# NAG Library Chapter Contents

## F08 (lapackeig)Least Squares and Eigenvalue Problems (LAPACK)

F08 (lapackeig) Chapter Introduction – a description of the Chapter and an overview of the algorithms available

 RoutineName Mark ofIntroduction Purpose f08aaf (dgels) Example Text Example Data 21 dgels nagf_lapackeig_dgels Solves a real linear least squares problem of full rank f08abf (dgeqrt) Example Text Example Data 25 dgeqrt nagf_lapackeig_dgeqrt Performs a $QR$ factorization of real general rectangular matrix, with explicit blocking f08acf (dgemqrt) 25 dgemqrt nagf_lapackeig_dgemqrt Applies the orthogonal transformation determined by f08abf (dgeqrt) f08aef (dgeqrf) Example Text Example Data 16 dgeqrf nagf_lapackeig_dgeqrf Performs a $QR$ factorization of real general rectangular matrix f08aff (dorgqr) Example Text Example Data 16 dorgqr nagf_lapackeig_dorgqr Forms all or part of orthogonal $Q$ from $QR$ factorization determined by f08aef (dgeqrf), f08bef (dgeqpf) or f08bff (dgeqp3) f08agf (dormqr) 16 dormqr nagf_lapackeig_dormqr Applies an orthogonal transformation determined by f08aef (dgeqrf), f08bef (dgeqpf) or f08bff (dgeqp3) f08ahf (dgelqf) Example Text Example Data 16 dgelqf nagf_lapackeig_dgelqf Performs a $LQ$ factorization of real general rectangular matrix f08ajf (dorglq) Example Text Example Data 16 dorglq nagf_lapackeig_dorglq Forms all or part of orthogonal $Q$ from $LQ$ factorization determined by f08ahf (dgelqf) f08akf (dormlq) 16 dormlq nagf_lapackeig_dormlq Applies the orthogonal transformation determined by f08ahf (dgelqf) f08anf (zgels) Example Text Example Data 21 zgels nagf_lapackeig_zgels Solves a complex linear least problem of full rank f08apf (zgeqrt) Example Text Example Data 25 zgeqrt nagf_lapackeig_zgeqrt Performs a $QR$ factorization of complex general rectangular matrix using recursive algorithm f08aqf (zgemqrt) 25 zgemqrt nagf_lapackeig_zgemqrt Applies the unitary transformation determined by f08apf (zgeqrt) f08asf (zgeqrf) Example Text Example Data 16 zgeqrf nagf_lapackeig_zgeqrf Performs a $QR$ factorization of complex general rectangular matrix f08atf (zungqr) Example Text Example Data 16 zungqr nagf_lapackeig_zungqr Forms all or part of unitary $Q$ from $QR$ factorization determined by f08asf (zgeqrf), f08bsf (zgeqpf) or f08btf (zgeqp3) f08auf (zunmqr) 16 zunmqr nagf_lapackeig_zunmqr Applies a unitary transformation determined by f08asf (zgeqrf), f08bsf (zgeqpf) or f08btf (zgeqp3) f08avf (zgelqf) Example Text Example Data 16 zgelqf nagf_lapackeig_zgelqf Performs a $LQ$ factorization of complex general rectangular matrix f08awf (zunglq) Example Text Example Data 16 zunglq nagf_lapackeig_zunglq Forms all or part of unitary $Q$ from $LQ$ factorization determined by f08avf (zgelqf) f08axf (zunmlq) 16 zunmlq nagf_lapackeig_zunmlq Applies the unitary transformation determined by f08avf (zgelqf) f08baf (dgelsy) Example Text Example Data 21 dgelsy nagf_lapackeig_dgelsy Computes the minimum-norm solution to a real linear least squares problem f08bbf (dtpqrt) Example Text Example Data 25 dtpqrt nagf_lapackeig_dtpqrt $QR$ factorization of real general triangular-pentagonal matrix f08bcf (dtpmqrt) 25 dtpmqrt nagf_lapackeig_dtpmqrt Applies the orthogonal transformation determined by f08bbf (dtpqrt) f08bef (dgeqpf) Example Text Example Data 16 dgeqpf nagf_lapackeig_dgeqpf $QR$ factorization, with column pivoting, of real general rectangular matrix f08bff (dgeqp3) Example Text Example Data 22 dgeqp3 nagf_lapackeig_dgeqp3 $QR$ factorization, with column pivoting, using BLAS-3, of real general rectangular matrix f08bhf (dtzrzf) Example Text Example Data 22 dtzrzf nagf_lapackeig_dtzrzf Reduces a real upper trapezoidal matrix to upper triangular form f08bkf (dormrz) 22 dormrz nagf_lapackeig_dormrz Applies the orthogonal transformation determined by f08bhf (dtzrzf) f08bnf (zgelsy) Example Text Example Data 21 zgelsy nagf_lapackeig_zgelsy Computes the minimum-norm solution to a complex linear least squares problem f08bpf (ztpqrt) Example Text Example Data 25 ztpqrt nagf_lapackeig_ztpqrt $QR$ factorization of complex triangular-pentagonal matrix f08bqf (ztpmqrt) 25 ztpmqrt nagf_lapackeig_ztpmqrt Applies the unitary transformation determined by f08bpf (ztpqrt) f08bsf (zgeqpf) Example Text Example Data 16 zgeqpf nagf_lapackeig_zgeqpf $QR$ factorization, with column pivoting, of complex general rectangular matrix f08btf (zgeqp3) Example Text Example Data 22 zgeqp3 nagf_lapackeig_zgeqp3 $QR$ factorization, with column pivoting, using BLAS-3, of complex general rectangular matrix