Procedures

Left-adjust the character sequence represented by the string. The length of the character sequence remains unchanged.

Right-adjust the character sequence represented by the string. The length of the character sequence remains unchanged.

Returns a boolean scalar where two arrays are element-wise equal within a tolerance. (Specification)

Sets the bits in set1 to the bitwise and of the original bits in set1 and set2. The sets must have the same number of bits otherwise the result is undefined. (Specification)

Sets the bits in set1 to the bitwise and of the original bits in set1 with the bitwise negation of set2. The sets must have the same number of bits otherwise the result is undefined.

arange creates a one-dimensional array of the integer/real type with fixed-spaced values of given spacing, within a given interval. (Specification)

arg computes the phase angle in the interval (-π,π]. (Specification)

(Specification)

argd computes the phase angle of degree version in the interval (-180.0,180.0]. (Specification)

argpi computes the phase angle of circular version in the interval (-1.0,1.0]. (Specification)

Used to define assignment for bitset_large. (Specification)

Assign a character sequence to a string.

ASUM takes the sum of the absolute values.

AXPY constant times a vector plus a vector.

BBCSD computes the CS decomposition of a unitary matrix in bidiagonal-block form, [ B11 | B12 0 0 ] [ 0 | 0 -I 0 ] X = [----------------] [ B21 | B22 0 0 ] [ 0 | 0 0 I ] [ C | -S 0 0 ]

BDSDC computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT, using a divide and conquer method, where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and U and VT are orthogonal matrices of left and right singular vectors, respectively. BDSDC can be used to compute all singular values, and optionally, singular vectors or singular vectors in compact form. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. See DLASD3 for details. The code currently calls DLASDQ if singular values only are desired. However, it can be slightly modified to compute singular values using the divide and conquer method.

BDSQR computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B using the implicit zero-shift QR algorithm. The SVD of B has the form B = Q * S * PH where S is the diagonal matrix of singular values, Q is an orthogonal matrix of left singular vectors, and P is an orthogonal matrix of right singular vectors. If left singular vectors are requested, this subroutine actually returns U*Q instead of Q, and, if right singular vectors are requested, this subroutine returns PHVT instead of PH, for given complex input matrices U and VT. When U and VT are the unitary matrices that reduce a general matrix A to bidiagonal form: A = UBVT, as computed by CGEBRD, then A = (UQ) * S * (PH*VT) is the SVD of A. Optionally, the subroutine may also compute QH*C for a given complex input matrix C. See "Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, no. 5, pp. 873-912, Sept 1990) and "Accurate singular values and differential qd algorithms," by B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics Department, University of California at Berkeley, July 1992 for a detailed description of the algorithm.

Returns an instance of type 'stringlist_index_type' representing backward index Specifications

Returns the number of bit positions in self.

Version experimental

Normal Distribution Cumulative Distribution Function (Specification)

Get uniform distribution cumulative distribution function (cdf) for integer, real and complex variables. (Specification)

Return the character sequence represented by the string.

Checks the value of a logical condition (Specification)

Computes the Cholesky factorization $A = L \cdot L^T$, or $A = U^T \cdot U$. (Specification)

Computes the Cholesky factorization $A = L \cdot L^T$, or $A = U^T \cdot U$. (Specification)

Remove trailing characters in set from string. If no character set is provided trailing whitespace is removed.

Computes the solution of the equality constrained least-squares problem

Computes the size of the workspace required by the constrained least-squares solver. (Specification)

COPY copies a vector x to a vector y.

Copies the contents of the key, old_key, to the key, new_key (Specifications)

Pearson correlation of array elements (Specification)

Returns the number of times substring 'pattern' has appeared in the input string 'string' Specifications

Covariance of array elements (Specification)

Computes the cross product of two vectors, returning a rank-1 and size-3 array (Specification)

deg2rad converts phase angles from degrees to radians. (Specification)

This interface provides methods to convert a matrix of one of the types defined by stdlib_specialmatrices to a standard rank-2 array. (Specifications)

Computes the determinant of a square matrix (Specification)

Creates a diagonal array or extract the diagonal elements of an array (Specification)

Computes differences between adjacent elements of an array. (Specification)

DISNA computes the reciprocal condition numbers for the eigenvectors of a real symmetric or complex Hermitian matrix or for the left or right singular vectors of a general m-by-n matrix. The reciprocal condition number is the 'gap' between the corresponding eigenvalue or singular value and the nearest other one. The bound on the error, measured by angle in radians, in the I-th computed vector is given by DLAMCH( 'E' ) * ( ANORM / SEP( I ) ) where ANORM = 2-norm(A) = max( abs( D(j) ) ). SEP(I) is not allowed to be smaller than DLAMCH( 'E' )*ANORM in order to limit the size of the error bound. DISNA may also be used to compute error bounds for eigenvectors of the generalized symmetric definite eigenproblem.

Version experimental

First derivative Legendre polynomial

DOT forms the dot product of two vectors. uses unrolled loops for increments equal to one.

DOTC forms the dot product of two complex vectors DOTC = X^H * Y

DOTU forms the dot product of two complex vectors DOTU = X^T * Y

Solves the eigendecomposition $A \cdot \bar{v} - \lambda \cdot \bar{v}$ for square matrix $A$. (Specification)

Solves the eigendecomposition $A \cdot \bar{v} - \lambda \cdot \bar{v}$ for a real symmetric $A = A^T$ or complex Hermitian $A = A^H$ square matrix. (Specification)

Returns the eigenvalues $lambda$, $A \cdot \bar{v} - \lambda \cdot \bar{v}$, for square matrix $A$. (Specification)

Returns the eigenvalues $lambda$, $A \cdot \bar{v} - \lambda \cdot \bar{v}$, for a real symmetric $A = A^T$ or complex Hermitian $A = A^H$ square matrix. (Specification)

exponential linear unit function

gradient of the exponential linear unit function

Check whether a string ends with substring or not

Matrix exponential: function interface version : experimental

Creates a new bitset, new, from a range, start_pos to stop_pos, in bitset old. If start_pos is greater than stop_pos the new bitset is empty. If start_pos is less than zero or stop_pos is greater than bits(old)-1 then if status is present it has the value index_invalid_error and new is undefined, otherwise processing stops with an informative message. (Specification)

Constructs the identity matrix (Specification)

Return the positions of the false elements in array. Specification

Fast approximation of the erf function

Fast approximation of the tanh function

gradient of the hyperbolic tangent function

Maps the 64 bit integer key to an unsigned integer value with only nbits bits where nbits is less than 64 (Specification)

Maps the 32 bit integer key to an unsigned integer value with only nbits bits where nbits is less than 32 (Specification)

Returns an instance of type 'stringlist_index_type' representing forward index Specifications

Finds the starting index of substring 'pattern' in the input 'string' Specifications

FNV_1 interfaces (Specification)

FNV_1 interfaces (Specification)

Hashes a key with the FNV_1 algorithm Arguments: key - the key to be hashed

FNV_1A interfaces (Specification)

FNV_1A interfaces (Specification)

Hashes a key with the FNV_1a algorithm (Specifications)

Frees the memory in a key (Specifications)

Gamma function for integer and complex numbers

Computes Gauss-Legendre quadrature nodes and weights.

Computes Gauss-Legendre-Lobatto quadrature nodes and weights.

gaussian function

gradient of the gaussian function

GBBRD reduces a complex general m-by-n band matrix A to real upper bidiagonal form B by a unitary transformation: QH * A * P = B. The routine computes B, and optionally forms Q or PH, or computes Q*HC for a given matrix C.

GBCON estimates the reciprocal of the condition number of a complex general band matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by CGBTRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / ( norm(A) * norm(inv(A)) ).

GBEQU computes row and column scalings intended to equilibrate an M-by-N band matrix A and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest element in each row and column of the matrix B with elements B(i,j)=R(i)A(i,j)C(j) have absolute value 1. R(i) and C(j) are restricted to be between SMLNUM = smallest safe number and BIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of A but works well in practice.

GBEQUB computes row and column scalings intended to equilibrate an M-by-N matrix A and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest element in each row and column of the matrix B with elements B(i,j)=R(i)A(i,j)C(j) have an absolute value of at most the radix. R(i) and C(j) are restricted to be a power of the radix between SMLNUM = smallest safe number and BIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of A but works well in practice. This routine differs from CGEEQU by restricting the scaling factors to a power of the radix. Barring over- and underflow, scaling by these factors introduces no additional rounding errors. However, the scaled entries' magnitudes are no longer approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).

GBMV performs one of the matrix-vector operations y := alphaAx + betay, or y := alphaATx + betay, or y := alpha*AHx + betay, 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.

GBRFS improves the computed solution to a system of linear equations when the coefficient matrix is banded, and provides error bounds and backward error estimates for the solution.

GBSV computes the solution to a complex system of linear equations A * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices. The LU decomposition with partial pivoting and row interchanges is used to factor A as A = L * U, where L is a product of permutation and unit lower triangular matrices with KL subdiagonals, and U is upper triangular with KL+KU superdiagonals. The factored form of A is then used to solve the system of equations A * X = B.

GBTRF computes an LU factorization of a complex m-by-n band matrix A using partial pivoting with row interchanges. This is the blocked version of the algorithm, calling Level 3 BLAS.

GBTRS solves a system of linear equations A * X = B, AT * X = B, or AH * X = B with a general band matrix A using the LU factorization computed by CGBTRF.

Returns the greatest common divisor of two integers (Specification)

GEBAK forms the right or left eigenvectors of a complex general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by CGEBAL.

GEBAL balances a general complex matrix A. This involves, first, permuting A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. Balancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors.

GEBRD 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.

GECON estimates the reciprocal of the condition number of a general complex matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by CGETRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / ( norm(A) * norm(inv(A)) ).

GEEQU computes row and column scalings intended to equilibrate an M-by-N matrix A and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest element in each row and column of the matrix B with elements B(i,j)=R(i)A(i,j)C(j) have absolute value 1. R(i) and C(j) are restricted to be between SMLNUM = smallest safe number and BIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of A but works well in practice.

GEEQUB computes row and column scalings intended to equilibrate an M-by-N matrix A and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest element in each row and column of the matrix B with elements B(i,j)=R(i)A(i,j)C(j) have an absolute value of at most the radix. R(i) and C(j) are restricted to be a power of the radix between SMLNUM = smallest safe number and BIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of A but works well in practice. This routine differs from CGEEQU by restricting the scaling factors to a power of the radix. Barring over- and underflow, scaling by these factors introduces no additional rounding errors. However, the scaled entries' magnitudes are no longer approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).

GEES computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = ZT(Z**H). Optionally, it also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left. The leading columns of Z then form an orthonormal basis for the invariant subspace corresponding to the selected eigenvalues. A complex matrix is in Schur form if it is upper triangular.

GEEV computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. The right eigenvector v(j) of A satisfies A * v(j) = lambda(j) * v(j) where lambda(j) is its eigenvalue. The left eigenvector u(j) of A satisfies u(j)H * A = lambda(j) * u(j)H where u(j)**H denotes the conjugate transpose of u(j). The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real.

GEHRD reduces a complex general matrix A to upper Hessenberg form H by an unitary similarity transformation: Q**H * A * Q = H .

GEJSV computes the singular value decomposition (SVD) of a complex M-by-N matrix [A], where M >= N. The SVD of [A] is written as [A] = [U] * [SIGMA] * [V]^*, where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N diagonal elements, [U] is an M-by-N (or M-by-M) unitary matrix, and [V] is an N-by-N unitary matrix. The diagonal elements of [SIGMA] are the singular values of [A]. The columns of [U] and [V] are the left and the right singular vectors of [A], respectively. The matrices [U] and [V] are computed and stored in the arrays U and V, respectively. The diagonal of [SIGMA] is computed and stored in the array SVA.

GELQ computes an LQ factorization of a complex M-by-N matrix A: A = ( L 0 ) * Q where: Q is a N-by-N orthogonal matrix; L is a lower-triangular M-by-M matrix; 0 is a M-by-(N-M) zero matrix, if M < N.

GELQF computes an LQ factorization of a complex M-by-N matrix A: A = ( L 0 ) * Q where: Q is a N-by-N orthogonal matrix; L is a lower-triangular M-by-M matrix; 0 is a M-by-(N-M) zero matrix, if M < N.

GELQT computes a blocked LQ factorization of a complex M-by-N matrix A using the compact WY representation of Q.

GELQT3 recursively computes a LQ factorization of a complex M-by-N matrix A, using the compact WY representation of Q. Based on the algorithm of Elmroth and Gustavson, IBM J. Res. Develop. Vol 44 No. 4 July 2000.

GELS solves overdetermined or underdetermined complex linear systems involving an M-by-N matrix A, or its conjugate-transpose, using a QR or LQ factorization of A. It is assumed that A has full rank. The following options are provided: 1. If TRANS = 'N' and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || B - AX ||. 2. If TRANS = 'N' and m < n: find the minimum norm solution of an underdetermined system A * X = B. 3. If TRANS = 'C' and m >= n: find the minimum norm solution of an underdetermined system AH * X = B. 4. If TRANS = 'C' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || B - A*H * X ||. Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X.

GELSD computes the minimum-norm solution to a real linear least squares problem: minimize 2-norm(| b - A*x |) using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient. Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X. The problem is solved in three steps: (1) Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a "bidiagonal least squares problem" (BLS) (2) Solve the BLS using a divide and conquer approach. (3) Apply back all the Householder transformations to solve the original least squares problem. The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

GELSS computes the minimum norm solution to a complex linear least squares problem: Minimize 2-norm(| b - A*x |). using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient. Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X. The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value.

GELSY computes the minimum-norm solution to a complex linear least squares problem: minimize || A * X - B || using a complete orthogonal factorization of A. A is an M-by-N matrix which may be rank-deficient. Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X. The routine first computes a QR factorization with column pivoting: A * P = Q * [ R11 R12 ] [ 0 R22 ] with R11 defined as the largest leading submatrix whose estimated condition number is less than 1/RCOND. The order of R11, RANK, is the effective rank of A. Then, R22 is considered to be negligible, and R12 is annihilated by unitary transformations from the right, arriving at the complete orthogonal factorization: A * P = Q * [ T11 0 ] * Z [ 0 0 ] The minimum-norm solution is then X = P * ZH [ inv(T11)*Q1H*B ] [ 0 ] where Q1 consists of the first RANK columns of Q. This routine is basically identical to the original xGELSX except three differences: o The permutation of matrix B (the right hand side) is faster and more simple. o The call to the subroutine xGEQPF has been substituted by the the call to the subroutine xGEQP3. This subroutine is a Blas-3 version of the QR factorization with column pivoting. o Matrix B (the right hand side) is updated with Blas-3.

Gaussian error linear unit function

Approximated gaussian error linear unit function

Gradient of the approximated gaussian error linear unit function

Gradient of the gaussian error linear unit function

GEMLQ overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': QH * C C * QH where Q is a complex unitary matrix defined as the product of blocked elementary reflectors computed by short wide LQ factorization (CGELQ)

GEMLQT overwrites the general complex M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q C C Q TRANS = 'C': QH C C QH where Q is a complex unitary matrix defined as the product of K elementary reflectors: Q = H(1) H(2) . . . H(K) = I - V T V**H generated using the compact WY representation as returned by CGELQT. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'.

GEMM performs one of the matrix-matrix operations C := alphaop( A )op( B ) + betaC, where op( X ) is one of op( X ) = X or op( X ) = XT or op( X ) = X*H, 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.

GEMQR overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': QH * C C * QH where Q is a complex unitary matrix defined as the product of blocked elementary reflectors computed by tall skinny QR factorization (CGEQR)

GEMQRT overwrites the general complex M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q C C Q TRANS = 'C': QH C C QH where Q is a complex orthogonal matrix defined as the product of K elementary reflectors: Q = H(1) H(2) . . . H(K) = I - V T V**H generated using the compact WY representation as returned by CGEQRT. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'.

GEMV performs one of the matrix-vector operations y := alphaAx + betay, or y := alphaATx + betay, or y := alpha*AHx + betay, where alpha and beta are scalars, x and y are vectors and A is an m by n matrix.

GEQLF computes a QL factorization of a complex M-by-N matrix A: A = Q * L.

GEQP3 computes a QR factorization with column pivoting of a real or complex M-by-N matrix A:

GEQR computes a QR factorization of a complex M-by-N matrix A: A = Q * ( R ), ( 0 ) where: Q is a M-by-M orthogonal matrix; R is an upper-triangular N-by-N matrix; 0 is a (M-N)-by-N zero matrix, if M > N.

