Auxiliary Routines

Routine naming conventions, mathematical notation, and matrix storage schemes used for LAPACK auxiliary routines are the same as for the driver and computational routines described in previous chapters.

The table below summarizes information about the available LAPACK auxiliary routines.

LAPACK Auxiliary Routines

Routine Name

Data Types

Description

?lacgv

c, z

Conjugates a complex vector.

?lacrm

c, z

Multiplies a complex matrix by a square real matrix.

?lacrt

c, z

Performs a linear transformation of a pair of complex vectors.

?laesy

c, z

Computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix.

?rot

c, z

Applies a plane rotation with real cosine and complex sine to a pair of complex vectors.

?spmv

c, z

Computes a matrix-vector product for complex vectors using a complex symmetric packed matrix

?spr

c, z

Performs the symmetrical rank-1 update of a complex symmetric packed matrix.

?syconv

s, c, d, z

Converts a symmetric matrix given by a triangular matrix factorization into two matrices and vice versa.

?symv

c, z

Computes a matrix-vector product for a complex symmetric matrix.

?syr

c, z

Performs the symmetric rank-1 update of a complex symmetric matrix.

i?max1

c, z

Finds the index of the vector element whose real part has maximum absolute value.

?sum1

sc, dz

Forms the 1-norm of the complex vector using the true absolute value.

?gbtf2

s, d, c, z

Computes the LU factorization of a general band matrix using the unblocked version of the algorithm.

?gebd2

s, d, c, z

Reduces a general matrix to bidiagonal form using an unblocked algorithm.

?gehd2

s, d, c, z

Reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

?gelq2

s, d, c, z

Computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.

?geql2

s, d, c, z

Computes the QL factorization of a general rectangular matrix using an unblocked algorithm.

?geqr2

s, d, c, z

Computes the QR factorization of a general rectangular matrix using an unblocked algorithm.

?geqr2p

s, d, c, z

Computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.

?geqrt2

s, d, c, z

Computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.

?geqrt3

s, d, c, z

Recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.

?gerq2

s, d, c, z

Computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.

?gesc2

s, d, c, z

Solves a system of linear equations using the LU factorization with complete pivoting computed by ?getc2.

?getc2

s, d, c, z

Computes the LU factorization with complete pivoting of the general n-by-n matrix.

?getf2

s, d, c, z

Computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).

?gtts2

s, d, c, z

Solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by ?gttrf.

?isnan

s, d,

Tests input for NaN.

?laisnan

s, d,

Tests input for NaN by comparing two arguments for inequality.

?labrd

s, d, c, z

Reduces the first nb rows and columns of a general matrix to a bidiagonal form.

?lacn2

s, d, c, z

Estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.

?lacon

s, d, c, z

Estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.

?lacpy

s, d, c, z

Copies all or part of one two-dimensional array to another.

?ladiv

s, d, c, z

Performs complex division in real arithmetic, avoiding unnecessary overflow.

?lae2

s, d

Computes the eigenvalues of a 2-by-2 symmetric matrix.

?laebz

s, d

Computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine ?stebz.

?laed0

s, d, c, z

Used by ?stedc. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.

?laed1

s, d

Used by sstedc/dstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.

?laed2

s, d

Used by sstedc/dstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.

?laed3

s, d

Used by sstedc/dstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.

?laed4

s, d

Used by sstedc/dstedc. Finds a single root of the secular equation.

?laed5

s, d

Used by sstedc/dstedc. Solves the 2-by-2 secular equation.

?laed6

s, d

Used by sstedc/dstedc. Computes one Newton step in solution of the secular equation.

?laed7

s, d, c, z

Used by ?stedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

?laed8

s, d, c, z

Used by ?stedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

?laed9

s, d

Used by sstedc/dstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.

?laeda

s, d

Used by ?stedc. Computes the Z vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense.

?laein

s, d, c, z

Computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.

?laev2

s, d, c, z

Computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.

?laexc

s, d

Swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation.

?lag2

s, d

Computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.

?lags2

s, d

Computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel.

?lagtf

s, d

Computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges.

?lagtm

s, d, c, z

Performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1.

?lagts

s, d

Solves the system of equations (T-λI)x = y or (T-λI)Tx = y,where T is a general tridiagonal matrix and λ a scalar, using the LU factorization computed by ?lagtf.