f08bvf (ztzrzf) Example Text Example Data 22 ztzrzf nagf_lapackeig_ztzrzf Reduces a complex upper trapezoidal matrix to upper triangular form f08bxf (zunmrz) 22 zunmrz nagf_lapackeig_zunmrz Applies the unitary transformation determined by f08bvf (ztzrzf) f08cef (dgeqlf) Example Text Example Data 22 dgeqlf nagf_lapackeig_dgeqlf $QL$ factorization of real general rectangular matrix f08cff (dorgql) Example Text Example Data 22 dorgql nagf_lapackeig_dorgql Form all or part of orthogonal $Q$ from $QL$ factorization determined by f08cef (dgeqlf) f08cgf (dormql) 22 dormql nagf_lapackeig_dormql Applies the orthogonal transformation determined by f08cef (dgeqlf) f08chf (dgerqf) Example Text Example Data 22 dgerqf nagf_lapackeig_dgerqf $RQ$ factorization of real general rectangular matrix f08cjf (dorgrq) Example Text Example Data 22 dorgrq nagf_lapackeig_dorgrq Form all or part of orthogonal $Q$ from $RQ$ factorization determined by f08chf (dgerqf) f08ckf (dormrq) 22 dormrq nagf_lapackeig_dormrq Applies the orthogonal transformation determined by f08chf (dgerqf) f08csf (zgeqlf) Example Text Example Data 22 zgeqlf nagf_lapackeig_zgeqlf $QL$ factorization of complex general rectangular matrix f08ctf (zungql) Example Text Example Data 22 zungql nagf_lapackeig_zungql Form all or part of unitary $Q$ from $QL$ factorization determined by f08csf (zgeqlf) f08cuf (zunmql) 22 zunmql nagf_lapackeig_zunmql Applies the unitary transformation determined by f08csf (zgeqlf) f08cvf (zgerqf) Example Text Example Data 22 zgerqf nagf_lapackeig_zgerqf $RQ$ factorization of complex general rectangular matrix f08cwf (zungrq) Example Text Example Data 22 zungrq nagf_lapackeig_zungrq Form all or part of unitary $Q$ from $RQ$ factorization determined by f08cvf (zgerqf) f08cxf (zunmrq) 22 zunmrq nagf_lapackeig_zunmrq Applies the unitary transformation determined by f08cvf (zgerqf) f08faf (dsyev) Example Text Example Data 21 dsyev nagf_lapackeig_dsyev Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix f08fbf (dsyevx) Example Text Example Data 21 dsyevx nagf_lapackeig_dsyevx Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix f08fcf (dsyevd) Example Text Example Data 19 dsyevd nagf_lapackeig_dsyevd Computes all eigenvalues and, optionally, all eigenvectors of real symmetric matrix (divide-and-conquer) f08fdf (dsyevr) Example Text Example Data 21 dsyevr nagf_lapackeig_dsyevr Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix (Relatively Robust Representations) f08fef (dsytrd) Example Text Example Data 16 dsytrd nagf_lapackeig_dsytrd Orthogonal reduction of real symmetric matrix to symmetric tridiagonal form f08fff (dorgtr) Example Text Example Data 16 dorgtr nagf_lapackeig_dorgtr Generate orthogonal transformation matrix from reduction to tridiagonal form determined by f08fef (dsytrd) f08fgf (dormtr) Example Text Example Data 16 dormtr nagf_lapackeig_dormtr Applies the orthogonal transformation determined by f08fef (dsytrd) f08flf (ddisna) 22 ddisna nagf_lapackeig_ddisna 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 f08fnf (zheev) Example Text Example Data 21 zheev nagf_lapackeig_zheev Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix f08fpf (zheevx) Example Text Example Data 21 zheevx nagf_lapackeig_zheevx Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix f08fqf (zheevd) Example Text Example Data 19 zheevd nagf_lapackeig_zheevd Computes all eigenvalues and, optionally, all eigenvectors of complex Hermitian matrix (divide-and-conquer) f08frf (zheevr) Example Text Example Data 21 zheevr nagf_lapackeig_zheevr Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix (Relatively Robust Representations) f08fsf (zhetrd) Example Text Example Data 16 zhetrd nagf_lapackeig_zhetrd Unitary reduction of complex Hermitian matrix to real symmetric tridiagonal form f08ftf (zungtr) Example Text Example Data 16 zungtr nagf_lapackeig_zungtr Generate unitary transformation matrix from reduction to tridiagonal form determined by f08fsf (zhetrd) f08fuf (zunmtr) Example Text Example Data 16 zunmtr nagf_lapackeig_zunmtr Applies the unitary transformation matrix determined by f08fsf (zhetrd) f08gaf (dspev) Example Text Example Data 21 dspev nagf_lapackeig_dspev Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage f08gbf (dspevx) Example Text Example Data 21 dspevx nagf_lapackeig_dspevx Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage f08gcf (dspevd) Example Text Example Data 19 dspevd nagf_lapackeig_dspevd Computes all eigenvalues and, optionally, all eigenvectors of real symmetric matrix, packed storage (divide-and-conquer or Pal–Walker–Kahan variant of the $QL$ or $QR$ algorithm) f08gef (dsptrd) Example Text Example Data 16 dsptrd nagf_lapackeig_dsptrd Orthogonal reduction of real symmetric matrix to symmetric tridiagonal form, packed storage f08gff (dopgtr) Example Text Example Data 16 dopgtr nagf_lapackeig_dopgtr Generate orthogonal transformation matrix from reduction to tridiagonal form determined by f08gef (dsptrd) f08ggf (dopmtr) Example Text Example Data 16 dopmtr nagf_lapackeig_dopmtr Applies the orthogonal transformation determined by f08gef (dsptrd) f08gnf (zhpev) Example Text Example Data 21 zhpev nagf_lapackeig_zhpev Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage f08gpf (zhpevx) Example Text Example Data 21 zhpevx nagf_lapackeig_zhpevx Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage f08gqf (zhpevd) Example Text Example Data 19 zhpevd nagf_lapackeig_zhpevd Computes all eigenvalues and, optionally, all eigenvectors of complex Hermitian matrix, packed storage (divide-and-conquer or Pal–Walker–Kahan variant of the $QL$ or $QR$ algorithm) f08gsf (zhptrd) Example Text Example Data 16 zhptrd nagf_lapackeig_zhptrd Performs a unitary reduction of complex Hermitian matrix to real symmetric tridiagonal form, packed storage f08gtf (zupgtr) Example Text Example Data 16 zupgtr nagf_lapackeig_zupgtr Generates a unitary transformation matrix from reduction to tridiagonal form determined by f08gsf (zhptrd) f08guf (zupmtr) Example Text Example Data 16 zupmtr nagf_lapackeig_zupmtr Applies the unitary transformation matrix determined by f08gsf (zhptrd) f08haf (dsbev) Example Text Example Data 21 dsbev nagf_lapackeig_dsbev Computes all eigenvalues and, optionally, eigenvectors of a real symmetric band matrix f08hbf (dsbevx) Example Text Example Data 21 dsbevx nagf_lapackeig_dsbevx Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix f08hcf (dsbevd) Example Text Example Data 19 dsbevd nagf_lapackeig_dsbevd Computes all eigenvalues and, optionally, all eigenvectors of real symmetric band matrix (divide-and-conquer or Pal–Walker–Kahan variant of the $QL$ or $QR$ algorithm) f08hef (dsbtrd) Example Text Example Data 16 dsbtrd nagf_lapackeig_dsbtrd Performs an orthogonal reduction of real symmetric band matrix to symmetric tridiagonal form f08hnf (zhbev) Example Text Example Data 21 zhbev nagf_lapackeig_zhbev Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix f08hpf (zhbevx) Example Text Example Data 21 zhbevx nagf_lapackeig_zhbevx Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix f08hqf (zhbevd) Example Text Example Data 19 zhbevd nagf_lapackeig_zhbevd Computes all eigenvalues and, optionally, all eigenvectors of complex Hermitian band matrix (divide-and-conquer) f08hsf (zhbtrd) Example Text Example Data 16 zhbtrd nagf_lapackeig_zhbtrd Performs a unitary reduction of complex Hermitian band matrix to real symmetric tridiagonal form f08jaf (dstev) Example Text Example Data 21 dstev nagf_lapackeig_dstev Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix f08jbf (dstevx) Example Text Example Data 21 dstevx nagf_lapackeig_dstevx Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix f08jcf (dstevd) Example Text Example Data 19 dstevd nagf_lapackeig_dstevd Computes all eigenvalues and, optionally, all eigenvectors of real symmetric tridiagonal matrix (divide-and-conquer) f08jdf (dstevr) Example Text Example Data 21 dstevr nagf_lapackeig_dstevr Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix (Relatively Robust Representations) f08jef (dsteqr) Example Text Example Data 16 dsteqr nagf_lapackeig_dsteqr Computes all eigenvalues and eigenvectors of real symmetric tridiagonal matrix, reduced from real symmetric matrix using the implicit $QL$ or $QR$ algorithm f08jff (dsterf) Example Text Example Data 16 dsterf nagf_lapackeig_dsterf Computes all eigenvalues of real symmetric tridiagonal matrix, root-free variant of the $QL$ or $QR$ algorithm f08jgf (dpteqr) Example Text Example Data 16 dpteqr nagf_lapackeig_dpteqr Computes all eigenvalues and eigenvectors of real symmetric positive definite tridiagonal matrix, reduced from real symmetric positive definite matrix f08jhf (dstedc) Example Text Example Data 22 dstedc nagf_lapackeig_dstedc Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a matrix reduced to this form (divide-and-conquer) f08jjf (dstebz) 16 dstebz nagf_lapackeig_dstebz Computes selected eigenvalues of real symmetric tridiagonal matrix by bisection f08jkf (dstein) 16 dstein nagf_lapackeig_dstein Computes selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in real array