GEQR2P computes a QR factorization of a complex m-by-n matrix A: A = Q * ( R ), ( 0 ) where: Q is a m-by-m orthogonal matrix; R is an upper-triangular n-by-n matrix with nonnegative diagonal entries; 0 is a (m-n)-by-n zero matrix, if m > n.

GEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * ( R ), ( 0 ) where: Q is a M-by-M orthogonal matrix; R is an upper-triangular N-by-N matrix; 0 is a (M-N)-by-N zero matrix, if M > N.

CGEQR2P computes a QR factorization of a complex M-by-N matrix A: A = Q * ( R ), ( 0 ) where: Q is a M-by-M orthogonal matrix; R is an upper-triangular N-by-N matrix with nonnegative diagonal entries; 0 is a (M-N)-by-N zero matrix, if M > N.

GEQRT computes a blocked QR factorization of a complex M-by-N matrix A using the compact WY representation of Q.

GEQRT2 computes a QR factorization of a complex M-by-N matrix A, using the compact WY representation of Q.

GEQRT3 recursively computes a QR factorization of a complex M-by-N matrix A, using the compact WY representation of Q. Based on the algorithm of Elmroth and Gustavson, IBM J. Res. Develop. Vol 44 No. 4 July 2000.

GER performs the rank 1 operation A := alphaxy**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.

GERC performs the rank 1 operation A := alphaxy**H + 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.

GERFS improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution.

GERQF computes an RQ factorization of a complex M-by-N matrix A: A = R * Q.

GERU performs the rank 1 operation A := alphaxy**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.

GESDD computes the singular value decomposition (SVD) of a complex M-by-N matrix A, optionally computing the left and/or right singular vectors, by using divide-and-conquer method. The SVD is written A = U * SIGMA * conjugate-transpose(V) where SIGMA is an M-by-N matrix which is zero except for its min(m,n) diagonal elements, U is an M-by-M unitary matrix, and V is an N-by-N unitary matrix. The diagonal elements of SIGMA are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A. Note that the routine returns VT = V**H, not V. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

GESV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices. The LU decomposition with partial pivoting and row interchanges is used to factor A as A = P * L * U, where P is a permutation matrix, L is unit lower triangular, and U is upper triangular. The factored form of A is then used to solve the system of equations A * X = B.

GESVD computes the singular value decomposition (SVD) of a complex M-by-N matrix A, optionally computing the left and/or right singular vectors. The SVD is written A = U * SIGMA * conjugate-transpose(V) where SIGMA is an M-by-N matrix which is zero except for its min(m,n) diagonal elements, U is an M-by-M unitary matrix, and V is an N-by-N unitary matrix. The diagonal elements of SIGMA are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A. Note that the routine returns V**H, not V.

GESVDQ computes the singular value decomposition (SVD) of a complex M-by-N matrix A, where M >= N. The SVD of A is written as [++] [xx] [x0] [xx] A = U * SIGMA * V^*, [++] = [xx] * [ox] * [xx] [++] [xx] where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal matrix, and V is an N-by-N unitary matrix. The diagonal elements of SIGMA are the singular values of A. The columns of U and V are the left and the right singular vectors of A, respectively.

GESVJ computes the singular value decomposition (SVD) of a complex M-by-N matrix A, where M >= N. The SVD of A is written as [++] [xx] [x0] [xx] A = U * SIGMA * V^*, [++] = [xx] * [ox] * [xx] [++] [xx] where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal matrix, and V is an N-by-N unitary matrix. The diagonal elements of SIGMA are the singular values of A. The columns of U and V are the left and the right singular vectors of A, respectively.

Read a whole line from a formatted unit into a string variable

Vector norm: subroutine interface version: experimental

Getter function to retrieve standard library version

GETRF computes an LU factorization of a general M-by-N matrix A using partial pivoting with row interchanges. The factorization has the form A = P * L * U where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n). This is the right-looking Level 3 BLAS version of the algorithm.

GETRF2 computes an LU factorization of a general M-by-N matrix A using partial pivoting with row interchanges. The factorization has the form A = P * L * U where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n). This is the recursive version of the algorithm. It divides the matrix into four submatrices: [ A11 | A12 ] where A11 is n1 by n1 and A22 is n2 by n2 A = [ -----|----- ] with n1 = min(m,n)/2 [ A21 | A22 ] n2 = n-n1 [ A11 ] The subroutine calls itself to factor [ --- ], [ A12 ] [ A12 ] do the swaps on [ --- ], solve A12, update A22, [ A22 ] then calls itself to factor A22 and do the swaps on A21.

GETRI computes the inverse of a matrix using the LU factorization computed by CGETRF. This method inverts U and then computes inv(A) by solving the system inv(A)*L = inv(U) for inv(A).

GETRS solves a system of linear equations A * X = B, AT * X = B, or AH * X = B with a general N-by-N matrix A using the LU factorization computed by CGETRF.

GETSLS solves overdetermined or underdetermined complex linear systems involving an M-by-N matrix A, using a tall skinny QR or short wide LQ factorization of A. It is assumed that A has full rank. The following options are provided: 1. If TRANS = 'N' and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || B - AX ||. 2. If TRANS = 'N' and m < n: find the minimum norm solution of an underdetermined system A * X = B. 3. If TRANS = 'C' and m >= n: find the minimum norm solution of an undetermined system AT * X = B. 4. If TRANS = 'C' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || B - A*T * X ||. Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X.

GETSQRHRT computes a NB2-sized column blocked QR-factorization of a complex M-by-N matrix A with M >= N, A = Q * R. The routine uses internally a NB1-sized column blocked and MB1-sized row blocked TSQR-factorization and perfors the reconstruction of the Householder vectors from the TSQR output. The routine also converts the R_tsqr factor from the TSQR-factorization output into the R factor that corresponds to the Householder QR-factorization, A = Q_tsqr * R_tsqr = Q * R. The output Q and R factors are stored in the same format as in CGEQRT (Q is in blocked compact WY-representation). See the documentation of CGEQRT for more details on the format.

GGBAK forms the right or left eigenvectors of a complex generalized eigenvalue problem Ax = lambdaB*x, by backward transformation on the computed eigenvectors of the balanced pair of matrices output by CGGBAL.

GGBAL balances a pair of general complex matrices (A,B). This involves, first, permuting A and B by similarity transformations to isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. Balancing may reduce the 1-norm of the matrices, and improve the accuracy of the computed eigenvalues and/or eigenvectors in the generalized eigenvalue problem Ax = lambdaB*x.

GGES computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized complex Schur form (S, T), and optionally left and/or right Schur vectors (VSL and VSR). This gives the generalized Schur factorization (A,B) = ( (VSL)S(VSR)H, (VSL)T(VSR)H ) where (VSR)*H is the conjugate-transpose of VSR. Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper triangular matrix S and the upper triangular matrix T. The leading columns of VSL and VSR then form an unitary basis for the corresponding left and right eigenspaces (deflating subspaces). (If only the generalized eigenvalues are needed, use the driver CGGEV instead, which is faster.) A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - wB is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both being zero. A pair of matrices (S,T) is in generalized complex Schur form if S and T are upper triangular and, in addition, the diagonal elements of T are non-negative real numbers.

GGEV computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A - lambdaB is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both being zero. The right generalized eigenvector v(j) corresponding to the generalized eigenvalue lambda(j) of (A,B) satisfies A * v(j) = lambda(j) * B * v(j). The left generalized eigenvector u(j) corresponding to the generalized eigenvalues lambda(j) of (A,B) satisfies u(j)H * A = lambda(j) * u(j)H * B where u(j)*H is the conjugate-transpose of u(j).

GGGLM solves a general Gauss-Markov linear model (GLM) problem: minimize || y ||_2 subject to d = Ax + By x where A is an N-by-M matrix, B is an N-by-P matrix, and d is a given N-vector. It is assumed that M <= N <= M+P, and rank(A) = M and rank( A B ) = N. Under these assumptions, the constrained equation is always consistent, and there is a unique solution x and a minimal 2-norm solution y, which is obtained using a generalized QR factorization of the matrices (A, B) given by A = Q(R), B = QTZ. (0) In particular, if matrix B is square nonsingular, then the problem GLM is equivalent to the following weighted linear least squares problem minimize || inv(B)(d-A*x) ||_2 x where inv(B) denotes the inverse of B.

GGHRD reduces a pair of complex matrices (A,B) to generalized upper Hessenberg form using unitary transformations, where A is a general matrix and B is upper triangular. The form of the generalized eigenvalue problem is Ax = lambdaBx, and B is typically made upper triangular by computing its QR factorization and moving the unitary matrix Q to the left side of the equation. This subroutine simultaneously reduces A to a Hessenberg matrix H: QHAZ = H and transforms B to another upper triangular matrix T: QHBZ = T in order to reduce the problem to its standard form Hy = lambdaTy where y = ZH*x. The unitary matrices Q and Z are determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that Q1 * A * Z1H = (Q1Q) * H * (Z1Z)H Q1 * B * Z1H = (Q1Q) * T * (Z1Z)H If Q1 is the unitary matrix from the QR factorization of B in the original equation Ax = lambdaB*x, then GGHRD reduces the original problem to generalized Hessenberg form.

GGLSE solves the linear equality-constrained least squares (LSE) problem: minimize || c - Ax ||_2 subject to Bx = d where A is an M-by-N matrix, B is a P-by-N matrix, c is a given M-vector, and d is a given P-vector. It is assumed that P <= N <= M+P, and rank(B) = P and rank( (A) ) = N. ( (B) ) These conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized RQ factorization of the matrices (B, A) given by B = (0 R)Q, A = ZT*Q.

GGQRF computes a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B: A = QR, B = QTZ, where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix, and R and T assume one of the forms: if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N, ( 0 ) N-M N M-N M where R11 is upper triangular, and if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P, P-N N ( T21 ) P P where T12 or T21 is upper triangular. In particular, if B is square and nonsingular, the GQR factorization of A and B implicitly gives the QR factorization of inv(B)A: inv(B)A = ZH * (inv(T)R) where inv(B) denotes the inverse of the matrix B, and Z' denotes the conjugate transpose of matrix Z.

GGRQF computes a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B: A = RQ, B = ZTQ, where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix, and R and T assume one of the forms: if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N, N-M M ( R21 ) N N where R12 or R21 is upper triangular, and if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P, ( 0 ) P-N P N-P N where T11 is upper triangular. In particular, if B is square and nonsingular, the GRQ factorization of A and B implicitly gives the RQ factorization of Ainv(B): Ainv(B) = (Rinv(T))ZH where inv(B) denotes the inverse of the matrix B, and Z*H denotes the conjugate transpose of the matrix Z.

GSVJ0 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.

GSVJ1 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 ~~~~~~~~~~~~~~~ GSVJ1 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.

GTCON estimates the reciprocal of the condition number of a complex tridiagonal matrix A using the LU factorization as computed by CGTTRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

GTRFS improves the computed solution to a system of linear equations when the coefficient matrix is tridiagonal, and provides error bounds and backward error estimates for the solution.

GTSV solves the equation AX = B, where A is an N-by-N tridiagonal matrix, by Gaussian elimination with partial pivoting. Note that the equation AT X = B may be solved by interchanging the order of the arguments DU and DL.

GTTRF computes an LU factorization of a complex tridiagonal matrix A using elimination with partial pivoting and row interchanges. The factorization has the form A = L * U where L is a product of permutation and unit lower bidiagonal matrices and U is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.

GTTRS solves one of the systems of equations A * X = B, AT * X = B, or AH * X = B, with a tridiagonal matrix A using the LU factorization computed by CGTTRF.

Wrapper function to handle GEES error codes

Process GEEV output flags

Process GGEV output flags

Process SYEV/HEEV output flags

HB2ST_KERNELS is an internal routine used by the CHETRD_HB2ST subroutine.

HBEV computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A.

HBEVD computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

HBGST reduces a complex Hermitian-definite banded generalized eigenproblem Ax = lambdaBx to standard form Cy = lambday, such that C has the same bandwidth as A. B must have been previously factorized as SHS by CPBSTF, using a split Cholesky factorization. A is overwritten by C = XHAX, where X = S(-1)*Q and Q is a unitary matrix chosen to preserve the bandwidth of A.

HBGV computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form Ax=(lambda)B*x. Here A and B are assumed to be Hermitian and banded, and B is also positive definite.

HBGVD computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form Ax=(lambda)B*x. Here A and B are assumed to be Hermitian and banded, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

HBMV performs the matrix-vector operation y := alphaAx + beta*y, where alpha and beta are scalars, x and y are n element vectors and A is an n by n hermitian band matrix, with k super-diagonals.

HBTRD reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T.

HECON estimates the reciprocal of the condition number of a complex Hermitian matrix A using the factorization A = UDUH or A = LDLH computed by CHETRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

HECON_ROOK estimates the reciprocal of the condition number of a complex Hermitian matrix A using the factorization A = UDUH or A = LDLH computed by CHETRF_ROOK. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

HEEQUB computes row and column scalings intended to equilibrate a Hermitian matrix A (with respect to the Euclidean norm) and reduce its condition number. The scale factors S are computed by the BIN algorithm (see references) so that the scaled matrix B with elements B(i,j) = S(i)A(i,j)S(j) has a condition number within a factor N of the smallest possible condition number over all possible diagonal scalings.

HEEV computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A.

HEEVD computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

HEEVR computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. HEEVR first reduces the matrix A to tridiagonal form T with a call to CHETRD. Then, whenever possible, HEEVR calls CSTEMR to compute the eigenspectrum using Relatively Robust Representations. CSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various "good" L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For each unreduced block (submatrix) of T, (a) Compute T - sigma I = L D L^T, so that L and D define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of D and L cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix T does not have this property in general. (b) Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see steps c) and d). (c) For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy. (d) For each eigenvalue with a large enough relative separation compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to (c) for any clusters that remain. The desired accuracy of the output can be specified by the input parameter ABSTOL. For more details, see CSTEMR's documentation and: - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices," Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154. - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997. Note 1 : HEEVR calls CSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. HEEVR calls SSTEBZ and CSTEIN on non-ieee machines and when partial spectrum requests are made. Normal execution of CSTEMR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner.

HEGST reduces a complex Hermitian-definite generalized eigenproblem to standard form. If ITYPE = 1, the problem is Ax = lambdaBx, and A is overwritten by inv(UH)Ainv(U) or inv(L)Ainv(LH) If ITYPE = 2 or 3, the problem is ABx = lambdax or BAx = lambdax, and A is overwritten by UAUH or LHAL. B must have been previously factorized as UHU or LL*H by CPOTRF.

HEGV computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form Ax=(lambda)Bx, ABx=(lambda)x, or BAx=(lambda)x. Here A and B are assumed to be Hermitian and B is also positive definite.

HEGVD computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form Ax=(lambda)Bx, ABx=(lambda)x, or BAx=(lambda)x. Here A and B are assumed to be Hermitian and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

HEMM performs one of the matrix-matrix operations C := alphaAB + betaC, or C := alphaBA + betaC, where alpha and beta are scalars, A is an hermitian matrix and B and C are m by n matrices.

HEMV performs the matrix-vector operation y := alphaAx + beta*y, where alpha and beta are scalars, x and y are n element vectors and A is an n by n hermitian matrix.

HER performs the hermitian rank 1 operation A := alphaxx**H + A, where alpha is a real scalar, x is an n element vector and A is an n by n hermitian matrix.

HER2 performs the hermitian rank 2 operation A := alphaxyH + conjg( alpha )yxH + A, where alpha is a scalar, x and y are n element vectors and A is an n by n hermitian matrix.

HER2K performs one of the hermitian rank 2k operations C := alphaABH + conjg( alpha )BAH + betaC, or C := alphaAHB + conjg( alpha )BHA + betaC, where alpha and beta are scalars with beta real, C is an n by n hermitian matrix and A and B are n by k matrices in the first case and k by n matrices in the second case.

HERFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian indefinite, and provides error bounds and backward error estimates for the solution.

HERK performs one of the hermitian rank k operations C := alphaAAH + betaC, or C := alphaAHA + betaC, where alpha and beta are real scalars, C is an n by n hermitian matrix and A is an n by k matrix in the first case and a k by n matrix in the second case.

This interface provides methods to compute the hermitian operation for the different matrix types defined by stdlib_specialmatrices. For real-valued matrices, this is equivalent to the standard transpose. Specifications

Computes the Hermitian version of a rank-2 matrix. For complex matrices, this returns conjg(transpose(a)). For real or integer matrices, this returns transpose(a).

