F08 Chapter Contents

F08 Introduction

f08aa – Solves an overdetermined or underdetermined real linear system

f08ae – QR factorization of real general rectangular matrix

f08af – Form all or part of orthogonal Q from QR factorization determined by f08ae or f08be

f08ag – Apply orthogonal transformation determined by f08ae or f08be

f08ah – LQ factorization of real general rectangular matrix

f08aj – Form all or part of orthogonal Q from LQ factorization determined by f08ah

f08ak – Apply orthogonal transformation determined by f08ah

f08an – Solves an overdetermined or underdetermined complex linear system

f08as – QR factorization of complex general rectangular matrix

f08at – Form all or part of unitary Q from QR factorization determined by f08as or f08bs

f08au – Apply unitary transformation determined by f08as or f08bs

f08av – LQ factorization of complex general rectangular matrix

f08aw – Form all or part of unitary Q from LQ factorization determined by f08av

f08ax – Apply unitary transformation determined by f08av

f08ba – Computes the minimum-norm solution to a real linear least-squares problem

f08be – QR factorization of real general rectangular matrix with column pivoting

f08bf – QR factorization of real general rectangular matrix with column pivoting, using BLAS-3

f08bh – Reduces a real upper trapezoidal matrix to upper triangular form

f08bk – Apply orthogonal transformation determined by f08bh

f08bn – Computes the minimum-norm solution to a complex linear least-squares problem

f08bs – QR factorization of complex general rectangular matrix with column pivoting

f08bt – QR factorization of complex general rectangular matrix with column pivoting, using BLAS-3

f08bv – Reduces a complex upper trapezoidal matrix to upper triangular form

f08bx – Apply unitary transformation determined by f08bv

f08ce – QL factorization of real general rectangular matrix

f08cf – Form all or part of orthogonal Q from QL factorization determined by f08ce

f08cg – Apply orthogonal transformation determined by f08ce

f08ch – RQ factorization of real general rectangular matrix

f08cj – Form all or part of orthogonal Q from RQ factorization determined by f08ch

f08ck – Apply orthogonal transformation determined by f08ch

f08cs – QL factorization of complex general rectangular matrix

f08ct – Form all or part of orthogonal Q from QL factorization determined by f08cs

f08cu – Apply unitary transformation determined by f08cs

f08cv – RQ factorization of complex general rectangular matrix

f08cw – Form all or part of orthogonal Q from RQ factorization determined by f08cv

f08cx – Apply unitary transformation determined by f08cv

f08fa – Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix

f08fb – Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix

f08fc – All eigenvalues and optionally all eigenvectors of real symmetric matrix (divide-and-conquer)

f08fd – Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix (Relatively Robust Representations)

f08fe – Orthogonal reduction of real symmetric matrix to symmetric tridiagonal form

f08ff – Generate orthogonal transformation matrix from reduction to tridiagonal form determined by f08fe

f08fg – Apply orthogonal transformation determined by f08fe

f08fl – 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 matrix

f08fn – Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix

f08fp – Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix

f08fq – All eigenvalues and optionally all eigenvectors of complex Hermitian matrix (divide-and-conquer)

f08fr – Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix (Relatively Robust Representations)

f08fs – Unitary reduction of complex Hermitian matrix to real symmetric tridiagonal form

f08ft – Generate unitary transformation matrix from reduction to tridiagonal form determined by f08fs

f08fu – Apply unitary transformation matrix determined by f08fs

f08ga – Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage

f08gb – Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage

f08gc – All eigenvalues and optionally all eigenvectors of real symmetric matrix, packed storage (divide-and-conquer)

f08ge – Orthogonal reduction of real symmetric matrix to symmetric tridiagonal form, packed storage

f08gf – Generate orthogonal transformation matrix from reduction to tridiagonal form determined by f08ge

f08gg – Apply orthogonal transformation determined by f08ge

f08gn – Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage

f08gp – Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage

f08gq – All eigenvalues and optionally all eigenvectors of complex Hermitian matrix, packed storage (divide-and-conquer)

f08gs – Unitary reduction of complex Hermitian matrix to real symmetric tridiagonal form, packed storage