f08jlf (dstegr) Example Text Example Data 22 dstegr nagf_lapackeig_dstegr Computes selected eigenvalues and, optionally, the corresponding eigenvectors of a real symmetric tridiagonal matrix or a symmetric matrix reduced to this form (Relatively Robust Representations) f08jsf (zsteqr) 16 zsteqr nagf_lapackeig_zsteqr Computes all eigenvalues and eigenvectors of real symmetric tridiagonal matrix, reduced from complex Hermitian matrix, using the implicit $QL$ or $QR$ algorithm f08juf (zpteqr) Example Text Example Data 16 zpteqr nagf_lapackeig_zpteqr Computes all eigenvalues and eigenvectors of real symmetric positive definite tridiagonal matrix, reduced from complex Hermitian positive definite matrix f08jvf (zstedc) Example Text Example Data 22 zstedc nagf_lapackeig_zstedc Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (divide-and-conquer) f08jxf (zstein) 16 zstein nagf_lapackeig_zstein Computes selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in complex array f08jyf (zstegr) Example Text Example Data 22 zstegr nagf_lapackeig_zstegr Computes selected eigenvalues and, optionally, the corresponding eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (Relatively Robust Representations) f08kaf (dgelss) Example Text Example Data 21 dgelss nagf_lapackeig_dgelss Computes the minimum-norm solution to a real linear least squares problem using singular value decomposition f08kbf (dgesvd) Example Text Example Data 21 dgesvd nagf_lapackeig_dgesvd Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors f08kcf (dgelsd) Example Text Example Data 21 dgelsd nagf_lapackeig_dgelsd Computes the minimum-norm solution to a real linear least squares problem using singular value decomposition (divide-and-conquer) f08kdf (dgesdd) Example Text Example Data 21 dgesdd nagf_lapackeig_dgesdd Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (divide-and-conquer) f08kef (dgebrd) Example Text Example Data 16 dgebrd nagf_lapackeig_dgebrd Performs an orthogonal reduction of real general rectangular matrix to bidiagonal form f08kff (dorgbr) Example Text Example Data 16 dorgbr nagf_lapackeig_dorgbr Generates an orthogonal transformation matrices from reduction to bidiagonal form determined by f08kef (dgebrd) f08kgf (dormbr) Example Text Example Data 16 dormbr nagf_lapackeig_dormbr Applies the orthogonal transformations from reduction to bidiagonal form determined by f08kef (dgebrd) f08khf (dgejsv) Example Text Example Data 23 dgejsv nagf_lapackeig_dgejsv Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (preconditioned Jacobi) f08kjf (dgesvj) Example Text Example Data 23 dgesvj nagf_lapackeig_dgesvj Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (fast Jacobi) f08knf (zgelss) Example Text Example Data 21 zgelss nagf_lapackeig_zgelss Computes the minimum-norm solution to a complex linear least squares problem using singular value decomposition f08kpf (zgesvd) Example Text Example Data 21 zgesvd nagf_lapackeig_zgesvd Computes the singular value decomposition of a complex matrix, optionally computing the left and/or right singular vectors f08kqf (zgelsd) Example Text Example Data 21 zgelsd nagf_lapackeig_zgelsd Computes the minimum-norm solution to a complex linear least squares problem using singular value decomposition (divide-and-conquer) f08krf (zgesdd) Example Text Example Data 21 zgesdd nagf_lapackeig_zgesdd Computes the singular value decomposition of a complex matrix, optionally computing the left and/or right singular vectors (divide-and-conquer) f08ksf (zgebrd) Example Text Example Data 16 zgebrd nagf_lapackeig_zgebrd Performs a unitary reduction of complex general rectangular matrix to bidiagonal form f08ktf (zungbr) Example Text Example Data 16 zungbr nagf_lapackeig_zungbr Generates unitary transformation matrices from the reduction to bidiagonal form determined by f08ksf (zgebrd) f08kuf (zunmbr) Example Text Example Data 16 zunmbr nagf_lapackeig_zunmbr Applies the unitary transformations from reduction to bidiagonal form determined by f08ksf (zgebrd) f08lef (dgbbrd) Example Text Example Data 19 dgbbrd nagf_lapackeig_dgbbrd Performs a reduction of real rectangular band matrix to upper bidiagonal form f08lsf (zgbbrd) Example Text Example Data 19 zgbbrd nagf_lapackeig_zgbbrd Reduction of complex rectangular band matrix to upper bidiagonal form f08mdf (dbdsdc) Example Text Example Data 22 dbdsdc nagf_lapackeig_dbdsdc Computes the singular value decomposition of a real bidiagonal matrix, optionally computing the singular vectors (divide-and-conquer) f08mef (dbdsqr) Example Text Example Data 16 dbdsqr