HESV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS matrices. The diagonal pivoting method is used to factor A as A = U * D * UH, if UPLO = 'U', or A = L * D * LH, if UPLO = 'L', where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then used to solve the system of equations A * X = B.

HESV_AA computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS matrices. Aasen's algorithm is used to factor A as A = UH * T * U, if UPLO = 'U', or A = L * T * LH, if UPLO = 'L', where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is Hermitian and tridiagonal. The factored form of A is then used to solve the system of equations A * X = B.

HESV_RK computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS matrices. The bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as A = PUD(UH)(PT), if UPLO = 'U', or A = PLD*(LH)(PT), if UPLO = 'L', where U (or L) is unit upper (or lower) triangular matrix, UH (or LH) is the conjugate of U (or L), P is a permutation matrix, P*T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. CHETRF_RK is called to compute the factorization of a complex Hermitian matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine CHETRS_3.

HESV_ROOK computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS matrices. The bounded Bunch-Kaufman ("rook") diagonal pivoting method is used to factor A as A = U * D * UT, if UPLO = 'U', or A = L * D * LT, if UPLO = 'L', where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. CHETRF_ROOK is called to compute the factorization of a complex Hermition matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting method. The factored form of A is then used to solve the system of equations A * X = B by calling CHETRS_ROOK (uses BLAS 2).

HESWAPR applies an elementary permutation on the rows and the columns of a hermitian matrix.

HETF2_RK computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method: A = PUD(UH)(PT) or A = PLD*(LH)(PT), where U (or L) is unit upper (or lower) triangular matrix, UH (or LH) is the conjugate of U (or L), P is a permutation matrix, P*T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the unblocked version of the algorithm, calling Level 2 BLAS. For more information see Further Details section.

HETF2_ROOK computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting method: A = UDUH or A = LDLH where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**H is the conjugate transpose of U, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the unblocked version of the algorithm, calling Level 2 BLAS.

HETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T.

HETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T.

HETRD_HE2HB reduces a complex Hermitian matrix A to complex Hermitian band-diagonal form AB by a unitary similarity transformation: Q**H * A * Q = AB.

HETRF computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is A = UDUH or A = LDLH where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the blocked version of the algorithm, calling Level 3 BLAS.

HETRF_AA computes the factorization of a complex hermitian matrix A using the Aasen's algorithm. The form of the factorization is A = UHTU or A = LTLH where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is a hermitian tridiagonal matrix. This is the blocked version of the algorithm, calling Level 3 BLAS.

HETRF_RK computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method: A = PUD(UH)(PT) or A = PLD*(LH)(PT), where U (or L) is unit upper (or lower) triangular matrix, UH (or LH) is the conjugate of U (or L), P is a permutation matrix, P*T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the blocked version of the algorithm, calling Level 3 BLAS. For more information see Further Details section.

HETRF_ROOK computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting method. The form of the factorization is A = UDUT or A = LDLT where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the blocked version of the algorithm, calling Level 3 BLAS.

HETRI computes the inverse of a complex Hermitian indefinite matrix A using the factorization A = UDUH or A = LDLH computed by CHETRF.

HETRI_ROOK computes the inverse of a complex Hermitian indefinite matrix A using the factorization A = UDUH or A = LDLH computed by CHETRF_ROOK.

HETRS solves a system of linear equations AX = B with a complex Hermitian matrix A using the factorization A = UDUH or A = LDL*H computed by CHETRF.

HETRS2 solves a system of linear equations AX = B with a complex Hermitian matrix A using the factorization A = UDUH or A = LDL*H computed by CHETRF and converted by CSYCONV.

HETRS_3 solves a system of linear equations A * X = B with a complex Hermitian matrix A using the factorization computed by CHETRF_RK or CHETRF_BK: A = PUD(UH)(PT) or A = PLD*(LH)(PT), where U (or L) is unit upper (or lower) triangular matrix, UH (or LH) is the conjugate of U (or L), P is a permutation matrix, P*T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This algorithm is using Level 3 BLAS.

HETRS_AA solves a system of linear equations AX = B with a complex hermitian matrix A using the factorization A = UHTU or A = LTL*H computed by CHETRF_AA.

HETRS_ROOK solves a system of linear equations AX = B with a complex Hermitian matrix A using the factorization A = UDUH or A = LDL*H computed by CHETRF_ROOK.

Level 3 BLAS like routine for C in RFP Format. HFRK performs one of the Hermitian rank--k operations C := alphaAAH + betaC, or C := alphaAHA + betaC, where alpha and beta are real scalars, C is an n--by--n Hermitian matrix and A is an n--by--k matrix in the first case and a k--by--n matrix in the second case.

HGEQZ computes the eigenvalues of a complex matrix pair (H,T), where H is an upper Hessenberg matrix and T is upper triangular, using the single-shift QZ method. Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a complex matrix pair (A,B): A = Q1HZ1H, B = Q1TZ1H, as computed by CGGHRD. If JOB='S', then the Hessenberg-triangular pair (H,T) is also reduced to generalized Schur form, H = QSZH, T = QPZH, where Q and Z are unitary matrices and S and P are upper triangular. Optionally, the unitary matrix Q from the generalized Schur factorization may be postmultiplied into an input matrix Q1, and the unitary matrix Z may be postmultiplied into an input matrix Z1. If Q1 and Z1 are the unitary matrices from CGGHRD that reduced the matrix pair (A,B) to generalized Hessenberg form, then the output matrices Q1Q and Z1Z are the unitary factors from the generalized Schur factorization of (A,B): A = (Q1Q)S(Z1Z)H, B = (Q1Q)P(Z1Z)H. To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, of (A,B)) are computed as a pair of complex values (alpha,beta). If beta is nonzero, lambda = alpha / beta is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP) Ax = lambdaBx and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the alternate form of the GNEP muAy = By. The values of alpha and beta for the i-th eigenvalue can be read directly from the generalized Schur form: alpha = S(i,i), beta = P(i,i). Ref: C.B. Moler Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973), pp. 241--256.

HPCON estimates the reciprocal of the condition number of a complex Hermitian packed matrix A using the factorization A = UDUH or A = LDLH computed by CHPTRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

HPEV computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix in packed storage.

HPEVD computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A in packed storage. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

HPGST reduces a complex Hermitian-definite generalized eigenproblem to standard form, using packed storage. If ITYPE = 1, the problem is Ax = lambdaBx, and A is overwritten by inv(UH)Ainv(U) or inv(L)Ainv(LH) If ITYPE = 2 or 3, the problem is ABx = lambdax or BAx = lambdax, and A is overwritten by UAUH or LHAL. B must have been previously factorized as UHU or LL*H by CPPTRF.

HPGV computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form Ax=(lambda)Bx, ABx=(lambda)x, or BAx=(lambda)x. Here A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite.

HPGVD computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form Ax=(lambda)Bx, ABx=(lambda)x, or BAx=(lambda)x. Here A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

HPMV performs the matrix-vector operation y := alphaAx + beta*y, where alpha and beta are scalars, x and y are n element vectors and A is an n by n hermitian matrix, supplied in packed form.

HPR performs the hermitian rank 1 operation A := alphaxx**H + A, where alpha is a real scalar, x is an n element vector and A is an n by n hermitian matrix, supplied in packed form.

HPR2 performs the hermitian rank 2 operation A := alphaxyH + conjg( alpha )yxH + A, where alpha is a scalar, x and y are n element vectors and A is an n by n hermitian matrix, supplied in packed form.

HPRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian indefinite and packed, and provides error bounds and backward error estimates for the solution.

HPSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian matrix stored in packed format and X and B are N-by-NRHS matrices. The diagonal pivoting method is used to factor A as A = U * D * UH, if UPLO = 'U', or A = L * D * LH, if UPLO = 'L', where U (or L) is a product of permutation and unit upper (lower) triangular matrices, D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then used to solve the system of equations A * X = B.

HPTRD reduces a complex Hermitian matrix A stored in packed form to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T.

HPTRF computes the factorization of a complex Hermitian packed matrix A using the Bunch-Kaufman diagonal pivoting method: A = UDUH or A = LDLH where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

HPTRI computes the inverse of a complex Hermitian indefinite matrix A in packed storage using the factorization A = UDUH or A = LDLH computed by CHPTRF.

HPTRS solves a system of linear equations AX = B with a complex Hermitian matrix A stored in packed format using the factorization A = UDUH or A = LDL*H computed by CHPTRF.

HSEIN uses inverse iteration to find specified right and/or left eigenvectors of a complex upper Hessenberg matrix H. The right eigenvector x and the left eigenvector y of the matrix H corresponding to an eigenvalue w are defined by: H * x = w * x, yh * H = w * yh where y**h denotes the conjugate transpose of the vector y.

HSEQR computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T ZH, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors. Optionally Z may be postmultiplied into an input unitary matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the unitary matrix Q: A = QHQH = (QZ)T(QZ)**H.

Code in ASCII collating sequence.

Character-to-integer conversion function.

Position of a substring within a string.

Inverse of a square matrix (Specification)

Inversion of a square matrix (Specification)

Checks whether c is an ASCII letter (A .. Z, a .. z).

Checks whether c is a letter or a number (0 .. 9, a .. z, A .. Z).

Checks whether or not c is in the ASCII character set - i.e. in the range 0 .. 0x7F.

Checks whether or not c is a blank character. That includes the only the space and tab characters

Returns a boolean scalar/array where two scalar/arrays are element-wise equal within a tolerance. (Specification)

Checks whether c is a control character.

Checks if a matrix (rank-2 array) is diagonal (Specification)

Checks whether c is a digit (0 .. 9).

Checks whether or not c is a printable character other than the space character.

Checks if a matrix (rank-2 array) is Hermitian (Specification)

Checks if a matrix (rank-2 array) is Hessenberg (Specification)

Checks whether c is a digit in base 16 (0 .. 9, A .. F, a .. f).

Checks whether c is a lowercase ASCII letter (a .. z).

Checks whether c is a digit in base 8 (0 .. 7).

Checks whether or not c is a printable character - including the space character.

Checks whether or not c is a punctuation character. That includes all ASCII characters which are not control characters, letters, digits, or whitespace.

Checks if a matrix (rank-2 array) is skew-symmetric (Specification)

Checks if a matrix (rank-2 array) is square (Specification)

Checks if a matrix (rank-2 array) is symmetric (Specification)

Checks if a matrix (rank-2 array) is triangular (Specification)

Checks whether c is an uppercase ASCII letter (A .. Z).

Checks whether or not c is a whitespace character. That includes the space, tab, vertical tab, form feed, carriage return, and linefeed characters.

ISNAN returns .TRUE. if its argument is NaN, and .FALSE. otherwise. To be replaced by the Fortran 2003 intrinsic in the future.

Joins an array of strings into a single string. The chunks are separated with a space, or an optional user-defined separator. Specifications

Computes the Kronecker product of two arrays of size M1xN1, and of M2xN2, returning an (M1M2)x(N1N2) array (Specification)

LA_GBAMV performs one of the matrix-vector operations y := alphaabs(A)abs(x) + betaabs(y), or y := alphaabs(A)Tabs(x) + betaabs(y), where alpha and beta are scalars, x and y are vectors and A is an m by n matrix. This function is primarily used in calculating error bounds. To protect against underflow during evaluation, components in the resulting vector are perturbed away from zero by (N+1) times the underflow threshold. To prevent unnecessarily large errors for block-structure embedded in general matrices, "symbolically" zero components are not perturbed. A zero entry is considered "symbolic" if all multiplications involved in computing that entry have at least one zero multiplicand.

LA_GBRCOND Estimates the Skeel condition number of op(A) * op2(C) where op2 is determined by CMODE as follows CMODE = 1 op2(C) = C CMODE = 0 op2(C) = I CMODE = -1 op2(C) = inv(C) The Skeel condition number cond(A) = norminf( |inv(A)||A| ) is computed by computing scaling factors R such that diag(R)Aop2(C) is row equilibrated and computing the standard infinity-norm condition number.

LA_GBRCOND_C Computes the infinity norm condition number of op(A) * inv(diag(C)) where C is a REAL vector.

LA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U). The "max absolute element" norm is used. If this is much less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable.

LA_GEAMV performs one of the matrix-vector operations y := alphaabs(A)abs(x) + betaabs(y), or y := alphaabs(A)Tabs(x) + betaabs(y), where alpha and beta are scalars, x and y are vectors and A is an m by n matrix. This function is primarily used in calculating error bounds. To protect against underflow during evaluation, components in the resulting vector are perturbed away from zero by (N+1) times the underflow threshold. To prevent unnecessarily large errors for block-structure embedded in general matrices, "symbolically" zero components are not perturbed. A zero entry is considered "symbolic" if all multiplications involved in computing that entry have at least one zero multiplicand.

LA_GERCOND estimates the Skeel condition number of op(A) * op2(C) where op2 is determined by CMODE as follows CMODE = 1 op2(C) = C CMODE = 0 op2(C) = I CMODE = -1 op2(C) = inv(C) The Skeel condition number cond(A) = norminf( |inv(A)||A| ) is computed by computing scaling factors R such that diag(R)Aop2(C) is row equilibrated and computing the standard infinity-norm condition number.

LA_GERCOND_C computes the infinity norm condition number of op(A) * inv(diag(C)) where C is a REAL vector.

LA_GERPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U). The "max absolute element" norm is used. If this is much less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable.

CLA_SYAMV performs the matrix-vector operation y := alphaabs(A)abs(x) + beta*abs(y), where alpha and beta are scalars, x and y are vectors and A is an n by n symmetric matrix. This function is primarily used in calculating error bounds. To protect against underflow during evaluation, components in the resulting vector are perturbed away from zero by (N+1) times the underflow threshold. To prevent unnecessarily large errors for block-structure embedded in general matrices, "symbolically" zero components are not perturbed. A zero entry is considered "symbolic" if all multiplications involved in computing that entry have at least one zero multiplicand.

LA_HERCOND_C computes the infinity norm condition number of op(A) * inv(diag(C)) where C is a REAL vector.

LA_HERPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U). The "max absolute element" norm is used. If this is much less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable.

LA_LIN_BERR computes componentwise relative backward error from the formula max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) where abs(Z) is the componentwise absolute value of the matrix or vector Z.

LA_PORCOND Estimates the Skeel condition number of op(A) * op2(C) where op2 is determined by CMODE as follows CMODE = 1 op2(C) = C CMODE = 0 op2(C) = I CMODE = -1 op2(C) = inv(C) The Skeel condition number cond(A) = norminf( |inv(A)||A| ) is computed by computing scaling factors R such that diag(R)Aop2(C) is row equilibrated and computing the standard infinity-norm condition number.

LA_PORCOND_C Computes the infinity norm condition number of op(A) * inv(diag(C)) where C is a REAL vector

LA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U). The "max absolute element" norm is used. If this is much less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable.

LA_SYAMV performs the matrix-vector operation y := alphaabs(A)abs(x) + beta*abs(y), where alpha and beta are scalars, x and y are vectors and A is an n by n symmetric matrix. This function is primarily used in calculating error bounds. To protect against underflow during evaluation, components in the resulting vector are perturbed away from zero by (N+1) times the underflow threshold. To prevent unnecessarily large errors for block-structure embedded in general matrices, "symbolically" zero components are not perturbed. A zero entry is considered "symbolic" if all multiplications involved in computing that entry have at least one zero multiplicand.

LA_SYRCOND estimates the Skeel condition number of op(A) * op2(C) where op2 is determined by CMODE as follows CMODE = 1 op2(C) = C CMODE = 0 op2(C) = I CMODE = -1 op2(C) = inv(C) The Skeel condition number cond(A) = norminf( |inv(A)||A| ) is computed by computing scaling factors R such that diag(R)Aop2(C) is row equilibrated and computing the standard infinity-norm condition number.

LA_SYRCOND_C Computes the infinity norm condition number of op(A) * inv(diag(C)) where C is a REAL vector.

LA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U). The "max absolute element" norm is used. If this is much less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable.

LA_WWADDW adds a vector W into a doubled-single vector (X, Y). This works for all extant IBM's hex and binary floating point arithmetic, but not for decimal.

LABAD takes as input the values computed by DLAMCH for underflow and overflow, and returns the square root of each of these values if the log of LARGE is sufficiently large. This subroutine is intended to identify machines with a large exponent range, such as the Crays, and redefine the underflow and overflow limits to be the square roots of the values computed by DLAMCH. This subroutine is needed because DLAMCH does not compensate for poor arithmetic in the upper half of the exponent range, as is found on a Cray.

LABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower real bidiagonal form by a unitary transformation Q**H * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form. This is an auxiliary routine called by CGEBRD

LACGV conjugates a complex vector of length N.

LACON estimates the 1-norm of a square, complex matrix A. Reverse communication is used for evaluating matrix-vector products.

LACPY copies all or part of a two-dimensional matrix A to another matrix B.

LACRM performs a very simple matrix-matrix multiplication: C := A * B, where A is M by N and complex; B is N by N and real; C is M by N and complex.

LACRT performs the operation ( c s )( x ) ==> ( x ) ( -s c )( y ) ( y ) where c and s are complex and the vectors x and y are complex.

LADIV_F := X / Y, where X and Y are complex. The computation of X / Y will not overflow on an intermediary step unless the results overflows.

LADIV_S performs complex division in real arithmetic a + ib p + iq = --------- c + i*d The algorithm is due to Michael Baudin and Robert L. Smith and can be found in the paper "A Robust Complex Division in Scilab"

LAEBZ contains the iteration loops which compute and use the function N(w), which is the count of eigenvalues of a symmetric tridiagonal matrix T less than or equal to its argument w. It performs a choice of two types of loops: IJOB=1, followed by IJOB=2: It takes as input a list of intervals and returns a list of sufficiently small intervals whose union contains the same eigenvalues as the union of the original intervals. The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP. The output interval (AB(j,1),AB(j,2)] will contain eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT. IJOB=3: It performs a binary search in each input interval (AB(j,1),AB(j,2)] for a point w(j) such that N(w(j))=NVAL(j), and uses C(j) as the starting point of the search. If such a w(j) is found, then on output AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output (AB(j,1),AB(j,2)] will be a small interval containing the point where N(w) jumps through NVAL(j), unless that point lies outside the initial interval. Note that the intervals are in all cases half-open intervals, i.e., of the form (a,b] , which includes b but not a . To avoid underflow, the matrix should be scaled so that its largest element is no greater than overflow(1/2) * underflow(1/4) in absolute value. To assure the most accurate computation of small eigenvalues, the matrix should be scaled to be not much smaller than that, either. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966 Note: the arguments are, in general, not checked for unreasonable values.

Using the divide and conquer method, LAED0: computes all eigenvalues of a symmetric tridiagonal matrix which is one diagonal block of those from reducing a dense or band Hermitian matrix and corresponding eigenvectors of the dense or band matrix.

LAED1 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. This routine is used only for the eigenproblem which requires all eigenvalues and eigenvectors of a tridiagonal matrix. DLAED7 handles the case in which eigenvalues only or eigenvalues and eigenvectors of a full symmetric matrix (which was reduced to tridiagonal form) are desired. T = Q(in) ( D(in) + RHO * ZZT ) QT(in) = Q(out) * D(out) * QT(out) where Z = QTu, u is a vector of length N with ones in the CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages: The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLAED2. The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine DLAED4 (as called by DLAED3). This routine also calculates the eigenvectors of the current problem. The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.

This subroutine computes the I-th updated eigenvalue of a symmetric rank-one modification to a diagonal matrix whose elements are given in the array d, and that D(i) < D(j) for i < j and that RHO > 0. This is arranged by the calling routine, and is no loss in generality. The rank-one modified system is thus diag( D ) + RHO * Z * Z_transpose. where we assume the Euclidean norm of Z is 1. The method consists of approximating the rational functions in the secular equation by simpler interpolating rational functions.

This subroutine computes the I-th eigenvalue of a symmetric rank-one modification of a 2-by-2 diagonal matrix diag( D ) + RHO * Z * transpose(Z) . The diagonal elements in the array D are assumed to satisfy D(i) < D(j) for i < j . We also assume RHO > 0 and that the Euclidean norm of the vector Z is one.

LAED6 computes the positive or negative root (closest to the origin) of z(1) z(2) z(3) f(x) = rho + --------- + ---------- + --------- d(1)-x d(2)-x d(3)-x It is assumed that if ORGATI = .true. the root is between d(2) and d(3); otherwise it is between d(1) and d(2) This routine will be called by DLAED4 when necessary. In most cases, the root sought is the smallest in magnitude, though it might not be in some extremely rare situations.

LAED7 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. This routine is used only for the eigenproblem which requires all eigenvalues and optionally eigenvectors of a dense or banded Hermitian matrix that has been reduced to tridiagonal form. T = Q(in) ( D(in) + RHO * ZZH ) QH(in) = Q(out) * D(out) * QH(out) where Z = Q*Hu, u is a vector of length N with ones in the CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages: The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine SLAED2. The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine SLAED4 (as called by SLAED3). This routine also calculates the eigenvectors of the current problem. The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.

LAED8 merges the two sets of eigenvalues together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more eigenvalues are close together or if there is a tiny element in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one.

LAED9 finds the roots of the secular equation, as defined by the values in D, Z, and RHO, between KSTART and KSTOP. It makes the appropriate calls to DLAED4 and then stores the new matrix of eigenvectors for use in calculating the next level of Z vectors.

LAEDA computes the Z vector corresponding to the merge step in the CURLVLth step of the merge process with TLVLS steps for the CURPBMth problem.

LAEIN uses inverse iteration to find a right or left eigenvector corresponding to the eigenvalue W of a complex upper Hessenberg matrix H.

LAESY computes the eigendecomposition of a 2-by-2 symmetric matrix ( ( A, B );( B, C ) ) provided the norm of the matrix of eigenvectors is larger than some threshold value. RT1 is the eigenvalue of larger absolute value, and RT2 of smaller absolute value. If the eigenvectors are computed, then on return ( CS1, SN1 ) is the unit eigenvector for RT1, hence [ CS1 SN1 ] . [ A B ] . [ CS1 -SN1 ] = [ RT1 0 ] [ -SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ]

LAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in an upper quasi-triangular matrix T by an orthogonal similarity transformation. T must be in Schur canonical form, that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign.

LAGTF factorizes the matrix (T - lambdaI), where T is an n by n tridiagonal matrix and lambda is a scalar, as T - lambdaI = PLU, where P is a permutation matrix, L is a unit lower tridiagonal matrix with at most one non-zero sub-diagonal elements per column and U is an upper triangular matrix with at most two non-zero super-diagonal elements per column. The factorization is obtained by Gaussian elimination with partial pivoting and implicit row scaling. The parameter LAMBDA is included in the routine so that LAGTF may be used, in conjunction with DLAGTS, to obtain eigenvectors of T by inverse iteration.

LAGTM performs a matrix-vector product of the form B := alpha * A * X + beta * B where A is a tridiagonal matrix of order N, B and X are N by NRHS matrices, and alpha and beta are real scalars, each of which may be 0., 1., or -1.

LAGTS may be used to solve one of the systems of equations (T - lambdaI)x = y or (T - lambdaI)Tx = y, where T is an n by n tridiagonal matrix, for x, following the factorization of (T - lambdaI) as (T - lambdaI) = PLU , by routine DLAGTF. The choice of equation to be solved is controlled by the argument JOB, and in each case there is an option to perturb zero or very small diagonal elements of U, this option being intended for use in applications such as inverse iteration.

LAHEF computes a partial factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method. The partial factorization has the form: A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: ( 0 U22 ) ( 0 D ) ( U12H U22H ) A = ( L11 0 ) ( D 0 ) ( L11H L21H ) if UPLO = 'L' ( L21 I ) ( 0 A22 ) ( 0 I ) where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. Note that U**H denotes the conjugate transpose of U. LAHEF is an auxiliary routine called by CHETRF. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

LAHEF_AA factorizes a panel of a complex hermitian matrix A using the Aasen's algorithm. The panel consists of a set of NB rows of A when UPLO is U, or a set of NB columns when UPLO is L. In order to factorize the panel, the Aasen's algorithm requires the last row, or column, of the previous panel. The first row, or column, of A is set to be the first row, or column, of an identity matrix, which is used to factorize the first panel. The resulting J-th row of U, or J-th column of L, is stored in the (J-1)-th row, or column, of A (without the unit diagonals), while the diagonal and subdiagonal of A are overwritten by those of T.

LAHEF_RK computes a partial factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method. The partial factorization has the form: A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: ( 0 U22 ) ( 0 D ) ( U12H U22H ) A = ( L11 0 ) ( D 0 ) ( L11H L21H ) if UPLO = 'L', ( L21 I ) ( 0 A22 ) ( 0 I ) where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. LAHEF_RK is an auxiliary routine called by CHETRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

LAHEF_ROOK computes a partial factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting method. The partial factorization has the form: A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: ( 0 U22 ) ( 0 D ) ( U12H U22H ) A = ( L11 0 ) ( D 0 ) ( L11H L21H ) if UPLO = 'L' ( L21 I ) ( 0 A22 ) ( 0 I ) where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. Note that U**H denotes the conjugate transpose of U. LAHEF_ROOK is an auxiliary routine called by CHETRF_ROOK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

LAHQR is an auxiliary routine called by CHSEQR to update the eigenvalues and Schur decomposition already computed by CHSEQR, by dealing with the Hessenberg submatrix in rows and columns ILO to IHI.

LAIC1 applies one step of incremental condition estimation in its simplest version: Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j lower triangular matrix L, such that twonorm(Lx) = sest Then LAIC1 computes sestpr, s, c such that the vector [ sx ] xhat = [ c ] is an approximate singular vector of [ L 0 ] Lhat = [ wH gamma ] in the sense that twonorm(Lhat*xhat) = sestpr. Depending on JOB, an estimate for the largest or smallest singular value is computed. Note that [s c]H and sestpr2 is an eigenpair of the system diag(sest*sest, 0) + [alpha gamma] * [ conjg(alpha) ] [ conjg(gamma) ] where alpha = xH*w.

This routine is not for general use. It exists solely to avoid over-optimization in DISNAN. LAISNAN checks for NaNs by comparing its two arguments for inequality. NaN is the only floating-point value where NaN != NaN returns .TRUE. To check for NaNs, pass the same variable as both arguments. A compiler must assume that the two arguments are not the same variable, and the test will not be optimized away. Interprocedural or whole-program optimization may delete this test. The ISNAN functions will be replaced by the correct Fortran 03 intrinsic once the intrinsic is widely available.

LALS0 applies back the multiplying factors of either the left or the right singular vector matrix of a diagonal matrix appended by a row to the right hand side matrix B in solving the least squares problem using the divide-and-conquer SVD approach. For the left singular vector matrix, three types of orthogonal matrices are involved: (1L) Givens rotations: the number of such rotations is GIVPTR; the pairs of columns/rows they were applied to are stored in GIVCOL; and the C- and S-values of these rotations are stored in GIVNUM. (2L) Permutation. The (NL+1)-st row of B is to be moved to the first row, and for J=2:N, PERM(J)-th row of B is to be moved to the J-th row. (3L) The left singular vector matrix of the remaining matrix. For the right singular vector matrix, four types of orthogonal matrices are involved: (1R) The right singular vector matrix of the remaining matrix. (2R) If SQRE = 1, one extra Givens rotation to generate the right null space. (3R) The inverse transformation of (2L). (4R) The inverse transformation of (1L).

LALSA is an itermediate step in solving the least squares problem by computing the SVD of the coefficient matrix in compact form (The singular vectors are computed as products of simple orthorgonal matrices.). If ICOMPQ = 0, LALSA applies the inverse of the left singular vector matrix of an upper bidiagonal matrix to the right hand side; and if ICOMPQ = 1, LALSA applies the right singular vector matrix to the right hand side. The singular vector matrices were generated in compact form by LALSA.

LALSD uses the singular value decomposition of A to solve the least squares problem of finding X to minimize the Euclidean norm of each column of A*X-B, where A is N-by-N upper bidiagonal, and X and B are N-by-NRHS. The solution X overwrites B. The singular values of A smaller than RCOND times the largest singular value are treated as zero in solving the least squares problem; in this case a minimum norm solution is returned. The actual singular values are returned in D in ascending order. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

LAMRG will create a permutation list which will merge the elements of A (which is composed of two independently sorted sets) into a single set which is sorted in ascending order.

LAMSWLQ overwrites the general complex M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': QH * C C * QH where Q is a complex unitary matrix defined as the product of blocked elementary reflectors computed by short wide LQ factorization (CLASWLQ)

LAMTSQR overwrites the general complex M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': QH * C C * QH where Q is a complex unitary matrix defined as the product of blocked elementary reflectors computed by tall skinny QR factorization (CLATSQR)

LANEG computes the Sturm count, the number of negative pivots encountered while factoring tridiagonal T - sigma I = L D L^T. This implementation works directly on the factors without forming the tridiagonal matrix T. The Sturm count is also the number of eigenvalues of T less than sigma. This routine is called from DLARRB. The current routine does not use the PIVMIN parameter but rather requires IEEE-754 propagation of Infinities and NaNs. This routine also has no input range restrictions but does require default exception handling such that x/0 produces Inf when x is non-zero, and Inf/Inf produces NaN. For more information, see: Marques, Riedy, and Voemel, "Benefits of IEEE-754 Features in Modern Symmetric Tridiagonal Eigensolvers," SIAM Journal on Scientific Computing, v28, n5, 2006. DOI 10.1137/050641624 (Tech report version in LAWN 172 with the same title.)

LANGB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n band matrix A, with kl sub-diagonals and ku super-diagonals.

LANGE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex matrix A.

LANGT returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex tridiagonal matrix A.

LANHB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n hermitian band matrix A, with k super-diagonals.

LANHE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex hermitian matrix A.

LANHF returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix A in RFP format.

LANHP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex hermitian matrix A, supplied in packed form.

LANHS returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg matrix A.

LANHT returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian tridiagonal matrix A.

LANSB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n symmetric band matrix A, with k super-diagonals.

LANSF returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix A in RFP format.

LANSP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix A, supplied in packed form.

LANST returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix A.

LANSY returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix A.

LANTB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n triangular band matrix A, with ( k + 1 ) diagonals.

LANTP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix A, supplied in packed form.

LANTR returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix A.

LAORHR_COL_GETRFNP computes the modified LU factorization without pivoting of a real general M-by-N matrix A. The factorization has the form: A - S = L * U, where: S is a m-by-n diagonal sign matrix with the diagonal D, so that D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing i-1 steps of Gaussian elimination. This means that the diagonal element at each step of "modified" Gaussian elimination is at least one in absolute value (so that division-by-zero not not possible during the division by the diagonal element); L is a M-by-N lower triangular matrix with unit diagonal elements (lower trapezoidal if M > N); and U is a M-by-N upper triangular matrix (upper trapezoidal if M < N). This routine is an auxiliary routine used in the Householder reconstruction routine DORHR_COL. In DORHR_COL, this routine is applied to an M-by-N matrix A with orthonormal columns, where each element is bounded by one in absolute value. With the choice of the matrix S above, one can show that the diagonal element at each step of Gaussian elimination is the largest (in absolute value) in the column on or below the diagonal, so that no pivoting is required for numerical stability [1]. For more details on the Householder reconstruction algorithm, including the modified LU factorization, see [1]. This is the blocked right-looking version of the algorithm, calling Level 3 BLAS to update the submatrix. To factorize a block, this routine calls the recursive routine LAORHR_COL_GETRFNP2. [1] "Reconstructing Householder vectors from tall-skinny QR", G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen, E. Solomonik, J. Parallel Distrib. Comput., vol. 85, pp. 3-31, 2015.

