# NAG Library Chapter Contents

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

F08 (lapack) 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_lapack_dgels Solves a real linear least squares problem of full rank f08abf (dgeqrt) Example Text Example Data 25 DGEQRT nagf_lapack_dgeqrt Performs a $QR$ factorization of real general rectangular matrix, with explicit blocking f08acf (dgemqrt) 25 DGEMQRT nagf_lapack_dgemqrt Applies the orthogonal transformation determined by f08abf (dgeqrt) f08aef (dgeqrf) Example Text Example Data 16 DGEQRF nagf_lapack_dgeqrf Performs a $QR$ factorization of real general rectangular matrix f08aff (dorgqr) Example Text Example Data 16 DORGQR nagf_lapack_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_lapack_dormqr Applies an orthogonal transformation determined by f08aef (dgeqrf), f08bef (dgeqpf) or f08bff (dgeqp3) f08ahf (dgelqf) Example Text Example Data 16 DGELQF nagf_lapack_dgelqf Performs a $LQ$ factorization of real general rectangular matrix f08ajf (dorglq) Example Text Example Data 16 DORGLQ nagf_lapack_dorglq Forms all or part of orthogonal $Q$ from $LQ$ factorization determined by f08ahf (dgelqf) f08akf (dormlq) 16 DORMLQ nagf_lapack_dormlq Applies the orthogonal transformation determined by f08ahf (dgelqf) f08anf (zgels) Example Text Example Data 21 ZGELS nagf_lapack_zgels Solves a complex linear least problem of full rank f08apf (zgeqrt) Example Text Example Data 25 ZGEQRT nagf_lapack_zgeqrt Performs a $QR$ factorization of complex general rectangular matrix using recursive algorithm f08aqf (zgemqrt) 25 ZGEMQRT nagf_lapack_zgemqrt Applies the unitary transformation determined by f08apf (zgeqrt) f08asf (zgeqrf) Example Text Example Data 16 ZGEQRF nagf_lapack_zgeqrf Performs a $QR$ factorization of complex general rectangular matrix f08atf (zungqr) Example Text Example Data 16 ZUNGQR nagf_lapack_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_lapack_zunmqr Applies a unitary transformation determined by f08asf (zgeqrf), f08bsf (zgeqpf) or f08btf (zgeqp3) f08avf (zgelqf) Example Text Example Data 16 ZGELQF nagf_lapack_zgelqf Performs a $LQ$ factorization of complex general rectangular matrix f08awf (zunglq) Example Text Example Data 16 ZUNGLQ nagf_lapack_zunglq Forms all or part of unitary $Q$ from $LQ$ factorization determined by f08avf (zgelqf) f08axf (zunmlq) 16 ZUNMLQ nagf_lapack_zunmlq Applies the unitary transformation determined by f08avf (zgelqf) f08baf (dgelsy) Example Text Example Data 21 DGELSY nagf_lapack_dgelsy Computes the minimum-norm solution to a real linear least squares problem f08bbf (dtpqrt) Example Text Example Data 25 DTPQRT nagf_lapack_dtpqrt $QR$ factorization of real general triangular-pentagonal matrix f08bcf (dtpmqrt) 25 DTPMQRT nagf_lapack_dtpmqrt Applies the orthogonal transformation determined by f08bbf (dtpqrt) f08bef (dgeqpf) Example Text Example Data 16 DGEQPF nagf_lapack_dgeqpf $QR$ factorization, with column pivoting, of real general rectangular matrix f08bff (dgeqp3) Example Text Example Data 22 DGEQP3 nagf_lapack_dgeqp3 $QR$ factorization, with column pivoting, using BLAS-3, of real general rectangular matrix f08bhf (dtzrzf) Example Text Example Data 22 DTZRZF nagf_lapack_dtzrzf Reduces a real upper trapezoidal matrix to upper triangular form f08bkf (dormrz) 22 DORMRZ nagf_lapack_dormrz Applies the orthogonal transformation determined by f08bhf (dtzrzf) f08bnf (zgelsy) Example Text Example Data 21 ZGELSY nagf_lapack_zgelsy Computes the minimum-norm solution to a complex linear least squares problem f08bpf (ztpqrt) Example Text Example Data 25 ZTPQRT nagf_lapack_ztpqrt $QR$ factorization of complex triangular-pentagonal matrix f08bqf (ztpmqrt) 25 ZTPMQRT nagf_lapack_ztpmqrt Applies the unitary transformation determined by f08bpf (ztpqrt) f08bsf (zgeqpf) Example Text Example Data 16 ZGEQPF nagf_lapack_zgeqpf $QR$ factorization, with column pivoting, of