?lagv2

s, d

Computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular.

?lahqr

s, d, c, z

Computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

?lahrd

s, d, c, z

Reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

?lahr2

s, d, c, z

Reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

?laic1

s, d, c, z

Applies one step of incremental condition estimation.

?laln2

s, d

Solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.

?lals0

s, d, c, z

Applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by ?gelsd.

?lalsa

s, d, c, z

Computes the SVD of the coefficient matrix in compact form. Used by ?gelsd.

?lalsd

s, d, c, z

Uses the singular value decomposition of A to solve the least squares problem.

?lamrg

s, d

Creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in ascending order.

?laneg

s, d

Computes the Sturm count.

?langb

s, d, c, z

Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of general band matrix.

?lange

s, d, c, z

Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general rectangular matrix.

?langt

s, d, c, z

Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix.

?lanhs

s, d, c, z

Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix.

?lansb

s, d, c, z

Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.

?lanhb

c, z

Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian band matrix.

?lansp

s, d, c, z

Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form.

?lanhp

c, z

Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix supplied in packed form.

?lanst/?lanht

s, d/c, z

Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric or complex Hermitian tridiagonal matrix.

?lansy

s, d, c, z

Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real/complex symmetric matrix.

?lanhe

c, z

Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix.

?lantb

s, d, c, z

Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix.

?lantp

s, d, c, z

Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form.

?lantr

s, d, c, z

Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.

?lanv2

s, d

Computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form.

?lapll

s, d, c, z

Measures the linear dependence of two vectors.

?lapmr

s, d, c, z

Rearranges rows of a matrix as specified by a permutation vector.

?lapmt

s, d, c, z

Performs a forward or backward permutation of the columns of a matrix.

?lapy2

s, d

Returns sqrt(x2+y2).

?lapy3

s, d

Returns sqrt(x2+y2+z2).

?laqgb

s, d, c, z

Scales a general band matrix, using row and column scaling factors computed by ?gbequ.

?laqge

s, d, c, z

Scales a general rectangular matrix, using row and column scaling factors computed by ?geequ.

?laqhb

c, z

Scales a Hermitian band matrix, using scaling factors computed by ?pbequ.

?laqp2

s, d, c, z

Computes a QR factorization with column pivoting of the matrix block.

?laqps

s, d, c, z

Computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.

?laqr0

s, d, c, z

Computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

?laqr1

s, d, c, z

Sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.

?laqr2

s, d, c, z

Performs the orthogonal/unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

?laqr3

s, d, c, z

Performs the orthogonal/unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

?laqr4

s, d, c, z

Computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

?laqr5

s, d, c, z

Performs a single small-bulge multi-shift QR sweep.

?laqsb

s, d, c, z

Scales a symmetric/Hermitian band matrix, using scaling factors computed by ?pbequ.

?laqsp

s, d, c, z

Scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by ?ppequ.

?laqsy

s, d, c, z

Scales a symmetric/Hermitian matrix, using scaling factors computed by ?poequ.

?laqtr

s, d

Solves a real quasi-triangular system of equations, or a complex quasi-triangular system of special form, in real arithmetic.

?lar1v

s, d, c, z

Computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.

?lar2v

s, d, c, z

Applies a vector of plane rotations with real cosines and real/complex sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.

?larf

s, d, c, z

Applies an elementary reflector to a general rectangular matrix.

?larfb

s, d, c, z

Applies a block reflector or its transpose/conjugate-transpose to a general rectangular matrix.

?larfg

s, d, c, z

Generates an elementary reflector (Householder matrix).

?larfgp

s, d, c, z

Generates an elementary reflector (Householder matrix) with non-negatibe beta.

?larft

s, d, c, z

Forms the triangular factor T of a block reflector H = I - vtvH

?larfx

s, d, c, z

Applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order 10.

?largv

s, d, c, z

Generates a vector of plane rotations with real cosines and real/complex sines.

?larnv

s, d, c, z

Returns a vector of random numbers from a uniform or normal distribution.

?larra

s, d

Computes the splitting points with the specified threshold.

?larrb

s, d

Provides limited bisection to locate eigenvalues for more accuracy.