LAORHR_COL_GETRFNP2 computes the modified LU factorization without pivoting of a real general M-by-N matrix A. The factorization has the form: A - S = L * U, where: S is a m-by-n diagonal sign matrix with the diagonal D, so that D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing i-1 steps of Gaussian elimination. This means that the diagonal element at each step of "modified" Gaussian elimination is at least one in absolute value (so that division-by-zero not possible during the division by the diagonal element); L is a M-by-N lower triangular matrix with unit diagonal elements (lower trapezoidal if M > N); and U is a M-by-N upper triangular matrix (upper trapezoidal if M < N). This routine is an auxiliary routine used in the Householder reconstruction routine DORHR_COL. In DORHR_COL, this routine is applied to an M-by-N matrix A with orthonormal columns, where each element is bounded by one in absolute value. With the choice of the matrix S above, one can show that the diagonal element at each step of Gaussian elimination is the largest (in absolute value) in the column on or below the diagonal, so that no pivoting is required for numerical stability [1]. For more details on the Householder reconstruction algorithm, including the modified LU factorization, see [1]. This is the recursive version of the LU factorization algorithm. Denote A - S by B. The algorithm divides the matrix B into four submatrices: [ B11 | B12 ] where B11 is n1 by n1, B = [ -----|----- ] B21 is (m-n1) by n1, [ B21 | B22 ] B12 is n1 by n2, B22 is (m-n1) by n2, with n1 = min(m,n)/2, n2 = n-n1. The subroutine calls itself to factor B11, solves for B21, solves for B12, updates B22, then calls itself to factor B22. For more details on the recursive LU algorithm, see [2]. LAORHR_COL_GETRFNP2 is called to factorize a block by the blocked routine DLAORHR_COL_GETRFNP, which uses blocked code calling Level 3 BLAS to update the submatrix. However, LAORHR_COL_GETRFNP2 is self-sufficient and can be used without DLAORHR_COL_GETRFNP. [1] "Reconstructing Householder vectors from tall-skinny QR", G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen, E. Solomonik, J. Parallel Distrib. Comput., vol. 85, pp. 3-31, 2015. [2] "Recursion leads to automatic variable blocking for dense linear algebra algorithms", F. Gustavson, IBM J. of Res. and Dev., vol. 41, no. 6, pp. 737-755, 1997.

Given two column vectors X and Y, let A = ( X Y ). The subroutine first computes the QR factorization of A = Q*R, and then computes the SVD of the 2-by-2 upper triangular matrix R. The smaller singular value of R is returned in SSMIN, which is used as the measurement of the linear dependency of the vectors X and Y.

LAPMR rearranges the rows of the M by N matrix X as specified by the permutation K(1),K(2),...,K(M) of the integers 1,...,M. If FORWRD = .TRUE., forward permutation: X(K(I),) is moved X(I,) for I = 1,2,...,M. If FORWRD = .FALSE., backward permutation: X(I,) is moved to X(K(I),) for I = 1,2,...,M.

LAPMT rearranges the columns of the M by N matrix X as specified by the permutation K(1),K(2),...,K(N) of the integers 1,...,N. If FORWRD = .TRUE., forward permutation: X(,K(J)) is moved X(,J) for J = 1,2,...,N. If FORWRD = .FALSE., backward permutation: X(,J) is moved to X(,K(J)) for J = 1,2,...,N.

LAQGB equilibrates a general M by N band matrix A with KL subdiagonals and KU superdiagonals using the row and scaling factors in the vectors R and C.

LAQGE equilibrates a general M by N matrix A using the row and column scaling factors in the vectors R and C.

LAQHB equilibrates an Hermitian band matrix A using the scaling factors in the vector S.

LAQHE equilibrates a Hermitian matrix A using the scaling factors in the vector S.

LAQHP equilibrates a Hermitian matrix A using the scaling factors in the vector S.

LAQPS computes a step of QR factorization with column pivoting of a complex M-by-N matrix A by using Blas-3. It tries to factorize NB columns from A starting from the row OFFSET+1, and updates all of the matrix with Blas-3 xGEMM. In some cases, due to catastrophic cancellations, it cannot factorize NB columns. Hence, the actual number of factorized columns is returned in KB. Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

LAQR0 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T ZH, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors. Optionally Z may be postmultiplied into an input unitary matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the unitary matrix Q: A = QHQH = (QZ)H(QZ)**H.

Given a 2-by-2 or 3-by-3 matrix H, LAQR1: sets v to a scalar multiple of the first column of the product () K = (H - s1I)(H - s2I) scaling to avoid overflows and most underflows. This is useful for starting double implicit shift bulges in the QR algorithm.

LAQR4 implements one level of recursion for CLAQR0. It is a complete implementation of the small bulge multi-shift QR algorithm. It may be called by CLAQR0 and, for large enough deflation window size, it may be called by CLAQR3. This subroutine is identical to CLAQR0 except that it calls CLAQR2 instead of CLAQR3. LAQR4 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T ZH, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors. Optionally Z may be postmultiplied into an input unitary matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the unitary matrix Q: A = QHQH = (QZ)H(QZ)**H.

LAQR5 called by CLAQR0 performs a single small-bulge multi-shift QR sweep.

LAQSB equilibrates a symmetric band matrix A using the scaling factors in the vector S.

LAQSP equilibrates a symmetric matrix A using the scaling factors in the vector S.

LAQSY equilibrates a symmetric matrix A using the scaling factors in the vector S.

LAQTR solves the real quasi-triangular system op(T)p = scalec, if LREAL = .TRUE. or the complex quasi-triangular systems op(T + iB)(p+iq) = scale(c+id), if LREAL = .FALSE. in real arithmetic, where T is upper quasi-triangular. If LREAL = .FALSE., then the first diagonal block of T must be 1 by 1, B is the specially structured matrix B = [ b(1) b(2) ... b(n) ] [ w ] [ w ] [ . ] [ w ] op(A) = A or AT, AT denotes the transpose of matrix A. On input, X = [ c ]. On output, X = [ p ]. [ d ] [ q ] This subroutine is designed for the condition number estimation in routine DTRSNA.

LAQZ0 computes the eigenvalues of a matrix pair (H,T), where H is an upper Hessenberg matrix and T is upper triangular, using the double-shift QZ method. Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a matrix pair (A,B): A = Q1HZ1H, B = Q1TZ1H, as computed by CGGHRD. If JOB='S', then the Hessenberg-triangular pair (H,T) is also reduced to generalized Schur form, H = QSZH, T = QPZH, where Q and Z are unitary matrices, P and S are an upper triangular matrices. Optionally, the unitary matrix Q from the generalized Schur factorization may be postmultiplied into an input matrix Q1, and the unitary matrix Z may be postmultiplied into an input matrix Z1. If Q1 and Z1 are the unitary matrices from CGGHRD that reduced the matrix pair (A,B) to generalized upper Hessenberg form, then the output matrices Q1Q and Z1Z are the unitary factors from the generalized Schur factorization of (A,B): A = (Q1Q)S(Z1Z)H, B = (Q1Q)P(Z1Z)H. To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, of (A,B)) are computed as a pair of values (alpha,beta), where alpha is complex and beta real. If beta is nonzero, lambda = alpha / beta is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP) Ax = lambdaBx and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the alternate form of the GNEP muAy = By. Eigenvalues can be read directly from the generalized Schur form: alpha = S(i,i), beta = P(i,i). Ref: C.B. Moler Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973), pp. 241--256. Ref: B. Kagstrom, D. Kressner, "Multishift Variants of the QZ Algorithm with Aggressive Early Deflation", SIAM J. Numer. Anal., 29(2006), pp. 199--227. Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril "A multishift, multipole rational QZ method with agressive early deflation"

LAQZ1 chases a 1x1 shift bulge in a matrix pencil down a single position

LAQZ4 Executes a single multishift QZ sweep

LAR1V computes the (scaled) r-th column of the inverse of the sumbmatrix in rows B1 through BN of the tridiagonal matrix L D LT - sigma I. When sigma is close to an eigenvalue, the computed vector is an accurate eigenvector. Usually, r corresponds to the index where the eigenvector is largest in magnitude. The following steps accomplish this computation : (a) Stationary qd transform, L D LT - sigma I = L(+) D(+) L(+)T, (b) Progressive qd transform, L D LT - sigma I = U(-) D(-) U(-)T, (c) Computation of the diagonal elements of the inverse of L D LT - sigma I by combining the above transforms, and choosing r as the index where the diagonal of the inverse is (one of the) largest in magnitude. (d) Computation of the (scaled) r-th column of the inverse using the twisted factorization obtained by combining the top part of the the stationary and the bottom part of the progressive transform.

LAR2V applies a vector of complex plane rotations with real cosines from both sides to a sequence of 2-by-2 complex Hermitian matrices, defined by the elements of the vectors x, y and z. For i = 1,2,...,n ( x(i) z(i) ) := ( conjg(z(i)) y(i) ) ( c(i) conjg(s(i)) ) ( x(i) z(i) ) ( c(i) -conjg(s(i)) ) ( -s(i) c(i) ) ( conjg(z(i)) y(i) ) ( s(i) c(i) )

LARCM performs a very simple matrix-matrix multiplication: C := A * B, where A is M by M and real; B is M by N and complex; C is M by N and complex.

LARF applies a complex elementary reflector H to a complex M-by-N matrix C, from either the left or the right. H is represented in the form H = I - tau * v * vH where tau is a complex scalar and v is a complex vector. If tau = 0, then H is taken to be the unit matrix. To apply HH (the conjugate transpose of H), supply conjg(tau) instead tau.

LARFB applies a complex block reflector H or its transpose H**H to a complex M-by-N matrix C, from either the left or the right.

LARFB_GETT applies a complex Householder block reflector H from the left to a complex (K+M)-by-N "triangular-pentagonal" matrix composed of two block matrices: an upper trapezoidal K-by-N matrix A stored in the array A, and a rectangular M-by-(N-K) matrix B, stored in the array B. The block reflector H is stored in a compact WY-representation, where the elementary reflectors are in the arrays A, B and T. See Further Details section.

LARFG generates a complex elementary reflector H of order n, such that HH * ( alpha ) = ( beta ), HH * H = I. ( x ) ( 0 ) where alpha and beta are scalars, with beta real, and x is an (n-1)-element complex vector. H is represented in the form H = I - tau * ( 1 ) * ( 1 v**H ) , ( v ) where tau is a complex scalar and v is a complex (n-1)-element vector. Note that H is not hermitian. If the elements of x are all zero and alpha is real, then tau = 0 and H is taken to be the unit matrix. Otherwise 1 <= real(tau) <= 2 and abs(tau-1) <= 1 .

LARFGP generates a complex elementary reflector H of order n, such that HH * ( alpha ) = ( beta ), HH * H = I. ( x ) ( 0 ) where alpha and beta are scalars, beta is real and non-negative, and x is an (n-1)-element complex vector. H is represented in the form H = I - tau * ( 1 ) * ( 1 v**H ) , ( v ) where tau is a complex scalar and v is a complex (n-1)-element vector. Note that H is not hermitian. If the elements of x are all zero and alpha is real, then tau = 0 and H is taken to be the unit matrix.

LARFT forms the triangular factor T of a complex block reflector H of order n, which is defined as a product of k elementary reflectors. If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the array V, and H = I - V * T * VH If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the array V, and H = I - VH * T * V

LARFY applies an elementary reflector, or Householder matrix, H, to an n x n Hermitian matrix C, from both the left and the right. H is represented in the form H = I - tau * v * v' where tau is a scalar and v is a vector. If tau is zero, then H is taken to be the unit matrix.

LARGV generates a vector of complex plane rotations with real cosines, determined by elements of the complex vectors x and y. For i = 1,2,...,n ( c(i) s(i) ) ( x(i) ) = ( r(i) ) ( -conjg(s(i)) c(i) ) ( y(i) ) = ( 0 ) where c(i)2 + ABS(s(i))2 = 1 The following conventions are used (these are the same as in CLARTG, but differ from the BLAS1 routine CROTG): If y(i)=0, then c(i)=1 and s(i)=0. If x(i)=0, then c(i)=0 and s(i) is chosen so that r(i) is real.

LARNV returns a vector of n random complex numbers from a uniform or normal distribution.

Compute the splitting points with threshold SPLTOL. LARRA sets any "small" off-diagonal elements to zero.

Given the relatively robust representation(RRR) L D L^T, LARRB: does "limited" bisection to refine the eigenvalues of L D L^T, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial guesses for these eigenvalues are input in W, the corresponding estimate of the error in these guesses and their gaps are input in WERR and WGAP, respectively. During bisection, intervals [left, right] are maintained by storing their mid-points and semi-widths in the arrays W and WERR respectively.

Find the number of eigenvalues of the symmetric tridiagonal matrix T that are in the interval (VL,VU] if JOBT = 'T', and of L D L^T if JOBT = 'L'.

LARRD computes the eigenvalues of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from DSTEMR. The user may ask for all eigenvalues, all eigenvalues in the half-open interval (VL, VU], or the IL-th through IU-th eigenvalues. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow(1/2) * underflow(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.

To find the desired eigenvalues of a given real symmetric tridiagonal matrix T, LARRE: sets any "small" off-diagonal elements to zero, and for each unreduced block T_i, it finds (a) a suitable shift at one end of the block's spectrum, (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and (c) eigenvalues of each L_i D_i L_i^T. The representations and eigenvalues found are then used by DSTEMR to compute the eigenvectors of T. The accuracy varies depending on whether bisection is used to find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to conpute all and then discard any unwanted one. As an added benefit, LARRE also outputs the n Gerschgorin intervals for the matrices L_i D_i L_i^T.

Given the initial representation L D L^T and its cluster of close eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ... W( CLEND ), LARRF: finds a new relatively robust representation L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the eigenvalues of L(+) D(+) L(+)^T is relatively isolated.

Given the initial eigenvalue approximations of T, LARRJ: does bisection to refine the eigenvalues of T, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial guesses for these eigenvalues are input in W, the corresponding estimate of the error in these guesses in WERR. During bisection, intervals [left, right] are maintained by storing their mid-points and semi-widths in the arrays W and WERR respectively.

LARRK computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from DSTEMR. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow(1/2) * underflow(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.

Perform tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.

LARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and APPROXIMATIONS to the eigenvalues of L D LT. The input eigenvalues should have been computed by SLARRE.

LARTG generates a plane rotation so that [ C S ] . [ F ] = [ R ] [ -conjg(S) C ] [ G ] [ 0 ] where C is real and C2 + |S|2 = 1. The mathematical formulas used for C and S are sgn(x) = { x / |x|, x != 0 { 1, x = 0 R = sgn(F) * sqrt(|F|2 + |G|2) C = |F| / sqrt(|F|2 + |G|2) S = sgn(F) * conjg(G) / sqrt(|F|2 + |G|2) When F and G are real, the formulas simplify to C = F/R and S = G/R, and the returned values of C, S, and R should be identical to those returned by LARTG. The algorithm used to compute these quantities incorporates scaling to avoid overflow or underflow in computing the square root of the sum of squares. This is a faster version of the BLAS1 routine CROTG, except for the following differences: F and G are unchanged on return. If G=0, then C=1 and S=0. If F=0, then C=0 and S is chosen so that R is real. Below, wp=>sp stands for single precision from LA_CONSTANTS module.

LARTGP generates a plane rotation so that [ CS SN ] . [ F ] = [ R ] where CS2 + SN2 = 1. [ -SN CS ] [ G ] [ 0 ] This is a slower, more accurate version of the Level 1 BLAS routine DROTG, with the following other differences: F and G are unchanged on return. If G=0, then CS=(+/-)1 and SN=0. If F=0 and (G .ne. 0), then CS=0 and SN=(+/-)1. The sign is chosen so that R >= 0.

LARTGS generates a plane rotation designed to introduce a bulge in Golub-Reinsch-style implicit QR iteration for the bidiagonal SVD problem. X and Y are the top-row entries, and SIGMA is the shift. The computed CS and SN define a plane rotation satisfying [ CS SN ] . [ X^2 - SIGMA ] = [ R ], [ -SN CS ] [ X * Y ] [ 0 ] with R nonnegative. If X^2 - SIGMA and X * Y are 0, then the rotation is by PI/2.

LARTV applies a vector of complex plane rotations with real cosines to elements of the complex vectors x and y. For i = 1,2,...,n ( x(i) ) := ( c(i) s(i) ) ( x(i) ) ( y(i) ) ( -conjg(s(i)) c(i) ) ( y(i) )

LARUV returns a vector of n random real numbers from a uniform (0,1) distribution (n <= 128). This is an auxiliary routine called by DLARNV and ZLARNV.

LARZ applies a complex elementary reflector H to a complex M-by-N matrix C, from either the left or the right. H is represented in the form H = I - tau * v * vH where tau is a complex scalar and v is a complex vector. If tau = 0, then H is taken to be the unit matrix. To apply HH (the conjugate transpose of H), supply conjg(tau) instead tau. H is a product of k elementary reflectors as returned by CTZRZF.

LARZB applies a complex block reflector H or its transpose H**H to a complex distributed M-by-N C from the left or the right. Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

LARZT forms the triangular factor T of a complex block reflector H of order > n, which is defined as a product of k elementary reflectors. If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the array V, and H = I - V * T * VH If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the array V, and H = I - VH * T * V Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

LASCL multiplies the M by N complex matrix A by the real scalar CTO/CFROM. This is done without over/underflow as long as the final result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that A may be full, upper triangular, lower triangular, upper Hessenberg, or banded.

