kx + (n-1)*incx do j = n,1,-1 if (x(jx)/=zero) then if (nounit) x(jx) = x(jx)/ap(kk) temp = x(jx) ix = jx do k = kk - 1,kk - j + 1,-1 ix = ix - incx x(ix) = x(ix) - temp*ap(k) end do end if jx = jx - incx kk = kk - j end do end if else kk = 1 if (incx==1) then do j = 1,n if (x(j)/=zero) then if (nounit) x(j) = x(j)/ap(kk) temp = x(j) k = kk + 1 do i = j + 1,n x(i) = x(i) - temp*ap(k) k = k + 1 end do end if kk = kk + (n-j+1) end do else jx = kx do j = 1,n if (x(jx)/=zero) then if (nounit) x(jx) = x(jx)/ap(kk) temp = x(jx) ix = jx do k = kk + 1,kk + n - j ix = ix + incx x(ix) = x(ix) - temp*ap(k) end do end if jx = jx + incx kk = kk + (n-j+1) end do end if end if else ! form x := inv( a**t )*x. if (stdlib_lsame(uplo,'U')) then kk = 1 if (incx==1) then do j = 1,n temp = x(j) k = kk do i = 1,j - 1 temp = temp - ap(k)*x(i) k = k + 1 end do if (nounit) temp = temp/ap(kk+j-1) x(j) = temp kk = kk + j end do else jx = kx do j = 1,n temp = x(jx) ix = kx do k = kk,kk + j - 2 temp = temp - ap(k)*x(ix) ix = ix + incx end do if (nounit) temp = temp/ap(kk+j-1) x(jx) = temp jx = jx + incx kk = kk + j end do end if else kk = (n* (n+1))/2 if (incx==1) then do j = n,1,-1 temp = x(j) k = kk do i = n,j + 1,-1 temp = temp - ap(k)*x(i) k = k - 1 end do if (nounit) temp = temp/ap(kk-n+j) x(j) = temp kk = kk - (n-j+1) end do else kx = kx + (n-1)*incx jx = kx do j = n,1,-1 temp = x(jx) ix = kx do k = kk,kk - (n- (j+1)),-1 temp = temp - ap(k)*x(ix) ix = ix - incx end do if (nounit) temp = temp/ap(kk-n+j) x(jx) = temp jx = jx - incx kk = kk - (n-j+1) end do end if end if end if return end subroutine stdlib_dtpsv pure subroutine stdlib_dtrmm(side,uplo,transa,diag,m,n,alpha,a,lda,b,ldb) !! DTRMM performs one of the matrix-matrix operations !! B := alpha*op( A )*B, or B := alpha*B*op( A ), !! where alpha is a scalar, B is an m by n matrix, A is a unit, or !! non-unit, upper or lower triangular matrix and op( A ) is one of !! op( A ) = A or op( A ) = A**T. ! -- reference blas level3 routine -- ! -- reference blas is a software package provided by univ. of tennessee, -- ! -- univ. of california berkeley, univ. of colorado denver and nag ltd..-- ! Scalar Arguments real(dp), intent(in) :: alpha integer(ilp), intent(in) :: lda, ldb, m, n character, intent(in) :: diag, side, transa, uplo ! Array Arguments real(dp), intent(in) :: a(lda,*) real(dp), intent(inout) :: b(ldb,*) ! ===================================================================== ! Intrinsic Functions intrinsic :: max ! Local Scalars real(dp) :: temp integer(ilp) :: i, info, j, k, nrowa logical(lk) :: lside, nounit, upper ! test the input parameters. lside = stdlib_lsame(side,'L') if (lside) then nrowa = m else nrowa = n end if nounit = stdlib_lsame(diag,'N') upper = stdlib_lsame(uplo,'U') info = 0 if ((.not.lside) .and. (.not.stdlib_lsame(side,'R'))) then info = 1 else if ((.not.upper) .and. (.not.stdlib_lsame(uplo,'L'))) then info = 2 else if ((.not.stdlib_lsame(transa,'N')) .and.(.not.stdlib_lsame(transa,'T')) .and.(& .not.stdlib_lsame(transa,'C'))) then info = 3 else if ((.not.stdlib_lsame(diag,'U')) .and. (.not.stdlib_lsame(diag,'N'))) & then info = 4 else if (m<0) then info = 5 else if (n<0) then info = 6 else if (lda<max(1,nrowa)) then info = 9 else if (ldb<max(1,m)) then info = 11 end if if (info/=0) then call stdlib_xerbla('DTRMM ',info) return end if ! quick return if possible. if (m==0 .or. n==0) return ! and when alpha.eq.zero. if (alpha==zero) then do j = 1,n do i = 1,m b(i,j) = zero end do end do return end if ! start the operations. if (lside) then if (stdlib_lsame(transa,'N')) then ! form b := alpha*a*b. if (upper) then do j = 1,n do k = 1,m if (b(k,j)/=zero) then temp = alpha*b(k,j) do i = 1,k - 1 b(i,j) = b(i,j) + temp*a(i,k) end do if (nounit) temp = temp*a(k,k) b(k,j) = temp end if end do end do else do j = 1,n do k = m,1,-1 if (b(k,j)/=zero) then temp = alpha*b(k,j) b(k,j) = temp if (nounit) b(k,j) = b(k,j)*a(k,k) do i = k + 1,m b(i,j) = b(i,j) + temp*a(i,k) end do end if end do end do end if else ! form b := alpha*a**t*b. if (upper) then do j = 1,n do i = m,1,-1 temp = b(i,j) if (nounit) temp = temp*a(i,i) do k = 1,i - 1 temp = temp + a(k,i)*b(k,j) end do b(i,j) = alpha*temp end do end do else do j = 1,n do i = 1,m temp = b(i,j) if (nounit) temp = temp*a(i,i) do k = i + 1,m temp = temp + a(k,i)*b(k,j) end do b(i,j) = alpha*temp end do end do end if end if else if (stdlib_lsame(transa,'N')) then ! form b := alpha*b*a. if (upper) then do j = n,1,-1 temp = alpha if (nounit) temp = temp*a(j,j) do i = 1,m b(i,j) = temp*b(i,j) end do do k = 1,j - 1 if (a(k,j)/=zero) then temp = alpha*a(k,j) do i = 1,m b(i,j) = b(i,j) + temp*b(i,k) end do end if end do end do else do j = 1,n temp = alpha if (nounit) temp = temp*a(j,j