?larrc

s, d

Computes the number of eigenvalues of the symmetric tridiagonal matrix.

?larrd

s, d

Computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.

?larre

s, d

Given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.

?larrf

s, d

Finds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.

?larrj

s, d

Performs refinement of the initial estimates of the eigenvalues of the matrix T.

?larrk

s, d

Computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.

?larrr

s, d

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

?larrv

s, d, c, z

Computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

?lartg

s, d, c, z

Generates a plane rotation with real cosine and real/complex sine.

?lartgp

s, d

Generates a plane rotation so that the diagonal is nonnegative.

?lartgs

s, d

Generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem.

?lartv

s, d, c, z

Applies a vector of plane rotations with real cosines and real/complex sines to the elements of a pair of vectors.

?laruv

s, d

Returns a vector of n random real numbers from a uniform distribution.

?larz

s, d, c, z

Applies an elementary reflector (as returned by ?tzrzf) to a general matrix.

?larzb

s, d, c, z

Applies a block reflector or its transpose/conjugate-transpose to a general matrix.

?larzt

s, d, c, z

Forms the triangular factor T of a block reflector H = I - vtvH.

?las2

s, d

Computes singular values of a 2-by-2 triangular matrix.

?lascl

s, d, c, z

Multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.

?lasd0

s, d

Computes the singular values of a real upper bidiagonal n-by-m matrix B with diagonal d and off-diagonal e. Used by ?bdsdc.

?lasd1

s, d

Computes the SVD of an upper bidiagonal matrix B of the specified size. Used by ?bdsdc.

?lasd2

s, d

Merges the two sets of singular values together into a single sorted set. Used by ?bdsdc.

?lasd3

s, d

Finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by ?bdsdc.

?lasd4

s, d

Computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by ?bdsdc.

?lasd5

s, d

Computes the square root of the i-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix. Used by ?bdsdc.

?lasd6

s, d

Computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by ?bdsdc.

?lasd7

s, d

Merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. Used by ?bdsdc.

?lasd8

s, d

Finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles. Used by ?bdsdc.

?lasd9

s, d

Finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles. Used by ?bdsdc.

?lasda

s, d

Computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by ?bdsdc.

?lasdq

s, d

Computes the SVD of a real bidiagonal matrix with diagonal d and off-diagonal e. Used by ?bdsdc.

?lasdt

s, d

Creates a tree of subproblems for bidiagonal divide and conquer. Used by ?bdsdc.

?laset

s, d, c, z

Initializes the off-diagonal elements and the diagonal elements of a matrix to given values.

?lasq1

s, d

Computes the singular values of a real square bidiagonal matrix. Used by ?bdsqr.

?lasq2

s, d

Computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by ?bdsqr and ?stegr.

?lasq3

s, d

Checks for deflation, computes a shift and calls dqds. Used by ?bdsqr.

?lasq4

s, d

Computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by ?bdsqr.

?lasq5

s, d

Computes one dqds transform in ping-pong form. Used by ?bdsqr and ?stegr.

?lasq6

s, d

Computes one dqd transform in ping-pong form. Used by ?bdsqr and ?stegr.

?lasr

s, d, c, z

Applies a sequence of plane rotations to a general rectangular matrix.

?lasrt

s, d

Sorts numbers in increasing or decreasing order.

?lassq

s, d, c, z

Updates a sum of squares represented in scaled form.

?lasv2

s, d

Computes the singular value decomposition of a 2-by-2 triangular matrix.

?laswp

s, d, c, z

Performs a series of row interchanges on a general rectangular matrix.

?lasy2

s, d

Solves the Sylvester matrix equation where the matrices are of order 1 or 2.

?lasyf

s, d, c, z

Computes a partial factorization of a real/complex symmetric matrix, using the diagonal pivoting method.

?lahef

c, z

Computes a partial factorization of a complex Hermitian indefinite matrix, using the diagonal pivoting method.

?latbs

s, d, c, z

Solves a triangular banded system of equations.

?latdf

s, d, c, z

Uses the LU factorization of the n-by-n matrix computed by ?getc2 and computes a contribution to the reciprocal Dif-estimate.

?latps

s, d, c, z

Solves a triangular system of equations with the matrix held in packed storage.

?latrd

s, d, c, z

Reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal/unitary similarity transformation.