Using a divide and conquer approach, LASD0: computes the singular value decomposition (SVD) of a real upper bidiagonal N-by-M matrix B with diagonal D and offdiagonal E, where M = N + SQRE. The algorithm computes orthogonal matrices U and VT such that B = U * S * VT. The singular values S are overwritten on D. A related subroutine, DLASDA, computes only the singular values, and optionally, the singular vectors in compact form.

LASD1 computes the SVD of an upper bidiagonal N-by-M matrix B, where N = NL + NR + 1 and M = N + SQRE. LASD1 is called from DLASD0. A related subroutine DLASD7 handles the case in which the singular values (and the singular vectors in factored form) are desired. LASD1 computes the SVD as follows: ( D1(in) 0 0 0 ) B = U(in) * ( Z1T a Z2T b ) * VT(in) ( 0 0 D2(in) 0 ) = U(out) * ( D(out) 0) * VT(out) where ZT = (Z1T a Z2T b) = uT VT**T, and u is a vector of dimension M with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros elsewhere; and the entry b is empty if SQRE = 0. The left singular vectors of the original matrix are stored in U, and the transpose of the right singular vectors are stored in VT, and the singular values are in D. The algorithm consists of three stages: The first stage consists of deflating the size of the problem when there are multiple singular values or when there are zeros in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLASD2. The second stage consists of calculating the updated singular values. This is done by finding the square roots of the roots of the secular equation via the routine DLASD4 (as called by DLASD3). This routine also calculates the singular vectors of the current problem. The final stage consists of computing the updated singular vectors directly using the updated singular values. The singular vectors for the current problem are multiplied with the singular vectors from the overall problem.

This subroutine computes the square root of the I-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix whose entries are given as the squares of the corresponding entries in the array d, and that 0 <= D(i) < D(j) for i < j and that RHO > 0. This is arranged by the calling routine, and is no loss in generality. The rank-one modified system is thus diag( D ) * diag( D ) + RHO * Z * Z_transpose. where we assume the Euclidean norm of Z is 1. The method consists of approximating the rational functions in the secular equation by simpler interpolating rational functions.

This subroutine computes the square root of the I-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix diag( D ) * diag( D ) + RHO * Z * transpose(Z) . The diagonal entries in the array D are assumed to satisfy 0 <= D(i) < D(j) for i < j . We also assume RHO > 0 and that the Euclidean norm of the vector Z is one.

LASD6 computes the SVD of an updated upper bidiagonal matrix B obtained by merging two smaller ones by appending a row. This routine is used only for the problem which requires all singular values and optionally singular vector matrices in factored form. B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE. A related subroutine, DLASD1, handles the case in which all singular values and singular vectors of the bidiagonal matrix are desired. LASD6 computes the SVD as follows: ( D1(in) 0 0 0 ) B = U(in) * ( Z1T a Z2T b ) * VT(in) ( 0 0 D2(in) 0 ) = U(out) * ( D(out) 0) * VT(out) where ZT = (Z1T a Z2T b) = uT VT**T, and u is a vector of dimension M with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros elsewhere; and the entry b is empty if SQRE = 0. The singular values of B can be computed using D1, D2, the first components of all the right singular vectors of the lower block, and the last components of all the right singular vectors of the upper block. These components are stored and updated in VF and VL, respectively, in LASD6. Hence U and VT are not explicitly referenced. The singular values are stored in D. The algorithm consists of two stages: The first stage consists of deflating the size of the problem when there are multiple singular values or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLASD7. The second stage consists of calculating the updated singular values. This is done by finding the roots of the secular equation via the routine DLASD4 (as called by DLASD8). This routine also updates VF and VL and computes the distances between the updated singular values and the old singular values. LASD6 is called from DLASDA.

LASD7 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more singular values are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one. LASD7 is called from DLASD6.

LASD8 finds the square roots of the roots of the secular equation, as defined by the values in DSIGMA and Z. It makes the appropriate calls to DLASD4, and stores, for each element in D, the distance to its two nearest poles (elements in DSIGMA). It also updates the arrays VF and VL, the first and last components of all the right singular vectors of the original bidiagonal matrix. LASD8 is called from DLASD6.

Using a divide and conquer approach, LASDA: computes the singular value decomposition (SVD) of a real upper bidiagonal N-by-M matrix B with diagonal D and offdiagonal E, where M = N + SQRE. The algorithm computes the singular values in the SVD B = U * S * VT. The orthogonal matrices U and VT are optionally computed in compact form. A related subroutine, DLASD0, computes the singular values and the singular vectors in explicit form.

LASDQ computes the singular value decomposition (SVD) of a real (upper or lower) bidiagonal matrix with diagonal D and offdiagonal E, accumulating the transformations if desired. Letting B denote the input bidiagonal matrix, the algorithm computes orthogonal matrices Q and P such that B = Q * S * PT (PT denotes the transpose of P). The singular values S are overwritten on D. The input matrix U is changed to U * Q if desired. The input matrix VT is changed to PT * VT if desired. The input matrix C is changed to QT * C if desired. See "Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK Working Note #3, for a detailed description of the algorithm.

LASET initializes a 2-D array A to BETA on the diagonal and ALPHA on the offdiagonals.

LASQ1 computes the singular values of a real N-by-N bidiagonal matrix with diagonal D and off-diagonal E. The singular values are computed to high relative accuracy, in the absence of denormalization, underflow and overflow. The algorithm was first presented in "Accurate singular values and differential qd algorithms" by K. V. Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230, 1994, and the present implementation is described in "An implementation of the dqds Algorithm (Positive Case)", LAPACK Working Note.

LASQ4 computes an approximation TAU to the smallest eigenvalue using values of d from the previous transform.

LASQ5 computes one dqds transform in ping-pong form, one version for IEEE machines another for non IEEE machines.

LASQ6 computes one dqd (shift equal to zero) transform in ping-pong form, with protection against underflow and overflow.

LASR applies a sequence of real plane rotations to a complex matrix A, from either the left or the right. When SIDE = 'L', the transformation takes the form A := PA and when SIDE = 'R', the transformation takes the form A := APT where P is an orthogonal matrix consisting of a sequence of z plane rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R', and PT is the transpose of P. When DIRECT = 'F' (Forward sequence), then P = P(z-1) * ... * P(2) * P(1) and when DIRECT = 'B' (Backward sequence), then P = P(1) * P(2) * ... * P(z-1) where P(k) is a plane rotation matrix defined by the 2-by-2 rotation R(k) = ( c(k) s(k) ) = ( -s(k) c(k) ). When PIVOT = 'V' (Variable pivot), the rotation is performed for the plane (k,k+1), i.e., P(k) has the form P(k) = ( 1 ) ( ... ) ( 1 ) ( c(k) s(k) ) ( -s(k) c(k) ) ( 1 ) ( ... ) ( 1 ) where R(k) appears as a rank-2 modification to the identity matrix in rows and columns k and k+1. When PIVOT = 'T' (Top pivot), the rotation is performed for the plane (1,k+1), so P(k) has the form P(k) = ( c(k) s(k) ) ( 1 ) ( ... ) ( 1 ) ( -s(k) c(k) ) ( 1 ) ( ... ) ( 1 ) where R(k) appears in rows and columns 1 and k+1. Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is performed for the plane (k,z), giving P(k) the form P(k) = ( 1 ) ( ... ) ( 1 ) ( c(k) s(k) ) ( 1 ) ( ... ) ( 1 ) ( -s(k) c(k) ) where R(k) appears in rows and columns k and z. The rotations are performed without ever forming P(k) explicitly.

Sort the numbers in D in increasing order (if ID = 'I') or in decreasing order (if ID = 'D' ). Use Quick Sort, reverting to Insertion sort on arrays of size <= 20. Dimension of STACK limits N to about 2**32.

LASSQ returns the values scl and smsq such that ( scl2 )*smsq = x( 1 )2 +...+ x( n )2 + ( scale2 )sumsq, where x( i ) = X( 1 + ( i - 1 )INCX ). The value of sumsq is assumed to be non-negative. scale and sumsq must be supplied in SCALE and SUMSQ and scl and smsq are overwritten on SCALE and SUMSQ respectively. If scale * sqrt( sumsq ) > tbig then we require: scale >= sqrt( TINYEPS ) / sbig on entry, and if 0 < scale * sqrt( sumsq ) < tsml then we require: scale <= sqrt( HUGE ) / ssml on entry, where tbig -- upper threshold for values whose square is representable; sbig -- scaling constant for big numbers; \see la_constants.f90 tsml -- lower threshold for values whose square is representable; ssml -- scaling constant for small numbers; \see la_constants.f90 and TINYEPS -- tiniest representable number; HUGE -- biggest representable number.

LASWLQ computes a blocked Tall-Skinny LQ factorization of a complex M-by-N matrix A for M <= N: A = ( L 0 ) * Q, where: Q is a n-by-N orthogonal matrix, stored on exit in an implicit form in the elements above the diagonal of the array A and in the elements of the array T; L is a lower-triangular M-by-M matrix stored on exit in the elements on and below the diagonal of the array A. 0 is a M-by-(N-M) zero matrix, if M < N, and is not stored.

LASWP performs a series of row interchanges on the matrix A. One row interchange is initiated for each of rows K1 through K2 of A.

LASYF computes a partial factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The partial factorization has the form: A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: ( 0 U22 ) ( 0 D ) ( U12T U22T ) A = ( L11 0 ) ( D 0 ) ( L11T L21T ) if UPLO = 'L' ( L21 I ) ( 0 A22 ) ( 0 I ) where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. Note that U**T denotes the transpose of U. LASYF is an auxiliary routine called by CSYTRF. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

DLATRF_AA factorizes a panel of a complex symmetric matrix A using the Aasen's algorithm. The panel consists of a set of NB rows of A when UPLO is U, or a set of NB columns when UPLO is L. In order to factorize the panel, the Aasen's algorithm requires the last row, or column, of the previous panel. The first row, or column, of A is set to be the first row, or column, of an identity matrix, which is used to factorize the first panel. The resulting J-th row of U, or J-th column of L, is stored in the (J-1)-th row, or column, of A (without the unit diagonals), while the diagonal and subdiagonal of A are overwritten by those of T.

LASYF_RK computes a partial factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method. The partial factorization has the form: A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: ( 0 U22 ) ( 0 D ) ( U12T U22T ) A = ( L11 0 ) ( D 0 ) ( L11T L21T ) if UPLO = 'L', ( L21 I ) ( 0 A22 ) ( 0 I ) where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. LASYF_RK is an auxiliary routine called by CSYTRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

LASYF_ROOK computes a partial factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting method. The partial factorization has the form: A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: ( 0 U22 ) ( 0 D ) ( U12T U22T ) A = ( L11 0 ) ( D 0 ) ( L11T L21T ) if UPLO = 'L' ( L21 I ) ( 0 A22 ) ( 0 I ) where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. LASYF_ROOK is an auxiliary routine called by CSYTRF_ROOK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

LATBS solves one of the triangular systems A * x = sb, AT * x = sb, or AH * x = s*b, with scaling to prevent overflow, where A is an upper or lower triangular band matrix. Here AT denotes the transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine CTBSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

LATDF computes the contribution to the reciprocal Dif-estimate by solving for x in Z * x = b, where b is chosen such that the norm of x is as large as possible. It is assumed that LU decomposition of Z has been computed by CGETC2. On entry RHS = f holds the contribution from earlier solved sub-systems, and on return RHS = x. The factorization of Z returned by CGETC2 has the form Z = P * L * U * Q, where P and Q are permutation matrices. L is lower triangular with unit diagonal elements and U is upper triangular.

LATPS solves one of the triangular systems A * x = sb, AT * x = sb, or AH * x = s*b, with scaling to prevent overflow, where A is an upper or lower triangular matrix stored in packed form. Here AT denotes the transpose of A, A*H denotes the conjugate transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine CTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to Ax = 0 is returned.

LATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Q**H * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. If UPLO = 'U', LATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = 'L', LATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied. This is an auxiliary routine called by CHETRD.

LATRS solves one of the triangular systems A * x = sb, AT * x = sb, or AH * x = s*b, with scaling to prevent overflow. Here A is an upper or lower triangular matrix, AT denotes the transpose of A, A*H denotes the conjugate transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine CTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to Ax = 0 is returned.

LATRZ factors the M-by-(M+L) complex upper trapezoidal matrix [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z by means of unitary transformations, where Z is an (M+L)-by-(M+L) unitary matrix and, R and A1 are M-by-M upper triangular matrices.

LATSQR computes a blocked Tall-Skinny QR factorization of a complex M-by-N matrix A for M >= N: A = Q * ( R ), ( 0 ) where: Q is a M-by-M orthogonal matrix, stored on exit in an implicit form in the elements below the diagonal of the array A and in the elements of the array T; R is an upper-triangular N-by-N matrix, stored on exit in the elements on and above the diagonal of the array A. 0 is a (M-N)-by-N zero matrix, and is not stored.

LAUNHR_COL_GETRFNP computes the modified LU factorization without pivoting of a complex general M-by-N matrix A. The factorization has the form: A - S = L * U, where: S is a m-by-n diagonal sign matrix with the diagonal D, so that D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing i-1 steps of Gaussian elimination. This means that the diagonal element at each step of "modified" Gaussian elimination is at least one in absolute value (so that division-by-zero not not possible during the division by the diagonal element); L is a M-by-N lower triangular matrix with unit diagonal elements (lower trapezoidal if M > N); and U is a M-by-N upper triangular matrix (upper trapezoidal if M < N). This routine is an auxiliary routine used in the Householder reconstruction routine CUNHR_COL. In CUNHR_COL, this routine is applied to an M-by-N matrix A with orthonormal columns, where each element is bounded by one in absolute value. With the choice of the matrix S above, one can show that the diagonal element at each step of Gaussian elimination is the largest (in absolute value) in the column on or below the diagonal, so that no pivoting is required for numerical stability [1]. For more details on the Householder reconstruction algorithm, including the modified LU factorization, see [1]. This is the blocked right-looking version of the algorithm, calling Level 3 BLAS to update the submatrix. To factorize a block, this routine calls the recursive routine LAUNHR_COL_GETRFNP2. [1] "Reconstructing Householder vectors from tall-skinny QR", G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen, E. Solomonik, J. Parallel Distrib. Comput., vol. 85, pp. 3-31, 2015.

LAUNHR_COL_GETRFNP2 computes the modified LU factorization without pivoting of a complex general M-by-N matrix A. The factorization has the form: A - S = L * U, where: S is a m-by-n diagonal sign matrix with the diagonal D, so that D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing i-1 steps of Gaussian elimination. This means that the diagonal element at each step of "modified" Gaussian elimination is at least one in absolute value (so that division-by-zero not possible during the division by the diagonal element); L is a M-by-N lower triangular matrix with unit diagonal elements (lower trapezoidal if M > N); and U is a M-by-N upper triangular matrix (upper trapezoidal if M < N). This routine is an auxiliary routine used in the Householder reconstruction routine CUNHR_COL. In CUNHR_COL, this routine is applied to an M-by-N matrix A with orthonormal columns, where each element is bounded by one in absolute value. With the choice of the matrix S above, one can show that the diagonal element at each step of Gaussian elimination is the largest (in absolute value) in the column on or below the diagonal, so that no pivoting is required for numerical stability [1]. For more details on the Householder reconstruction algorithm, including the modified LU factorization, see [1]. This is the recursive version of the LU factorization algorithm. Denote A - S by B. The algorithm divides the matrix B into four submatrices: [ B11 | B12 ] where B11 is n1 by n1, B = [ -----|----- ] B21 is (m-n1) by n1, [ B21 | B22 ] B12 is n1 by n2, B22 is (m-n1) by n2, with n1 = min(m,n)/2, n2 = n-n1. The subroutine calls itself to factor B11, solves for B21, solves for B12, updates B22, then calls itself to factor B22. For more details on the recursive LU algorithm, see [2]. LAUNHR_COL_GETRFNP2 is called to factorize a block by the blocked routine CLAUNHR_COL_GETRFNP, which uses blocked code calling Level 3 BLAS to update the submatrix. However, LAUNHR_COL_GETRFNP2 is self-sufficient and can be used without CLAUNHR_COL_GETRFNP. [1] "Reconstructing Householder vectors from tall-skinny QR", G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen, E. Solomonik, J. Parallel Distrib. Comput., vol. 85, pp. 3-31, 2015. [2] "Recursion leads to automatic variable blocking for dense linear algebra algorithms", F. Gustavson, IBM J. of Res. and Dev., vol. 41, no. 6, pp. 737-755, 1997.