complex general rectangular matrix f08btf (zgeqp3) Example Text Example Data 22 ZGEQP3 nagf_lapack_zgeqp3 $QR$ factorization, with column pivoting, using BLAS-3, of complex general rectangular matrix f08bvf (ztzrzf) Example Text Example Data 22 ZTZRZF nagf_lapack_ztzrzf Reduces a complex upper trapezoidal matrix to upper triangular form f08bxf (zunmrz) 22 ZUNMRZ nagf_lapack_zunmrz Applies the unitary transformation determined by f08bvf (ztzrzf) f08cef (dgeqlf) Example Text Example Data 22 DGEQLF nagf_lapack_dgeqlf $QL$ factorization of real general rectangular matrix f08cff (dorgql) Example Text Example Data 22 DORGQL nagf_lapack_dorgql Form all or part of orthogonal $Q$ from $QL$ factorization determined by f08cef (dgeqlf) f08cgf (dormql) 22 DORMQL nagf_lapack_dormql Applies the orthogonal transformation determined by f08cef (dgeqlf) f08chf (dgerqf) Example Text Example Data 22 DGERQF nagf_lapack_dgerqf $RQ$ factorization of real general rectangular matrix f08cjf (dorgrq) Example Text Example Data 22 DORGRQ nagf_lapack_dorgrq Form all or part of orthogonal $Q$ from $RQ$ factorization determined by f08chf (dgerqf) f08ckf (dormrq) 22 DORMRQ nagf_lapack_dormrq Applies the orthogonal transformation determined by f08chf (dgerqf) f08csf (zgeqlf) Example Text Example Data 22 ZGEQLF nagf_lapack_zgeqlf $QL$ factorization of complex general rectangular matrix f08ctf (zungql) Example Text Example Data 22 ZUNGQL nagf_lapack_zungql Form all or part of unitary $Q$ from $QL$ factorization determined by f08csf (zgeqlf) f08cuf (zunmql) 22 ZUNMQL nagf_lapack_zunmql Applies the unitary transformation determined by f08csf (zgeqlf) f08cvf (zgerqf) Example Text Example Data 22 ZGERQF nagf_lapack_zgerqf $RQ$ factorization of complex general rectangular matrix f08cwf (zungrq) Example Text Example Data 22 ZUNGRQ nagf_lapack_zungrq Form all or part of unitary $Q$ from $RQ$ factorization determined by f08cvf (zgerqf) f08cxf (zunmrq) 22 ZUNMRQ nagf_lapack_zunmrq Applies the unitary transformation determined by f08cvf (zgerqf) f08faf (dsyev) Example Text Example Data 21 DSYEV nagf_lapack_dsyev Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix f08fbf (dsyevx) Example Text Example Data 21 DSYEVX nagf_lapack_dsyevx Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix f08fcf (dsyevd) Example Text Example Data 19 DSYEVD nagf_lapack_dsyevd Computes all eigenvalues and, optionally, all eigenvectors of real symmetric matrix (divide-and-conquer) f08fdf (dsyevr) Example Text Example Data 21 DSYEVR nagf_lapack_dsyevr Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix (Relatively Robust Representations) f08fef (dsytrd) Example Text Example Data 16 DSYTRD nagf_lapack_dsytrd Orthogonal reduction of real symmetric matrix to symmetric tridiagonal form f08fff (dorgtr) Example Text Example Data 16 DORGTR nagf_lapack_dorgtr Generate orthogonal transformation matrix from reduction to tridiagonal form determined by f08fef (dsytrd) f08fgf (dormtr) Example Text Example Data 16 DORMTR nagf_lapack_dormtr Applies the orthogonal transformation determined by f08fef (dsytrd) f08flf (ddisna) 22 DDISNA nagf_lapack_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_lapack_zheev Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix f08fpf (zheevx) Example Text Example Data 21 ZHEEVX nagf_lapack_zheevx Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix f08fqf (zheevd) Example Text Example Data 19 ZHEEVD nagf_lapack_zheevd Computes all eigenvalues and, optionally, all eigenvectors of complex Hermitian matrix (divide-and-conquer) f08frf (zheevr) Example Text Example Data 21 ZHEEVR nagf_lapack_zheevr Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix (Relatively Robust Representations) f08fsf (zhetrd) Example Text Example Data 16 ZHETRD nagf_lapack_zhetrd Unitary reduction of complex Hermitian matrix to real symmetric tridiagonal form f08ftf (zungtr) Example Text Example Data 16 