nagf_lapackeig_dbdsqr Performs an SVD of real bidiagonal matrix reduced from real general matrix f08msf (zbdsqr) 16 zbdsqr nagf_lapackeig_zbdsqr Performs an SVD of real bidiagonal matrix reduced from complex general matrix f08naf (dgeev) Example Text Example Data 21 dgeev nagf_lapackeig_dgeev Computes all eigenvalues and, optionally, left and/or right eigenvectors of a real nonsymmetric matrix f08nbf (dgeevx) Example Text Example Data 21 dgeevx nagf_lapackeig_dgeevx 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 f08nef (dgehrd) Example Text Example Data 16 dgehrd nagf_lapackeig_dgehrd Performs an orthogonal reduction of real general matrix to upper Hessenberg form f08nff (dorghr) Example Text Example Data 16 dorghr nagf_lapackeig_dorghr Generates an orthogonal transformation matrix from reduction to Hessenberg form determined by f08nef (dgehrd) f08ngf (dormhr) Example Text Example Data 16 dormhr nagf_lapackeig_dormhr Applies the orthogonal transformation matrix from reduction to Hessenberg form determined by f08nef (dgehrd) f08nhf (dgebal) Example Text Example Data 16 dgebal nagf_lapackeig_dgebal Balances a real general matrix f08njf (dgebak) 16 dgebak nagf_lapackeig_dgebak Transforms eigenvectors of real balanced matrix to those of original matrix supplied to f08nhf (dgebal) f08nnf (zgeev) Example Text Example Data 21 zgeev nagf_lapackeig_zgeev Computes all eigenvalues and, optionally, left and/or right eigenvectors of a complex nonsymmetric matrix f08npf (zgeevx) Example Text Example Data 21 zgeevx nagf_lapackeig_zgeevx 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 f08nsf (zgehrd) Example Text Example Data 16 zgehrd nagf_lapackeig_zgehrd Performs a unitary reduction of complex general matrix to upper Hessenberg form f08ntf (zunghr) Example Text Example Data 16 zunghr nagf_lapackeig_zunghr Generates a unitary transformation matrix from reduction to Hessenberg form determined by f08nsf (zgehrd) f08nuf (zunmhr) Example Text Example Data 16 zunmhr nagf_lapackeig_zunmhr Applies the unitary transformation matrix from reduction to Hessenberg form determined by f08nsf (zgehrd) f08nvf (zgebal) Example Text Example Data 16 zgebal nagf_lapackeig_zgebal Balances a complex general matrix f08nwf (zgebak) 16 zgebak nagf_lapackeig_zgebak Transforms eigenvectors of complex balanced matrix to those of original matrix supplied to f08nvf (zgebal) f08paf (dgees) Example Text Example Data 21 dgees nagf_lapackeig_dgees Computes for real square nonsymmetric matrix, the eigenvalues, the real Schur form, and, optionally, the matrix of Schur vectors f08pbf (dgeesx) Example Text Example Data 21 dgeesx nagf_lapackeig_dgeesx 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 f08pef (dhseqr) Example Text Example Data 16 dhseqr nagf_lapackeig_dhseqr Computes the eigenvalues and Schur factorization of real upper Hessenberg matrix reduced from real general matrix f08pkf (dhsein) 16 dhsein nagf_lapackeig_dhsein Computes selected right and/or left eigenvectors of real upper Hessenberg matrix by inverse iteration f08pnf (zgees) Example Text Example Data 21 zgees nagf_lapackeig_zgees Computes for complex square nonsymmetric matrix, the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors f08ppf (zgeesx) Example Text Example Data 21 zgeesx nagf_lapackeig_zgeesx Computes for real square nonsymmetric matrix, the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors; also computes a reciprocal condition number for the average of the selected eigenvalues and for the right invariant subspace corresponding to these eigenvalues f08psf (zhseqr) Example Text Example Data 16 zhseqr nagf_lapackeig_zhseqr Computes the eigenvalues and Schur factorization of complex upper Hessenberg matrix reduced from complex general matrix f08pxf (zhsein) 16 zhsein nagf_lapackeig_zhsein Computes selected right and/or left eigenvectors of complex upper Hessenberg matrix by inverse iteration f08qff (dtrexc) Example Text Example Data 16 dtrexc nagf_lapackeig_dtrexc Reorders a Schur factorization of real matrix using orthogonal similarity transformation f08qgf (dtrsen) Example Text Example Data 16 dtrsen nagf_lapackeig_dtrsen Reorders a Schur factorization of real matrix, form orthonormal basis of right invariant subspace for selected eigenvalues, with estimates of sensitivities f08qhf (dtrsyl) Example Text Example Data 16 dtrsyl nagf_lapackeig_dtrsyl Solves the real Sylvester matrix equation $AX+XB=C$, $A$ and $B$ are upper quasi-triangular or transposes f08qkf (dtrevc) 16 dtrevc nagf_lapackeig_dtrevc Computes left and right eigenvectors