?latrs

s, d, c, z

Solves a triangular system of equations with the scale factor set to prevent overflow.

?latrz

s, d, c, z

Factors an upper trapezoidal matrix by means of orthogonal/unitary transformations.

?lauu2

s, d, c, z

Computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm).

?lauum

s, d, c, z

Computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm).

?org2l/?ung2l

s, d/c, z

Generates all or part of the orthogonal/unitary matrix Q from a QL factorization determined by ?geqlf (unblocked algorithm).

?org2r/?ung2r

s, d/c, z

Generates all or part of the orthogonal/unitary matrix Q from a QR factorization determined by ?geqrf (unblocked algorithm).

?orgl2/?ungl2

s, d/c, z

Generates all or part of the orthogonal/unitary matrix Q from an LQ factorization determined by ?gelqf (unblocked algorithm).

?orgr2/?ungr2

s, d/c, z

Generates all or part of the orthogonal/unitary matrix Q from an RQ factorization determined by ?gerqf (unblocked algorithm).

?orm2l/?unm2l

s, d/c, z

Multiplies a general matrix by the orthogonal/unitary matrix from a QL factorization determined by ?geqlf (unblocked algorithm).

?orm2r/?unm2r

s, d/c, z

Multiplies a general matrix by the orthogonal/unitary matrix from a QR factorization determined by ?geqrf (unblocked algorithm).

?orml2/?unml2

s, d/c, z

Multiplies a general matrix by the orthogonal/unitary matrix from a LQ factorization determined by ?gelqf (unblocked algorithm).

?ormr2/?unmr2

s, d/c, z

Multiplies a general matrix by the orthogonal/unitary matrix from a RQ factorization determined by ?gerqf (unblocked algorithm).

?ormr3/?unmr3

s, d/c, z

Multiplies a general matrix by the orthogonal/unitary matrix from a RZ factorization determined by ?tzrzf (unblocked algorithm).

?pbtf2

s, d, c, z

Computes the Cholesky factorization of a symmetric/ Hermitian positive definite band matrix (unblocked algorithm).

?potf2

s, d, c, z

Computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).

?ptts2

s, d, c, z

Solves a tridiagonal system of the form AX=B using the L D LH factorization computed by ?pttrf.

?rscl

s, d, cs, zd

Multiplies a vector by the reciprocal of a real scalar.

?syswapr

s, d, c, z

Applies an elementary permutation on the rows and columns of a symmetric matrix.

?heswapr

c, z

Applies an elementary permutation on the rows and columns of a Hermitian matrix.

?sygs2/?hegs2

s, d/c, z

Reduces a symmetric/Hermitian positive-definite generalized eigenproblem to standard form, using the factorization results obtained from ?potrf (unblocked algorithm).

?sytd2/?hetd2

s, d/c, z

Reduces a symmetric/Hermitian matrix to real symmetric tridiagonal form by an orthogonal/unitary similarity transformation (unblocked algorithm).

?sytf2

s, d, c, z

Computes the factorization of a real/complex symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

?hetf2

c, z

Computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm).

?tgex2

s, d, c, z

Swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal/unitary equivalence transformation.

?tgsy2

s, d, c, z

Solves the generalized Sylvester equation (unblocked algorithm).

?trti2

s, d, c, z

Computes the inverse of a triangular matrix (unblocked algorithm).

clag2z

cz

Converts a complex single precision matrix to a complex double precision matrix.

dlag2s

ds

Converts a double precision matrix to a single precision matrix.

slag2d

sd

Converts a single precision matrix to a double precision matrix.

zlag2c

zc

Converts a complex double precision matrix to a complex single precision matrix.

?larfp

s, d, c, z

Generates a real or complex elementary reflector.

ila?lc

s, d, c, z

Scans a matrix for its last non-zero column.

ila?lr

s, d, c, z

Scans a matrix for its last non-zero row.

?gsvj0

s, d

Pre-processor for the routine ?gesvj.

?gsvj1

s, d

Pre-processor for the routine ?gesvj, applies Jacobi rotations targeting only particular pivots.

?sfrk

s, d

Performs a symmetric rank-k operation for matrix in RFP format.

?hfrk

c, z

Performs a Hermitian rank-k operation for matrix in RFP format.