ZUNGTR nagf_lapack_zungtr Generate unitary transformation matrix from reduction to tridiagonal form determined by f08fsf (zhetrd) f08fuf (zunmtr) Example Text Example Data 16 ZUNMTR nagf_lapack_zunmtr Applies the unitary transformation matrix determined by f08fsf (zhetrd) f08gaf (dspev) Example Text Example Data 21 DSPEV nagf_lapack_dspev Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage f08gbf (dspevx) Example Text Example Data 21 DSPEVX nagf_lapack_dspevx Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage f08gcf (dspevd) Example Text Example Data 19 DSPEVD nagf_lapack_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_lapack_dsptrd Orthogonal reduction of real symmetric matrix to symmetric tridiagonal form, packed storage f08gff (dopgtr) Example Text Example Data 16 DOPGTR nagf_lapack_dopgtr Generate orthogonal transformation matrix from reduction to tridiagonal form determined by f08gef (dsptrd) f08ggf (dopmtr) Example Text Example Data 16 DOPMTR nagf_lapack_dopmtr Applies the orthogonal transformation determined by f08gef (dsptrd) f08gnf (zhpev) Example Text Example Data 21 ZHPEV nagf_lapack_zhpev Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage f08gpf (zhpevx) Example Text Example Data 21 ZHPEVX nagf_lapack_zhpevx Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage f08gqf (zhpevd) Example Text Example Data 19 ZHPEVD nagf_lapack_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_lapack_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_lapack_zupgtr Generates a unitary transformation matrix from reduction to tridiagonal form determined by f08gsf (zhptrd) f08guf (zupmtr) Example Text Example Data 16 ZUPMTR nagf_lapack_zupmtr Applies the unitary transformation matrix determined by f08gsf (zhptrd) f08haf (dsbev) Example Text Example Data 21 DSBEV nagf_lapack_dsbev Computes all eigenvalues and, optionally, eigenvectors of a real symmetric band matrix f08hbf (dsbevx) Example Text Example Data 21 DSBEVX nagf_lapack_dsbevx Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix f08hcf (dsbevd) Example Text Example Data 19 DSBEVD nagf_lapack_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_lapack_dsbtrd Performs an orthogonal reduction of real symmetric band matrix to symmetric tridiagonal form f08hnf (zhbev) Example Text Example Data 21 ZHBEV nagf_lapack_zhbev Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix f08hpf (zhbevx) Example Text Example Data 21 ZHBEVX nagf_lapack_zhbevx Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix f08hqf (zhbevd) Example Text Example Data 19 ZHBEVD nagf_lapack_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_lapack_zhbtrd Performs a unitary reduction of complex Hermitian band matrix to real symmetric tridiagonal form f08jaf (dstev) Example Text Example Data 21 DSTEV nagf_lapack_dstev Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix f08jbf (dstevx) Example Text Example Data 21 DSTEVX nagf_lapack_dstevx Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix f08jcf (dstevd) Example Text Example Data 19 DSTEVD nagf_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_dstebz Computes selected eigenvalues of real symmetric tridiagonal matrix by bisection f08jkf (dstein) 16 DSTEIN nagf_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_dgebrd Performs an orthogonal reduction of real general rectangular matrix to bidiagonal form f08kff (dorgbr) Example Text Example Data 16 DORGBR nagf_lapack_dorgbr Generates an orthogonal transformation matrices from reduction to bidiagonal form determined by f08kef (dgebrd) f08kgf (dormbr) Example Text Example Data 16 DORMBR nagf_lapack_dormbr Applies the orthogonal transformations from reduction to bidiagonal form determined by f08kef (dgebrd) f08khf (dgejsv) Example Text Example Data 23 DGEJSV nagf_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_zgebrd Performs a unitary reduction of complex general rectangular matrix to