of real upper quasi-triangular matrix f08qlf (dtrsna) Example Text Example Data 16 dtrsna nagf_lapackeig_dtrsna Computes estimates of sensitivities of selected eigenvalues and eigenvectors of real upper quasi-triangular matrix f08qtf (ztrexc) Example Text Example Data 16 ztrexc nagf_lapackeig_ztrexc Reorders a Schur factorization of complex matrix using unitary similarity transformation f08quf (ztrsen) Example Text Example Data 16 ztrsen nagf_lapackeig_ztrsen Reorders a Schur factorization of complex matrix, form orthonormal basis of right invariant subspace for selected eigenvalues, with estimates of sensitivities f08qvf (ztrsyl) Example Text Example Data 16 ztrsyl nagf_lapackeig_ztrsyl Solves the complex Sylvester matrix equation $AX+XB=C$, $A$ and $B$ are upper triangular or conjugate-transposes f08qxf (ztrevc) 16 ztrevc nagf_lapackeig_ztrevc Computes left and right eigenvectors of complex upper triangular matrix f08qyf (ztrsna) Example Text Example Data 16 ztrsna nagf_lapackeig_ztrsna Computes estimates of sensitivities of selected eigenvalues and eigenvectors of complex upper triangular matrix f08raf (dorcsd) Example Text Example Data 25 dorcsd nagf_lapackeig_dorcsd Computes the CS decomposition of an orthogonal matrix partitioned into four real submatrices f08rnf (zuncsd) Example Text Example Data 25 zuncsd nagf_lapackeig_zuncsd Computes the CS decomposition of a unitary matrix partitioned into four complex submatrices f08saf (dsygv) Example Text Example Data 21 dsygv nagf_lapackeig_dsygv Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem f08sbf (dsygvx) Example Text Example Data 21 dsygvx nagf_lapackeig_dsygvx Computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem f08scf (dsygvd) Example Text Example Data 21 dsygvd nagf_lapackeig_dsygvd Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem (divide-and-conquer) f08sef (dsygst) Example Text Example Data 16 dsygst nagf_lapackeig_dsygst Performs a reduction to standard form of real symmetric-definite generalized eigenproblem $Ax=\lambda Bx$, $ABx=\lambda x$ or $BAx=\lambda x$, $B$ factorized by f07fdf (dpotrf) f08snf (zhegv) Example Text Example Data 21 zhegv nagf_lapackeig_zhegv Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem f08spf (zhegvx) Example Text Example Data 21 zhegvx nagf_lapackeig_zhegvx Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem f08sqf (zhegvd) Example Text Example Data 21 zhegvd nagf_lapackeig_zhegvd Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem (divide-and-conquer) f08ssf (zhegst) Example Text Example Data 16 zhegst nagf_lapackeig_zhegst Performs a reduction to standard form of complex Hermitian-definite generalized eigenproblem $Ax=\lambda Bx$, $ABx=\lambda x$ or $BAx=\lambda x$, $B$ factorized by f07frf (zpotrf) f08taf (dspgv) Example Text Example Data 21 dspgv nagf_lapackeig_dspgv Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage f08tbf (dspgvx) Example Text Example Data 21 dspgvx nagf_lapackeig_dspgvx Computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage f08tcf (dspgvd) Example Text Example Data 21 dspgvd nagf_lapackeig_dspgvd Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage (divide-and-conquer) f08tef (dspgst) Example Text Example Data 16 dspgst nagf_lapackeig_dspgst Performs a 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 f07gdf (dpptrf) f08tnf (zhpgv) Example Text Example Data 21 zhpgv nagf_lapackeig_zhpgv Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage f08tpf (zhpgvx) Example Text Example Data 21 zhpgvx nagf_lapackeig_zhpgvx Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage f08tqf (zhpgvd) Example Text Example Data 21 zhpgvd nagf_lapackeig_zhpgvd Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage (divide-and-conquer) f08tsf (zhpgst) Example Text Example Data 16 zhpgst nagf_lapackeig_zhpgst Performs a 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 f07grf (zpptrf) f08uaf (dsbgv) Example Text Example Data 21 dsbgv nagf_lapackeig_dsbgv Computes all the eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem f08ubf (dsbgvx) Example Text Example Data 21 dsbgvx nagf_lapackeig_dsbgvx Computes selected eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem f08ucf (dsbgvd) Example Text Example Data 21 dsbgvd nagf_lapackeig_dsbgvd Computes all the eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem (divide-and-conquer) f08uef (dsbgst) Example Text Example Data 19 dsbgst nagf_lapackeig_dsbgst Performs a 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 $A$ f08uff (dpbstf) 19 dpbstf nagf_lapackeig_dpbstf Computes a