?tfsm

s, d, c, z

Solves a matrix equation (one operand is a triangular matrix in RFP format).

?lansf

s, d

Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix in RFP format.

?lanhf

c, z

Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format.

?tfttp

s, d, c, z

Copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).

?tfttr

s, d, c, z

Copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR).

?tpqrt2

s, d, c, z

Computes a QR factorization of a real or complex "triangular-pentagonal" matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

?tprfb

s, d, c, z

Applies a real or complex "triangular-pentagonal" blocked reflector to a real or complex matrix, which is composed of two blocks.

?tpttf

s, d, c, z

Copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).

?tpttr

s, d, c, z

Copies a triangular matrix from the standard packed format (TP) to the standard full format (TR).

?trttf

s, d, c, z

Copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).

?trttp

s, d, c, z

Copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).

?pstf2

s, d, c, z

Computes the Cholesky factorization with complete pivoting of a real symmetric or complex Hermitian positive semi-definite matrix.

dlat2s

ds

Converts a double-precision triangular matrix to a single-precision triangular matrix.

zlat2c

zc

Converts a double complex triangular matrix to a complex triangular matrix.

?lacp2

c, z

Copies all or part of a real two-dimensional array to a complex array.

?la_gbamv

s, d, c, z

Performs a matrix-vector operation to calculate error bounds.

?la_gbrcond

s, d

Estimates the Skeel condition number for a general banded matrix.

?la_gbrcond_c

c, z

Computes the infinity norm condition number of op(A)*inv(diag(c)) for general banded matrices.

?la_gbrcond_x

c, z

Computes the infinity norm condition number of op(A)*diag(x) for general banded matrices.

?la_gbrfsx_extended

s, d, c, z

Improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

?la_gbrpvgrw

s, d, c, z

Computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix.

?la_geamv

s, d, c, z

Computes a matrix-vector product using a general matrix to calculate error bounds.

?la_gercond

s, d

Estimates the Skeel condition number for a general matrix.

?la_gercond_c

c, z

Computes the infinity norm condition number of op(A)*inv(diag(c)) for general matrices.

?la_gercond_x

c, z

Computes the infinity norm condition number of op(A)*diag(x) for general matrices.

?la_gerfsx_extended

s, d

Improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

?la_heamv

c, z

Computes a matrix-vector product using a Hermitian indefinite matrix to calculate error bounds.

?la_hercond_c

c, z

Computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian indefinite matrices.

?la_hercond_x

c, z

Computes the infinity norm condition number of op(A)*diag(x) for Hermitian indefinite matrices.

?la_herfsx_extended

c, z

Improves the computed solution to a system of linear equations for Hermitian indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

?la_lin_berr

s, d, c, z

Computes a component-wise relative backward error.

?la_porcond

s, d

Estimates the Skeel condition number for a symmetric positive-definite matrix.

?la_porcond_c

c, z

Computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian positive-definite matrices.

?la_porcond_x

c, z

Computes the infinity norm condition number of op(A)*diag(x) for Hermitian positive-definite matrices.

?la_porfsx_extended

s, d, c, z

Improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

?la_porpvgrw

s, d, c, z

Computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.

?laqhe

c, z

Scales a Hermitian matrix.

?laqhp

c, z

Scales a Hermitian matrix stored in packed form.

?larcm

c, z

Copies all or part of a real two-dimensional array to a complex array.

?la_rpvgrw

c, z

Multiplies a square real matrix by a complex matrix.

?larscl2

s, d, c, z

Performs reciprocal diagonal scaling on a vector.

?lascl2

s, d, c, z

Performs diagonal scaling on a vector.

?la_syamv

s, d, c, z

Computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.

?la_syrcond

s, d

Estimates the Skeel condition number for a symmetric indefinite matrix.

?la_syrcond_c

c, z

Computes the infinity norm condition number of op(A)*inv(diag(c)) for symmetric indefinite matrices.

?la_syrcond_x

c, z

Computes the infinity norm condition number of op(A)*diag(x) for symmetric indefinite matrices.

?la_syrfsx_extended

s, d, c, z

Improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

?la_syrpvgrw

s, d, c, z

Computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.

?la_wwaddw

s, d, c, z

Adds a vector into a doubled-single vector.