bidiagonal form f08ktf (zungbr) Example Text Example Data 16 ZUNGBR nagf_lapack_zungbr Generates unitary transformation matrices from the reduction to bidiagonal form determined by f08ksf (zgebrd) f08kuf (zunmbr) Example Text Example Data 16 ZUNMBR nagf_lapack_zunmbr Applies the unitary transformations from reduction to bidiagonal form determined by f08ksf (zgebrd) f08lef (dgbbrd) Example Text Example Data 19 DGBBRD nagf_lapack_dgbbrd Performs a reduction of real rectangular band matrix to upper bidiagonal form f08lsf (zgbbrd) Example Text Example Data 19 ZGBBRD nagf_lapack_zgbbrd Reduction of complex rectangular band matrix to upper bidiagonal form f08mdf (dbdsdc) Example Text Example Data 22 DBDSDC nagf_lapack_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_lapack_dbdsqr Performs an SVD of real bidiagonal matrix reduced from real general matrix f08msf (zbdsqr) 16 ZBDSQR nagf_lapack_zbdsqr Performs an SVD of real bidiagonal matrix reduced from complex general matrix f08naf (dgeev) Example Text Example Data 21 DGEEV nagf_lapack_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_lapack_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_lapack_dgehrd Performs an orthogonal reduction of real general matrix to upper Hessenberg form f08nff (dorghr) Example Text Example Data 16 DORGHR nagf_lapack_dorghr Generates an orthogonal transformation matrix from reduction to Hessenberg form determined by f08nef (dgehrd) f08ngf (dormhr) Example Text Example Data 16 DORMHR nagf_lapack_dormhr Applies the orthogonal transformation matrix from reduction to Hessenberg form determined by f08nef (dgehrd) f08nhf (dgebal) Example Text Example Data 16 DGEBAL nagf_lapack_dgebal Balances a real general matrix f08njf (dgebak) 16 DGEBAK nagf_lapack_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_lapack_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_lapack_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_lapack_zgehrd Performs a unitary reduction of complex general matrix to upper Hessenberg form f08ntf (zunghr) Example Text Example Data 16 ZUNGHR nagf_lapack_zunghr Generates a unitary transformation matrix from reduction to Hessenberg form determined by f08nsf (zgehrd) f08nuf (zunmhr) Example Text Example Data 16 ZUNMHR nagf_lapack_zunmhr Applies the unitary transformation matrix from reduction to Hessenberg form determined by f08nsf (zgehrd) f08nvf (zgebal) Example Text Example Data 16 ZGEBAL nagf_lapack_zgebal Balances a complex general matrix f08nwf (zgebak) 16 ZGEBAK nagf_lapack_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_lapack_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_lapack_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_lapack_dhseqr Computes the eigenvalues and Schur factorization of real upper Hessenberg matrix reduced from real general matrix f08pkf (dhsein) 16 DHSEIN nagf_lapack_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_lapack_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_lapack_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_lapack_zhseqr Computes the eigenvalues and Schur factorization of complex upper Hessenberg matrix reduced from complex general matrix f08pxf (zhsein) 16 ZHSEIN nagf_lapack_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_lapack_dtrexc Reorders a Schur factorization of real matrix using orthogonal similarity transformation f08qgf (dtrsen) Example Text Example Data 16 DTRSEN nagf_lapack_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_lapack_dtrsyl Solves the real Sylvester matrix equation $AX+XB=C$, $A$ and $B$ are upper quasi-triangular or transposes f08qkf (dtrevc) 16 DTREVC nagf_lapack_dtrevc Computes left and right eigenvectors of real upper quasi-triangular matrix f08qlf (dtrsna) Example Text Example Data 16 DTRSNA nagf_lapack_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_lapack_ztrexc Reorders a Schur factorization of complex matrix