f08gt – Generate unitary transformation matrix from reduction to tridiagonal form determined by f08gs

f08gu – Apply unitary transformation matrix determined by f08gs

f08ha – Computes all eigenvalues and, optionally, eigenvectors of a real symmetric band matrix

f08hb – Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix

f08hc – All eigenvalues and optionally all eigenvectors of real symmetric band matrix (divide-and-conquer)

f08he – Orthogonal reduction of real symmetric band matrix to symmetric tridiagonal form

f08hn – Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix

f08hp – Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix

f08hq – All eigenvalues and optionally all eigenvectors of complex Hermitian band matrix (divide-and-conquer)

f08hs – Unitary reduction of complex Hermitian band matrix to real symmetric tridiagonal form

f08ja – Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix

f08jb – Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix

f08jc – All eigenvalues and optionally all eigenvectors of real symmetric tridiagonal matrix (divide-and-conquer)

f08jd – Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix (Relatively Robust Representations)

f08je – All eigenvalues and eigenvectors of real symmetric tridiagonal matrix, reduced from real symmetric matrix using implicit QL or QR

f08jf – All eigenvalues of real symmetric tridiagonal matrix, root-free variant of QL or QR

f08jg – All eigenvalues and eigenvectors of real symmetric positive-definite tridiagonal matrix, reduced from real symmetric positive-definite matrix

f08jh – Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a matrix reduced to this form (divide-and-conquer)

f08jj – Selected eigenvalues of real symmetric tridiagonal matrix by bisection

f08jk – Selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in real array

f08jl – Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a symmetric matrix reduced to this form (Relatively Robust Representations)

f08js – All eigenvalues and eigenvectors of real symmetric tridiagonal matrix, reduced from complex Hermitian matrix, using implicit QL or QR

f08ju – All eigenvalues and eigenvectors of real symmetric positive-definite tridiagonal matrix, reduced from complex Hermitian positive-definite matrix

f08jv – Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (divide-and-conquer)

f08jx – Selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in complex array

f08jy – Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (Relatively Robust Representations)

f08ka – Computes the minimum-norm solution to a real linear least-squares problem using singular value decomposition

f08kb – Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors

f08kc – Computes the minimum-norm solution to a real linear least-squares problem using singular value decomposition (divide-and-conquer)

f08kd – Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (divide-and-conquer)

f08ke – Orthogonal reduction of real general rectangular matrix to bidiagonal form

f08kf – Generate orthogonal transformation matrices from reduction to bidiagonal form determined by f08ke

f08kg – Apply orthogonal transformations from reduction to bidiagonal form determined by f08ke

f08kn – Computes the minimum-norm solution to a complex linear least-squares problem using singular value decomposition

f08kp – Computes the singular value decomposition of a complex matrix, optionally computing the left and/or right singular vectors

f08kq – Computes the minimum-norm solution to a complex linear least-squares problem using singular value decomposition (divide-and-conquer)

f08kr – Computes the singular value decomposition of a complex matrix, optionally computing the left and/or right singular vectors (divide-and-conquer)

f08ks – Unitary reduction of complex general rectangular matrix to bidiagonal form

f08kt – Generate unitary transformation matrices from reduction to bidiagonal form determined by f08ks

f08ku – Apply unitary transformations from reduction to bidiagonal form determined by f08ks

f08le – Reduction of real rectangular band matrix to upper bidiagonal form

f08ls – Reduction of complex rectangular band matrix to upper bidiagonal form

f08md – Computes the singular value decomposition of a real bidiagonal matrix, optionally computing the singular vectors (divide-and-conquer)

f08me – SVD of real bidiagonal matrix reduced from real general matrix

f08ms – SVD of real bidiagonal matrix reduced from complex general matrix

f08na – Computes all eigenvalues and, optionally, left and/or right eigenvectors of a real nonsymmetric matrix

f08nb – Computes all eigenvalues and, optionally, left and/or right eigenvectors of a real nonsymmetric matrix; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors

f08ne – Orthogonal reduction of real general matrix to upper Hessenberg form

f08nf – Generate orthogonal transformation matrix from reduction to Hessenberg form determined by f08ne