split Cholesky factorization of real symmetric positive definite band matrix $A$ f08unf (zhbgv) Example Text Example Data 21 zhbgv nagf_lapackeig_zhbgv Computes all the eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem f08upf (zhbgvx) Example Text Example Data 21 zhbgvx nagf_lapackeig_zhbgvx Computes selected eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem f08uqf (zhbgvd) Example Text Example Data 21 zhbgvd nagf_lapackeig_zhbgvd Computes all the eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem (divide-and-conquer) f08usf (zhbgst) Example Text Example Data 19 zhbgst nagf_lapackeig_zhbgst Performs a 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 $A$ f08utf (zpbstf) 19 zpbstf nagf_lapackeig_zpbstf Computes a split Cholesky factorization of complex Hermitian positive definite band matrix $A$ f08vaf (dggsvd) Example Text Example Data 21 dggsvd nagf_lapackeig_dggsvd Computes the generalized singular value decomposition of a real matrix pair f08vcf (dggsvd3) Example Text Example Data 26.0 dggsvd3 nagf_lapackeig_dggsvd3 Computes, using BLAS-3, the generalized singular value decomposition of a real matrix pair f08vef (dggsvp) Example Text Example Data 22 dggsvp nagf_lapackeig_dggsvp Produces orthogonal matrices that simultaneously reduce the $m$ by $n$ matrix $A$ and the $p$ by $n$ matrix $B$ to upper triangular form f08vgf (dggsvp3) Example Text Example Data 26.0 dggsvp3 nagf_lapackeig_dggsvp3 Produces orthogonal matrices, using BLAS-3, that simultaneously reduce the $m$ by $n$ matrix $A$ and the $p$ by $n$ matrix $B$ to upper triangular form f08vnf (zggsvd) Example Text Example Data 21 zggsvd nagf_lapackeig_zggsvd Computes the generalized singular value decomposition of a complex matrix pair f08vqf (zggsvd3) Example Text Example Data 26.0 zggsvd3 nagf_lapackeig_zggsvd3 Computes, using BLAS-3, the generalized singular value decomposition of a complex matrix pair f08vsf (zggsvp) Example Text Example Data 22 zggsvp nagf_lapackeig_zggsvp Produces unitary matrices that simultaneously reduce the complex, $m$ by $n$, matrix $A$ and the complex, $p$ by $n$, matrix $B$ to upper triangular form f08vuf (zggsvp3) Example Text Example Data 26.0 zggsvp3 nagf_lapackeig_zggsvp3 Produces unitary matrices, using BLAS-3, that simultaneously reduce the complex, $m$ by $n$, matrix $A$ and the complex, $p$ by $n$, matrix $B$ to upper triangular form f08waf (dggev) Example Text Example Data 21 dggev nagf_lapackeig_dggev Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors f08wbf (dggevx) Example Text Example Data 21 dggevx nagf_lapackeig_dggevx 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 f08wcf (dggev3) Example Text Example Data 26.0 dggev3 nagf_lapackeig_dggev3 Computes, for a real nonsymmetric matrix pair, using BLAS-3, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors f08wef (dgghrd) 20 dgghrd nagf_lapackeig_dgghrd Performs an orthogonal reduction of a pair of real general matrices to generalized upper Hessenberg form f08wff (dgghd3) 26.0 dgghd3 nagf_lapackeig_dgghd3 Performs, using BLAS-3, an orthogonal reduction of a pair of real general matrices to generalized upper Hessenberg form f08whf (dggbal) 20 dggbal nagf_lapackeig_dggbal Balances a pair of real, square, matrices f08wjf (dggbak) 20 dggbak nagf_lapackeig_dggbak Transforms eigenvectors of a pair of real balanced matrices to those of original matrix pair supplied to f08whf (dggbal) f08wnf (zggev) Example Text Example Data 21 zggev nagf_lapackeig_zggev Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors f08wpf (zggevx) Example Text Example Data 21 zggevx nagf_lapackeig_zggevx 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 f08wqf (zggev3) Example Text Example Data 26.0 zggev3 nagf_lapackeig_zggev3 Computes, for a complex nonsymmetric matrix pair, using BLAS-3, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors f08wsf (zgghrd) 20 zgghrd nagf_lapackeig_zgghrd Performs a unitary reduction of a pair of complex general matrices to generalized upper Hessenberg form f08wtf (zgghd3) 26.0 zgghd3 nagf_lapackeig_zgghd3 Performs, using BLAS-3, a unitary reduction of a pair of complex general matrices to generalized upper Hessenberg form f08wvf (zggbal) 20 zggbal nagf_lapackeig_zggbal Balances a pair of complex, square, matrices f08wwf (zggbak) 20 zggbak nagf_lapackeig_zggbak Transforms eigenvectors of a pair of complex balanced matrices to those of original matrix pair supplied to f08wvf (zggbal) f08xaf (dgges) Example Text Example Data 21 dgges nagf_lapackeig_dgges 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 