LAUUM computes the product U * UH or LH * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in A. If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in A. This is the blocked form of the algorithm, calling Level 3 BLAS.

leaky Rectified linear unit function

Gradient of the leaky Rectified linear unit function

Legendre polynomial

Returns the length of the character sequence represented by the string.

Returns the length of the character sequence without trailing spaces represented by the string.

Lexically compare the order of two character sequences being greater equal, The left-hand side, the right-hand side or both character sequences can be represented by a string.

Lexically compare the order of two character sequences being greater, The left-hand side, the right-hand side or both character sequences can be represented by a string.

Flow control: on output flag present, return it; otherwise, halt on error

Create rank 1 array of linearly spaced elements If the number of elements is not specified, create an array with size 100. If n is a negative value, return an array with size 0. If n = 1, return an array whose only element is end (Specification)

Lexically compare the order of two character sequences being less equal, The left-hand side, the right-hand side or both character sequences can be represented by a string.

Lexically compare the order of two character sequences being less, The left-hand side, the right-hand side or both character sequences can be represented by a string.

Load multidimensional array in npy format (Specification)

Loads a 2D array from a text file (Specification)

Logarithm of factorial n!, integer variable

Logarithm of gamma function

Logarithm of lower incomplete gamma function

Logarithm of upper incomplete gamma function

softmax function. Available for ranks 1 to 4

Create rank 1 array of logarithmically spaced elements from basestart to baseend. If the number of elements is not specified, create an array with size 50. If n is a negative value, return an array with size 0. If n = 1, return an array whose only element is base**end. If no base is specified, logspace will default to using a base of 10

Lower incomplete gamma function

Computes the squares solution to system $A \cdot x = b$. (Specification)

Computes the integer, real [, complex] working space required by the least-squares solver (Specification)

Matrix exponential: subroutine interface version : experimental

Mean of array elements (Specification)

Median of array elements (Specification)

Computes a list of coordinate matrices from coordinate vectors. (Specification)

Matrix norms: function interface version: experimental

Central moment of array elements (Specification)

Moves the allocated character scalar from 'from' to 'to' Specifications

(Specification

(Specification)

(Specification)

NMHASH32 interfaces (Specification)

NMHASH32X interfaces (Specification)

Computes the vector norm of a generic-rank array $A$. (Specification)

NRM2 returns the euclidean norm of a vector via the function name, so that NRM2 := sqrt( x'*x )

Returns a 64 bit pseudo random integer, harvest, distributed uniformly over the odd integers of the 64 bit kind. (Specification)

Returns a 32 bit pseudo random integer, harvest, distributed uniformly over the odd integers of the int32 kind. (Specification)

Opens a file (Specification)

Overload the * for scalar-matrix multiplications for the different matrix types provided by stdlib_specialmatrices. Specifications

Overload the + operator for matrix-matrix addition. The two matrices need to be of the same type and kind. Specifications

Overload the - operator for matrix-matrix subtraction. The two matrices need to be of the same type and kind. Specifications

Determinant operator of a square matrix (Specification)

Inverse operator of a square matrix (Specification)

Pseudo-inverse operator of a matrix (Specification)

Concatenates stringlist with the input entity Returns a new stringlist Specifications

Concatenate two character sequences, the left-hand side, the right-hand side or both character sequences can be represented by a string.

Compares stringlist for inequality with the input entity Returns a logical Specifications

Returns .true. if not all bits in set1 and set2 have the same value, .false. otherwise. The sets must have the same number of bits otherwise the result is undefined. (Specification)

Compare two character sequences for inequality, the left-hand side, the right-hand side or both character sequences can be represented by a string.

Returns .true. if the bits in set1 and set2 differ and the highest order different bit is set to 0 in set1 and to 1 in set2, .false. otherwise. The sets must have the same number of bits otherwise the result is undefined. (Specification)

Compare two character sequences for being less, the left-hand side, the right-hand side or both character sequences can be represented by a string.

Returns .true. if the bits in set1 and set2 are the same or the highest order different bit is set to 0 in set1 and to 1 in set2, .false. otherwise. The sets must have the same number of bits otherwise the result is undefined. (Specification)

Compare two character sequences for being less than, the left-hand side, the right-hand side or both character sequences can be represented by a string.

Compares stringlist for equality with the input entity Returns a logical Specifications

Comparison operators

Returns .true. if all bits in set1 and set2 have the same value, .false. otherwise. The sets must have the same number of bits otherwise the result is undefined. (Specification)

Compare two character sequences for equality, the left-hand side, the right-hand side or both character sequences can be represented by a string.

Returns .true. if the bits in set1 and set2 differ and the highest order different bit is set to 1 in set1 and to 0 in set2, .false. otherwise. The sets must have the same number of bits otherwise the result is undefined. (Specification)

Compare two character sequences for being greater, the left-hand side, the right-hand side or both character sequences can be represented by a string.

Returns .true. if the bits in set1 and set2 are the same or the highest order different bit is set to 1 in set1 and to 0 in set2, .false. otherwise. The sets must have the same number of bits otherwise the result is undefined. (Specification)

Compare two character sequences for being greater than, the left-hand side, the right-hand side or both character sequences can be represented by a string.

OPGTR generates a real orthogonal matrix Q which is defined as the product of n-1 elementary reflectors H(i) of order n, as returned by DSPTRD using packed storage: if UPLO = 'U', Q = H(n-1) . . . H(2) H(1), if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).

OPMTR overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': QT * C C * QT where Q is a real orthogonal matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of nq-1 elementary reflectors, as returned by DSPTRD using packed storage: if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1); if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).

Fallback value for optional arguments (Specification)

Sets the bits in set1 to the bitwise or of the original bits in set1 and set2. The sets must have the same number of bits otherwise the result is undefined. (Specification)

ORBDB simultaneously bidiagonalizes the blocks of an M-by-M partitioned orthogonal matrix X: [ B11 | B12 0 0 ] [ X11 | X12 ] [ P1 | ] [ 0 | 0 -I 0 ] [ Q1 | ]**T X = [-----------] = [---------] [----------------] [---------] . [ X21 | X22 ] [ | P2 ] [ B21 | B22 0 0 ] [ | Q2 ] [ 0 | 0 0 I ] X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is not the case, then X must be transposed and/or permuted. This can be done in constant time using the TRANS and SIGNS options. See DORCSD for details.) The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by- (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are represented implicitly by Householder vectors. B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented implicitly by angles THETA, PHI.

ORBDB1 simultaneously bidiagonalizes the blocks of a tall and skinny matrix X with orthonomal columns: [ B11 ] [ X11 ] [ P1 | ] [ 0 ] [-----] = [---------] [-----] Q1**T . [ X21 ] [ | P2 ] [ B21 ] [ 0 ] X11 is P-by-Q, and X21 is (M-P)-by-Q. Q must be no larger than P, M-P, or M-Q. Routines DORBDB2, DORBDB3, and DORBDB4 handle cases in which Q is not the minimum dimension. The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P), and (M-Q)-by-(M-Q), respectively. They are represented implicitly by Householder vectors. B11 and B12 are Q-by-Q bidiagonal matrices represented implicitly by angles THETA, PHI.

ORBDB2 simultaneously bidiagonalizes the blocks of a tall and skinny matrix X with orthonomal columns: [ B11 ] [ X11 ] [ P1 | ] [ 0 ] [-----] = [---------] [-----] Q1**T . [ X21 ] [ | P2 ] [ B21 ] [ 0 ] X11 is P-by-Q, and X21 is (M-P)-by-Q. P must be no larger than M-P, Q, or M-Q. Routines DORBDB1, DORBDB3, and DORBDB4 handle cases in which P is not the minimum dimension. The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P), and (M-Q)-by-(M-Q), respectively. They are represented implicitly by Householder vectors. B11 and B12 are P-by-P bidiagonal matrices represented implicitly by angles THETA, PHI.

ORBDB3 simultaneously bidiagonalizes the blocks of a tall and skinny matrix X with orthonomal columns: [ B11 ] [ X11 ] [ P1 | ] [ 0 ] [-----] = [---------] [-----] Q1**T . [ X21 ] [ | P2 ] [ B21 ] [ 0 ] X11 is P-by-Q, and X21 is (M-P)-by-Q. M-P must be no larger than P, Q, or M-Q. Routines DORBDB1, DORBDB2, and DORBDB4 handle cases in which M-P is not the minimum dimension. The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P), and (M-Q)-by-(M-Q), respectively. They are represented implicitly by Householder vectors. B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented implicitly by angles THETA, PHI.

ORBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny matrix X with orthonomal columns: [ B11 ] [ X11 ] [ P1 | ] [ 0 ] [-----] = [---------] [-----] Q1**T . [ X21 ] [ | P2 ] [ B21 ] [ 0 ] X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P, M-P, or Q. Routines DORBDB1, DORBDB2, and DORBDB3 handle cases in which M-Q is not the minimum dimension. The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P), and (M-Q)-by-(M-Q), respectively. They are represented implicitly by Householder vectors. B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented implicitly by angles THETA, PHI.

ORBDB5 orthogonalizes the column vector X = [ X1 ] [ X2 ] with respect to the columns of Q = [ Q1 ] . [ Q2 ] The columns of Q must be orthonormal. If the projection is zero according to Kahan's "twice is enough" criterion, then some other vector from the orthogonal complement is returned. This vector is chosen in an arbitrary but deterministic way.

ORBDB6 orthogonalizes the column vector X = [ X1 ] [ X2 ] with respect to the columns of Q = [ Q1 ] . [ Q2 ] The columns of Q must be orthonormal. If the projection is zero according to Kahan's "twice is enough" criterion, then the zero vector is returned.

ORCSD computes the CS decomposition of an M-by-M partitioned orthogonal matrix X: [ I 0 0 | 0 0 0 ] [ 0 C 0 | 0 -S 0 ] [ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**T X = [-----------] = [---------] [---------------------] [---------] . [ X21 | X22 ] [ | U2 ] [ 0 0 0 | I 0 0 ] [ | V2 ] [ 0 S 0 | 0 C 0 ] [ 0 0 I | 0 0 0 ] X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P, (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in which R = MIN(P,M-P,Q,M-Q).

ORCSD2BY1 computes the CS decomposition of an M-by-Q matrix X with orthonormal columns that has been partitioned into a 2-by-1 block structure: [ I1 0 0 ] [ 0 C 0 ] [ X11 ] [ U1 | ] [ 0 0 0 ] X = [-----] = [---------] [----------] V1**T . [ X21 ] [ | U2 ] [ 0 0 0 ] [ 0 S 0 ] [ 0 0 I2] X11 is P-by-Q. The orthogonal matrices U1, U2, and V1 are P-by-P, (M-P)-by-(M-P), and Q-by-Q, respectively. C and S are R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in which R = MIN(P,M-P,Q,M-Q). I1 is a K1-by-K1 identity matrix and I2 is a K2-by-K2 identity matrix, where K1 = MAX(Q+P-M,0), K2 = MAX(Q-P,0).

The generic subroutine interface implementing the ORD_SORT algorithm, a translation to Fortran 2008, of the "Rust" sort algorithm found in slice.rs https://github.com/rust-lang/rust/blob/90eb44a5897c39e3dff9c7e48e3973671dcd9496/src/liballoc/slice.rs#L2159 ORD_SORT is a hybrid stable comparison algorithm combining merge sort, and insertion sort. (Specification)

ORG2L generates an m by n real matrix Q with orthonormal columns, which is defined as the last n columns of a product of k elementary reflectors of order m Q = H(k) . . . H(2) H(1) as returned by DGEQLF.

ORG2R generates an m by n real matrix Q with orthonormal columns, which is defined as the first n columns of a product of k elementary reflectors of order m Q = H(1) H(2) . . . H(k) as returned by DGEQRF.

ORGBR generates one of the real orthogonal matrices Q or PT determined by DGEBRD when reducing a real matrix A to bidiagonal form: A = Q * B * PT. Q and PT 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 ORGBR returns the first n columns of Q, where m >= n >= k; if m < k, Q = H(1) H(2) . . . H(m-1) and ORGBR returns Q as an M-by-M matrix. If VECT = 'P', A is assumed to have been a K-by-N matrix, and PT is of order N: if k < n, PT = G(k) . . . G(2) G(1) and ORGBR returns the first m rows of PT, where n >= m >= k; if k >= n, PT = G(n-1) . . . G(2) G(1) and ORGBR returns PT as an N-by-N matrix.

ORGHR generates a real orthogonal matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by DGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

ORGLQ generates an M-by-N real matrix Q with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N Q = H(k) . . . H(2) H(1) as returned by DGELQF.

ORGQL generates an M-by-N real matrix Q with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M Q = H(k) . . . H(2) H(1) as returned by DGEQLF.

ORGQR generates an M-by-N real matrix Q with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M Q = H(1) H(2) . . . H(k) as returned by DGEQRF.

ORGRQ generates an M-by-N real matrix Q with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N Q = H(1) H(2) . . . H(k) as returned by DGERQF.

ORGTR generates a real orthogonal matrix Q which is defined as the product of n-1 elementary reflectors of order N, as returned by DSYTRD: if UPLO = 'U', Q = H(n-1) . . . H(2) H(1), if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).

ORGTSQR generates an M-by-N real matrix Q_out with orthonormal columns, which are the first N columns of a product of real orthogonal matrices of order M which are returned by DLATSQR Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ). See the documentation for DLATSQR.

ORGTSQR_ROW generates an M-by-N real matrix Q_out with orthonormal columns from the output of DLATSQR. These N orthonormal columns are the first N columns of a product of complex unitary matrices Q(k)_in of order M, which are returned by DLATSQR in a special format. Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ). The input matrices Q(k)_in are stored in row and column blocks in A. See the documentation of DLATSQR for more details on the format of Q(k)_in, where each Q(k)_in is represented by block Householder transformations. This routine calls an auxiliary routine DLARFB_GETT, where the computation is performed on each individual block. The algorithm first sweeps NB-sized column blocks from the right to left starting in the bottom row block and continues to the top row block (hence _ROW in the routine name). This sweep is in reverse order of the order in which DLATSQR generates the output blocks.

ORHR_COL takes an M-by-N real matrix Q_in with orthonormal columns as input, stored in A, and performs Householder Reconstruction (HR), i.e. reconstructs Householder vectors V(i) implicitly representing another M-by-N matrix Q_out, with the property that Q_in = Q_out*S, where S is an N-by-N diagonal matrix with diagonal entries equal to +1 or -1. The Householder vectors (columns V(i) of V) are stored in A on output, and the diagonal entries of S are stored in D. Block reflectors are also returned in T (same output format as DGEQRT).

ORM2L overwrites the general real m by n matrix C with Q * C if SIDE = 'L' and TRANS = 'N', or QT * C if SIDE = 'L' and TRANS = 'T', or C * Q if SIDE = 'R' and TRANS = 'N', or C * QT if SIDE = 'R' and TRANS = 'T', where Q is a real orthogonal matrix defined as the product of k elementary reflectors Q = H(k) . . . H(2) H(1) as returned by DGEQLF. Q is of order m if SIDE = 'L' and of order n if SIDE = 'R'.

ORM2R overwrites the general real m by n matrix C with Q * C if SIDE = 'L' and TRANS = 'N', or QT* C if SIDE = 'L' and TRANS = 'T', or C * Q if SIDE = 'R' and TRANS = 'N', or C * QT if SIDE = 'R' and TRANS = 'T', where Q is a real orthogonal matrix defined as the product of k elementary reflectors Q = H(1) H(2) . . . H(k) as returned by DGEQRF. Q is of order m if SIDE = 'L' and of order n if SIDE = 'R'.

If VECT = 'Q', ORMBR: overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': QT * C C * QT If VECT = 'P', ORMBR overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': P * C C * P TRANS = 'T': PT * C C * PT Here Q and PT are the orthogonal matrices determined by DGEBRD when reducing a real matrix A to bidiagonal form: A = Q * B * PT. Q and PT 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 PT 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).

ORMHR overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': QT * C C * QT where Q is a real orthogonal matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of IHI-ILO elementary reflectors, as returned by DGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

ORMLQ overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': QT * C C * QT where Q is a real orthogonal matrix defined as the product of k elementary reflectors Q = H(k) . . . H(2) H(1) as returned by DGELQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'.