f08ng – Apply orthogonal transformation matrix from reduction to Hessenberg form determined by f08ne

f08nh – Balance real general matrix

f08nj – Transform eigenvectors of real balanced matrix to those of original matrix supplied to f08nh

f08nn – Computes all eigenvalues and, optionally, left and/or right eigenvectors of a complex nonsymmetric matrix

f08np – Computes all eigenvalues and, optionally, left and/or right eigenvectors of a complex nonsymmetric matrix; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors

f08ns – Unitary reduction of complex general matrix to upper Hessenberg form

f08nt – Generate unitary transformation matrix from reduction to Hessenberg form determined by f08ns

f08nu – Apply unitary transformation matrix from reduction to Hessenberg form determined by f08ns

f08nv – Balance complex general matrix

f08nw – Transform eigenvectors of complex balanced matrix to those of original matrix supplied to f08nv

f08pa – Computes for real square nonsymmetric matrix, the eigenvalues, the real Schur form, and, optionally, the matrix of Schur vectors

f08pb – Computes for real square nonsymmetric matrix, the eigenvalues, the real Schur form, and, optionally, the matrix of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues

f08pe – Eigenvalues and Schur factorization of real upper Hessenberg matrix reduced from real general matrix

f08pk – Selected right and/or left eigenvectors of real upper Hessenberg matrix by inverse iteration

f08pn – Computes for complex square nonsymmetric matrix, the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors

f08pp – Computes for real square nonsymmetric matrix, the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues

f08ps – Eigenvalues and Schur factorization of complex upper Hessenberg matrix reduced from complex general matrix

f08px – Selected right and/or left eigenvectors of complex upper Hessenberg matrix by inverse iteration

f08qf – Reorder Schur factorization of real matrix using orthogonal similarity transformation

f08qg – Reorder Schur factorization of real matrix, form orthonormal basis of right invariant subspace for selected eigenvalues, with estimates of sensitivities

f08qh – Solve real Sylvester matrix equation AX+XB=C, A and B are upper quasi-triangular or transposes

f08qk – Left and right eigenvectors of real upper quasi-triangular matrix

f08ql – Estimates of sensitivities of selected eigenvalues and eigenvectors of real upper quasi-triangular matrix

f08qt – Reorder Schur factorization of complex matrix using unitary similarity transformation

f08qu – Reorder Schur factorization of complex matrix, form orthonormal basis of right invariant subspace for selected eigenvalues, with estimates of sensitivities

f08qv – Solve complex Sylvester matrix equation AX+XB=C, A and B are upper triangular or conjugate-transposes

f08qx – Left and right eigenvectors of complex upper triangular matrix

f08qy – Estimates of sensitivities of selected eigenvalues and eigenvectors of complex upper triangular matrix

f08sa – Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem

f08sb – Computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem

f08sc – Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem (divide-and-conquer)

f08se – Reduction to standard form of real symmetric-definite generalized eigenproblem Ax= lambda Bx, ABx= lambda x or BAx= lambda x, B factorized by f07fd

f08sn – Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem

f08sp – Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem

f08sq – Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem (divide-and-conquer)

f08ss – Reduction to standard form of complex Hermitian-definite generalized eigenproblem Ax= lambda Bx, ABx= lambda x or BAx= lambda x, B factorized by f07fr

f08ta – Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage

f08tb – Computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage

f08tc – Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage (divide-and-conquer)

f08te – Reduction to standard form of real symmetric-definite generalized eigenproblem Ax= lambda Bx, ABx= lambda x or BAx= lambda x, packed storage, B factorized by f07gd

f08tn – Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage

f08tp – Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage

f08tq – Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage (divide-and-conquer)

f08ts – Reduction to standard form of complex Hermitian-definite generalized eigenproblem Ax= lambda Bx, ABx= lambda x or BAx= lambda x, packed storage, B factorized by f07gr

f08ua – Computes all the eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem

f08ub – Computes selected eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem

f08uc – Computes all the eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem (divide-and-conquer)