f08xbf (dggesx) Example Text Example Data 21 dggesx nagf_lapackeig_dggesx 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 f08xcf (dgges3) Example Text Example Data 26.0 dgges3 nagf_lapackeig_dgges3 Computes, for a real nonsymmetric matrix pair, using BLAS-3, the generalized eigenvalues, the generalized real Schur form and, optionally, the left and/or right matrices of Schur vectors f08xef (dhgeqz) Example Text Example Data 20 dhgeqz nagf_lapackeig_dhgeqz Computes eigenvalues and generalized Schur factorization of real generalized upper Hessenberg form reduced from a pair of real general matrices f08xnf (zgges) Example Text Example Data 21 zgges nagf_lapackeig_zgges 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 f08xpf (zggesx) Example Text Example Data 21 zggesx nagf_lapackeig_zggesx 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 f08xqf (zgges3) Example Text Example Data 26.0 zgges3 nagf_lapackeig_zgges3 Computes, for a complex nonsymmetric matrix pair, using BLAS-3, the generalized eigenvalues, the generalized complex Schur form and, optionally, the left and/or right matrices of Schur vectors f08xsf (zhgeqz) Example Text Example Data 20 zhgeqz nagf_lapackeig_zhgeqz Eigenvalues and generalized Schur factorization of complex generalized upper Hessenberg form reduced from a pair of complex, square, matrices f08yef (dtgsja) Example Text Example Data 22 dtgsja nagf_lapackeig_dtgsja Computes the generalized singular value decomposition of a real upper triangular (or trapezoidal) matrix pair f08yff (dtgexc) Example Text Example Data 22 dtgexc nagf_lapackeig_dtgexc Reorders the generalized real Schur decomposition of a real matrix pair using an orthogonal equivalence transformation f08ygf (dtgsen) Example Text Example Data 22 dtgsen nagf_lapackeig_dtgsen 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 f08yhf (dtgsyl) Example Text Example Data 22 dtgsyl nagf_lapackeig_dtgsyl Solves the real-valued, generalized, quasi-trangular, Sylvester equation f08ykf (dtgevc) Example Text Example Data 20 dtgevc nagf_lapackeig_dtgevc Computes right and left generalized eigenvectors of the matrix pair $\left(A,B\right)$ which is assumed to be in generalized upper Schur form f08ylf (dtgsna) Example Text Example Data 22 dtgsna nagf_lapackeig_dtgsna Estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a real matrix pair in generalized real Schur canonical form f08ysf (ztgsja) Example Text Example Data 22 ztgsja nagf_lapackeig_ztgsja Computes the generalized singular value decomposition of a complex upper triangular (or trapezoidal) matrix pair f08ytf (ztgexc) Example Text Example Data 22 ztgexc nagf_lapackeig_ztgexc Reorders the generalized Schur decomposition of a complex matrix pair using an unitary equivalence transformation f08yuf (ztgsen) Example Text Example Data 22 ztgsen nagf_lapackeig_ztgsen 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 f08yvf (ztgsyl) Example Text Example Data 22 ztgsyl nagf_lapackeig_ztgsyl Solves the complex generalized Sylvester equation f08yxf (ztgevc) Example Text Example Data 20 ztgevc nagf_lapackeig_ztgevc Computes left and right eigenvectors of a pair of complex upper triangular matrices f08yyf (ztgsna) Example Text Example Data 22 ztgsna nagf_lapackeig_ztgsna Estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a complex matrix pair in generalized Schur canonical form f08zaf (dgglse) Example Text Example Data 21 dgglse nagf_lapackeig_dgglse Solves the real linear equality-constrained least squares (LSE) problem f08zbf (dggglm) Example Text Example Data 21 dggglm nagf_lapackeig_dggglm Solves a real general Gauss–Markov linear model (GLM) problem f08zef (dggqrf) Example Text Example Data 22 dggqrf nagf_lapackeig_dggqrf Computes a generalized $QR$ factorization of a real matrix pair f08zff (dggrqf) Example Text Example Data 22 dggrqf nagf_lapackeig_dggrqf Computes a generalized $RQ$ factorization of a real matrix pair f08znf (zgglse) Example Text Example Data 21 zgglse nagf_lapackeig_zgglse Solves the complex linear equality-constrained least squares (LSE) problem f08zpf (zggglm) Example Text Example Data 21 zggglm nagf_lapackeig_zggglm Solves a complex general Gauss–Markov linear model (GLM) problem f08zsf (zggqrf) Example Text Example Data 22 zggqrf nagf_lapackeig_zggqrf Computes a generalized $QR$ factorization of a complex matrix pair f08ztf (zggrqf) Example Text Example Data 22 zggrqf nagf_lapackeig_zggrqf Computes a generalized $RQ$ factorization of a complex matrix pair
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017