ORMQL overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': QT * C C * QT where Q is a real orthogonal matrix defined as the product of k elementary reflectors Q = H(k) . . . H(2) H(1) as returned by DGEQLF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'.

ORMQR overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': QT * C C * QT where Q is a real orthogonal matrix defined as the product of k elementary reflectors Q = H(1) H(2) . . . H(k) as returned by DGEQRF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'.

ORMRQ overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': QT * C C * QT where Q is a real orthogonal matrix defined as the product of k elementary reflectors Q = H(1) H(2) . . . H(k) as returned by DGERQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'.

ORMRZ overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': QT * C C * QT where Q is a real orthogonal matrix defined as the product of k elementary reflectors Q = H(1) H(2) . . . H(k) as returned by DTZRZF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'.

ORMTR overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': QT * C C * QT where Q is a real orthogonal matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of nq-1 elementary reflectors, as returned by DSYTRD: if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1); if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).

Computes the outer product of two vectors, returning a rank-2 array (Specification)

Left pad the input string Specifications

Right pad the input string Specifications

PBCON estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite band matrix using the Cholesky factorization A = UHU or A = LLH computed by CPBTRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

PBEQU computes row and column scalings intended to equilibrate a Hermitian positive definite band matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)A(i,j)S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings.

PBRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and banded, and provides error bounds and backward error estimates for the solution.

PBSTF computes a split Cholesky factorization of a complex Hermitian positive definite band matrix A. This routine is designed to be used in conjunction with CHBGST. The factorization has the form A = S*HS where S is a band matrix of the same bandwidth as A and the following structure: S = ( U ) ( M L ) where U is upper triangular of order m = (n+kd)/2, and L is lower triangular of order n-m.

PBSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite band matrix and X and B are N-by-NRHS matrices. The Cholesky decomposition is used to factor A as A = UH * U, if UPLO = 'U', or A = L * LH, if UPLO = 'L', where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations A * X = B.

PBTRF computes the Cholesky factorization of a complex Hermitian positive definite band matrix A. The factorization has the form A = UH * U, if UPLO = 'U', or A = L * LH, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular.

PBTRS solves a system of linear equations AX = B with a Hermitian positive definite band matrix A using the Cholesky factorization A = UHU or A = LL*H computed by CPBTRF.

Version experimental

Normal Distribution Probability Density Function (Specification)

Get uniform distribution probability density (pdf) for integer, real and complex variables. (Specification)

PENGY_HASH interfaces (Specification)

PFTRF computes the Cholesky factorization of a complex Hermitian positive definite matrix A. The factorization has the form A = UH * U, if UPLO = 'U', or A = L * LH, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular. This is the block version of the algorithm, calling Level 3 BLAS.

PFTRI computes the inverse of a complex Hermitian positive definite matrix A using the Cholesky factorization A = UHU or A = LLH computed by CPFTRF.

PFTRS solves a system of linear equations AX = B with a Hermitian positive definite matrix A using the Cholesky factorization A = UHU or A = LL*H computed by CPFTRF.

Pseudo-inverse of a matrix (Specification)

POCON estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite matrix using the Cholesky factorization A = UHU or A = LLH computed by CPOTRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

POEQU computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)A(i,j)S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings.

POEQUB computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)A(i,j)S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. This routine differs from CPOEQU by restricting the scaling factors to a power of the radix. Barring over- and underflow, scaling by these factors introduces no additional rounding errors. However, the scaled diagonal entries are no longer approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).

PORFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite, and provides error bounds and backward error estimates for the solution.

POSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite matrix and X and B are N-by-NRHS matrices. The Cholesky decomposition is used to factor A as A = UH* U, if UPLO = 'U', or A = L * LH, if UPLO = 'L', where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used to solve the system of equations A * X = B.

POTRF computes the Cholesky factorization of a complex Hermitian positive definite matrix A. The factorization has the form A = UH * U, if UPLO = 'U', or A = L * LH, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular. This is the block version of the algorithm, calling Level 3 BLAS.

POTRF2 computes the Cholesky factorization of a Hermitian positive definite matrix A using the recursive algorithm. The factorization has the form A = UH * U, if UPLO = 'U', or A = L * LH, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular. This is the recursive version of the algorithm. It divides the matrix into four submatrices: [ A11 | A12 ] where A11 is n1 by n1 and A22 is n2 by n2 A = [ -----|----- ] with n1 = n/2 [ A21 | A22 ] n2 = n-n1 The subroutine calls itself to factor A11. Update and scale A21 or A12, update A22 then calls itself to factor A22.

POTRI computes the inverse of a complex Hermitian positive definite matrix A using the Cholesky factorization A = UHU or A = LLH computed by CPOTRF.

POTRS solves a system of linear equations AX = B with a Hermitian positive definite matrix A using the Cholesky factorization A = UHU or A = LL*H computed by CPOTRF.

PPCON estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite packed matrix using the Cholesky factorization A = UHU or A = LLH computed by CPPTRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

PPEQU computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A in packed storage and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j)=S(i)A(i,j)S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings.

PPRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and packed, and provides error bounds and backward error estimates for the solution.

PPSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite matrix stored in packed format and X and B are N-by-NRHS matrices. The Cholesky decomposition is used to factor A as A = UH * U, if UPLO = 'U', or A = L * LH, if UPLO = 'L', where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used to solve the system of equations A * X = B.

PPTRF computes the Cholesky factorization of a complex Hermitian positive definite matrix A stored in packed format. The factorization has the form A = UH * U, if UPLO = 'U', or A = L * LH, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular.

PPTRI computes the inverse of a complex Hermitian positive definite matrix A using the Cholesky factorization A = UHU or A = LLH computed by CPPTRF.

PPTRS solves a system of linear equations AX = B with a Hermitian positive definite matrix A in packed storage using the Cholesky factorization A = UHU or A = LL*H computed by CPPTRF.

Computation of the Moore-Penrose pseudo-inverse (Specification)

PSTRF computes the Cholesky factorization with complete pivoting of a complex Hermitian positive semidefinite matrix A. The factorization has the form PT * A * P = UH * U , if UPLO = 'U', PT * A * P = L * LH, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular, and P is stored as vector PIV. This algorithm does not attempt to check that A is positive semidefinite. This version of the algorithm calls level 3 BLAS.

PTCON computes the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite tridiagonal matrix using the factorization A = LDLH or A = UHDU computed by CPTTRF. Norm(inv(A)) is computed by a direct method, and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

PTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using SPTTRF and then calling CBDSQR to compute the singular values of the bidiagonal factor. This routine computes the eigenvalues of the positive definite tridiagonal matrix to high relative accuracy. This means that if the eigenvalues range over many orders of magnitude in size, then the small eigenvalues and corresponding eigenvectors will be computed more accurately than, for example, with the standard QR method. The eigenvectors of a full or band positive definite Hermitian matrix can also be found if CHETRD, CHPTRD, or CHBTRD has been used to reduce this matrix to tridiagonal form. (The reduction to tridiagonal form, however, may preclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix, if these eigenvalues range over many orders of magnitude.)

PTRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and tridiagonal, and provides error bounds and backward error estimates for the solution.

PTSV computes the solution to a complex system of linear equations AX = B, where A is an N-by-N Hermitian positive definite tridiagonal matrix, and X and B are N-by-NRHS matrices. A is factored as A = LDL*H, and the factored form of A is then used to solve the system of equations.

PTTRF computes the LDLH factorization of a complex Hermitian positive definite tridiagonal matrix A. The factorization may also be regarded as having the form A = UH DU.

PTTRS solves a tridiagonal system of the form A * X = B using the factorization A = UHDU or A = LDLH computed by CPTTRF. D is a diagonal matrix specified in the vector D, U (or L) is a unit bidiagonal matrix whose superdiagonal (subdiagonal) is specified in the vector E, and X and B are N by NRHS matrices.

Computes the QR factorization of matrix $A = Q R$. (Specification)

Computes the working array space required by the QR factorization solver (Specification)

rad2deg converts phase angles from radians to degrees. (Specification)

The generic subroutine interface implementing the LSD radix sort algorithm, see https://en.wikipedia.org/wiki/Radix_sort for more details. It is always O(N) in sorting random data, but need a O(N) buffer. (Specification)

Version experimental

Read a character sequence from a connected unformatted unit into the string.

Read a character sequence from a connected unformatted unit into the string.

Regularized (normalized) lower incomplete gamma function, P

Regularized (normalized) upper incomplete gamma function, Q

Rectified linear unit function

Gradient rectified linear unit function

Repeats the character sequence hold by the string by the number of specified copies.

Replaces all the occurrences of substring 'pattern' in the input 'string' with the replacement 'replacement' Version: experimental

Reverses the character sequence hold by the input string

Reverse the character order in the input character variable (Specification)

ROT applies a plane rotation, where the cos (C) is real and the sin (S) is complex, and the vectors CX and CY are complex.

ROT applies a plane rotation.

The computation uses the formulas |x| = sqrt( Re(x)2 + Im(x)2 ) sgn(x) = x / |x| if x /= 0 = 1 if x = 0 c = |a| / sqrt(|a|2 + |b|2) s = sgn(a) * conjg(b) / sqrt(|a|2 + |b|2) When a and b are real and r /= 0, the formulas simplify to r = sgn(a)sqrt(|a|2 + |b|*2) c = a / r s = b / r the same as in SROTG when |a| > |b|. When |b| >= |a|, the sign of c and s will be different from those computed by SROTG if the signs of a and b are not the same.

ROTM 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+IINCX), 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 ROTMG for a description of data storage in DPARAM.

ROTMG 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.

RSCL multiplies an n-element real vector x by the real scalar 1/a. This is done without overflow or underflow as long as the final result x/a does not overflow or underflow.

Version experimental

Normal Distribution Random Variates (Specification)

Get uniformly distributed random variate for integer, real and complex variables. (Specification)

Save multidimensional array in npy format (Specification)

Saves a 2D array into a text file (Specification)

SB2ST_KERNELS is an internal routine used by the DSYTRD_SB2ST subroutine.

SBEV computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A.

SBEVD computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

SBGST reduces a real symmetric-definite banded generalized eigenproblem Ax = lambdaBx to standard form Cy = lambday, such that C has the same bandwidth as A. B must have been previously factorized as STS by DPBSTF, using a split Cholesky factorization. A is overwritten by C = XTAX, where X = S(-1)*Q and Q is an orthogonal matrix chosen to preserve the bandwidth of A.

SBGV computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form Ax=(lambda)B*x. Here A and B are assumed to be symmetric and banded, and B is also positive definite.

SBGVD computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form Ax=(lambda)B*x. Here A and B are assumed to be symmetric and banded, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

SBMV performs the matrix-vector operation y := alphaAx + 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.

SBTRD reduces a real symmetric band matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T.

SCAL scales a vector by a constant.

Scan a string for the presence of a set of characters. Scans a string for any of the characters in a set of characters.

Computes the Schur decomposition of matrix $A = Z T Z^H$. (Specification)

Computes the working array space required by the Schur decomposition solver (Specification)

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 SDOT = sum for I = 0 to N-1 of SX(LX+IINCX) * SY(LY+IINCY), where LX = 1 if INCX >= 0, else LX = 1+(1-N)*INCX, and LY is defined in a similar way using INCY.

Hashes a key with the NMHASH32 hash algorithm (Specifications)

Hashes a key with the NMHASH32X hash algorithm (Specifications) Arguments: key - the key to be hashed seed - the seed (unused) for the hashing algorithm

Hashes a key with the waterhash algorithm (Specifications)

(Specification)

Scaled Exponential Linear Unit

Scaled Exponential Linear Unit

Level 3 BLAS like routine for C in RFP Format. SFRK performs one of the symmetric rank--k operations C := alphaAAT + betaC, or C := alphaATA + betaC, where alpha and beta are real 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.

Fisher-Yates shuffle algorithm for a rank one array of integer, real and complex variables. (Specification)

Sigmoid function

Gradient of the sigmoid function

Sigmoid Linear Unit function

Gradient of the Sigmoid Linear Unit function

Integrates sampled values using Simpson's rule (Specification)

Integrates sampled values using trapezoidal rule weights for given abscissas (Specification)

Extracts characters from the input string to return a new string

softmax function. Available for ranks 1 to 4

Gradient of the softmax function. Available for ranks 1 to 4

softplus function

Gradient of the softplus function

Solves the linear system $A \cdot x = b$ for the unknown vector $x$ from a square matrix $A$. (Specification)

Computes the solution of the equality constrained least-squares problem

Computes the squares solution to system $A \cdot x = b$. (Specification)

Solves the linear system $A \cdot x = b$ for the unknown vector $x$ from a square matrix $A$. (Specification)

The generic subroutine interface implementing the SORT algorithm, based on the introsort of David Musser. (Specification)

The generic subroutine interface implementing the SORT_ADJ algorithm, based on the "Rust" sort algorithm found in slice.rs https://github.com/rust-lang/rust/blob/90eb44a5897c39e3dff9c7e48e3973671dcd9496/src/liballoc/slice.rs#L2159 but modified to return an array of indices that would provide a stable sort of the rank one ARRAY input. (Specification)

The generic subroutine interface implementing the SORT_INDEX algorithm, based on the "Rust" sort algorithm found in slice.rs https://github.com/rust-lang/rust/blob/90eb44a5897c39e3dff9c7e48e3973671dcd9496/src/liballoc/slice.rs#L2159 but modified to return an array of indices that would provide a stable sort of the rank one ARRAY input. (Specification)

SPCON estimates the reciprocal of the condition number (in the 1-norm) of a complex symmetric packed matrix A using the factorization A = UDUT or A = LDLT computed by CSPTRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

SPEV computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage.

SPEVD computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

SPGST reduces a real symmetric-definite generalized eigenproblem to standard form, using packed storage. If ITYPE = 1, the problem is Ax = lambdaBx, and A is overwritten by inv(UT)Ainv(U) or inv(L)Ainv(LT) If ITYPE = 2 or 3, the problem is ABx = lambdax or BAx = lambdax, and A is overwritten by UAUT or LTAL. B must have been previously factorized as UTU or LL*T by DPPTRF.

SPGV computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form Ax=(lambda)Bx, ABx=(lambda)x, or BAx=(lambda)x. Here A and B are assumed to be symmetric, stored in packed format, and B is also positive definite.

SPGVD computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form Ax=(lambda)Bx, ABx=(lambda)x, or BAx=(lambda)x. Here A and B are assumed to be symmetric, stored in packed format, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

(Specifications) This interface provides methods to compute the matrix-vector product

SPMV performs the matrix-vector operation y := alphaAx + 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.

SPMV performs the matrix-vector operation y := alphaAx + 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.

SPOOKY_HASH interfaces (Specification)

SPR performs the symmetric rank 1 operation A := alphaxx**H + A, where alpha is a complex scalar, x is an n element vector and A is an n by n symmetric matrix, supplied in packed form.

SPR performs the symmetric rank 1 operation A := alphaxx**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.

SPR2 performs the symmetric rank 2 operation A := alphaxyT + alphayxT + 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.

SPRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite and packed, and provides error bounds and backward error estimates for the solution.

SPSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices. The diagonal pivoting method is used to factor A as A = U * D * UT, if UPLO = 'U', or A = L * D * LT, if UPLO = 'L', where U (or L) is a product of permutation and unit upper (lower) triangular matrices, D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then used to solve the system of equations A * X = B.

SPTRD reduces a real symmetric matrix A stored in packed form to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T.

SPTRF computes the factorization of a complex symmetric matrix A stored in packed format using the Bunch-Kaufman diagonal pivoting method: A = UDUT or A = LDLT where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

SPTRI computes the inverse of a complex symmetric indefinite matrix A in packed storage using the factorization A = UDUT or A = LDLT computed by CSPTRF.

SPTRS solves a system of linear equations AX = B with a complex symmetric matrix A stored in packed format using the factorization A = UDUT or A = LDL*T computed by CSPTRF.

SROT applies a plane rotation, where the cos and sin (c and s) are real and the vectors cx and cy are complex. jack dongarra, linpack, 3/11/78.

SSCAL scales a complex vector by a real constant.

Check whether a string starts with substring or not

This subroutine translates from a BLAST-specified integer constant to the character string specifying a transposition operation. CHLA_TRANSTYPE returns an CHARACTER*1. If CHLA_TRANSTYPE: is 'X', then input is not an integer indicating a transposition operator. Otherwise CHLA_TRANSTYPE returns the constant value corresponding to TRANS.