f08ue – Reduction of real symmetric-definite banded generalized eigenproblem Ax= lambda Bx to standard form Cy= lambda y, such that C has the same bandwidth as A

f08uf – Computes a split Cholesky factorization of real symmetric positive-definite band matrix A

f08un – Computes all the eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem

f08up – Computes selected eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem

f08uq – Computes all the eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem (divide-and-conquer)

f08us – Reduction of complex Hermitian-definite banded generalized eigenproblem Ax= lambda Bx to standard form Cy= lambda y, such that C has the same bandwidth as A

f08ut – Computes a split Cholesky factorization of complex Hermitian positive-definite band matrix A

f08va – Computes the generalized singular value decomposition of a real matrix pair

f08ve – Computes orthogonal matrices as processing steps for computing the generalized singular value decomposition of a real matrix pair

f08vn – Computes the generalized singular value decomposition of a complex matrix pair

f08vs – Computes orthogonal matrices as processing steps for computing the generalized singular value decomposition of a complex matrix pair

f08wa – Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors

f08wb – Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors

f08we – Orthogonal reduction of a pair of real general matrices to generalized upper Hessenberg form

f08wh – Balance a pair of real general matrices

f08wj – Transform eigenvectors of a pair of real balanced matrices to those of original matrix pair supplied to f08wh

f08wn – Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors

f08wp – Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors

f08ws – Unitary reduction of a pair of complex general matrices to generalized upper Hessenberg form

f08wv – Balance a pair of complex general matrices

f08ww – Transform eigenvectors of a pair of complex balanced matrices to those of original matrix pair supplied to f08wv

f08xa – Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, the generalized real Schur form and, optionally, the left and/or right matrices of Schur vectors

f08xb – Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, the generalized real Schur form and, optionally, the left and/or right matrices of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues

f08xe – Eigenvalues and generalized Schur factorization of real generalized upper Hessenberg form reduced from a pair of real general matrices

f08xn – Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, the generalized complex Schur form and, optionally, the left and/or right matrices of Schur vectors

f08xp – Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, the generalized complex Schur form and, optionally, the left and/or right matrices of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues

f08xs – Eigenvalues and generalized Schur factorization of complex generalized upper Hessenberg form reduced from a pair of complex general matrices

f08ye – Computes the generalized singular value decomposition of a real upper triangular (or trapezoidal) matrix pair

f08yf – Reorders the generalized real Schur decomposition of a real matrix pair using an orthogonal equivalence transformation

f08yg – Reorders the generalized real Schur decomposition of a real matrix pair using an orthogonal equivalence transformation, computes the generalized eigenvalues of the reordered pair and, optionally, computes the estimates of reciprocal condition numbers for eigenvalues and eigenspaces

f08yh – Solves the real-valued generalized Sylvester equation

f08yk – Left and right eigenvectors of a pair of real upper quasi-triangular matrices

f08yl – Estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a real matrix pair in generalized real Schur canonical form

f08ys – Computes the generalized singular value decomposition of a complex upper triangular (or trapezoidal) matrix pair

f08yt – Reorders the generalized Schur decomposition of a complex matrix pair using an unitary equivalence transformation

f08yu – Reorders the generalized Schur decomposition of a complex matrix pair using an unitary equivalence transformation, computes the generalized eigenvalues of the reordered pair and, optionally, computes the estimates of reciprocal condition numbers for eigenvalues and eigenspaces

f08yv – Solves the complex generalized Sylvester equation

f08yx – Left and right eigenvectors of a pair of complex upper triangular matrices

f08yy – Estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a complex matrix pair in generalized Schur canonical form

f08za – Solves the real linear equality-constrained least-squares (LSE) problem

f08zb – Solves a real general Gauss–Markov linear model (GLM) problem

f08ze – Computes a generalized QR factorization of a real matrix pair

f08zf – Computes a generalized RQ factorization of a real matrix pair

f08zn – Solves the complex linear equality-constrained least-squares (LSE) problem

f08zp – Solves a complex general Gauss–Markov linear model (GLM) problem

f08zs – Computes a generalized QR factorization of a complex matrix pair

f08zt – Computes a generalized RQ factorization of a complex matrix pair