using unitary similarity transformation f08quf (ztrsen) Example Text Example Data 16 ZTRSEN nagf_lapack_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_lapack_ztrsyl Solves the complex Sylvester matrix equation $AX+XB=C$, $A$ and $B$ are upper triangular or conjugate-transposes f08qxf (ztrevc) 16 ZTREVC nagf_lapack_ztrevc Computes left and right eigenvectors of complex upper triangular matrix f08qyf (ztrsna) Example Text Example Data 16 ZTRSNA nagf_lapack_ztrsna Computes estimates of sensitivities of selected eigenvalues and eigenvectors of complex upper triangular matrix f08raf (dorcsd) Example Text Example Data 25 DORCSD nagf_lapack_dorcsd Computes the CS decomposition of an orthogonal matrix partitioned into four real submatrices f08rnf (zuncsd) Example Text Example Data 25 ZUNCSD nagf_lapack_zuncsd Computes the CS decomposition of a unitary matrix partitioned into four complex submatrices f08saf (dsygv) Example Text Example Data 21 DSYGV nagf_lapack_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_lapack_dsygvx Computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem f08scf (dsygvd) Example Text Example Data 21 DSYGVD nagf_lapack_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_lapack_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_lapack_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_lapack_zhegvx Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem f08sqf (zhegvd) Example Text Example Data 21 ZHEGVD nagf_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_dpbstf Computes a split Cholesky factorization of real symmetric positive definite band matrix $A$ f08unf (zhbgv) Example Text Example Data 21 ZHBGV nagf_lapack_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_lapack_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_lapack_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_lapack_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_lapack_zpbstf Computes a split Cholesky factorization of complex Hermitian positive definite band matrix $A$ f08vaf (dggsvd) Example Text Example Data 21 DGGSVD nagf_lapack_dggsvd Computes the generalized singular value decomposition of a real matrix pair f08vcf (dggsvd3) Example Text Example Data 26.0 DGGSVD3 nagf_lapack_dggsvd3 Computes, using BLAS-3, the generalized singular value decomposition of a real matrix pair f08vef (dggsvp) Example Text Example Data 22 DGGSVP nagf_lapack_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_lapack_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_lapack_zggsvd Computes the generalized singular value decomposition of a complex matrix pair f08vqf (zggsvd3) Example Text Example Data 26.0 ZGGSVD3 nagf_lapack_zggsvd3 Computes, using BLAS-3, the generalized singular value decomposition of a complex matrix pair f08vsf (zggsvp) Example Text Example Data 22 ZGGSVP nagf_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_dgghrd Performs an orthogonal reduction of a pair of real general matrices to generalized upper Hessenberg form f08wff (dgghd3) 26.0 DGGHD3 nagf_lapack_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_lapack_dggbal Balances a pair of real, square, matrices f08wjf (dggbak) 20 DGGBAK nagf_lapack_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_lapack_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_lapack_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_lapack_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_lapack_zgghrd Performs a unitary reduction of a pair of complex general matrices to generalized upper Hessenberg form f08wtf (zgghd3) 26.0 ZGGHD3 nagf_lapack_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_lapack_zggbal Balances a pair of complex, square, matrices f08wwf (zggbak) 20 ZGGBAK nagf_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_dtgsyl Solves the real-valued, generalized, quasi-trangular, Sylvester equation f08ykf (dtgevc) Example Text Example Data 20 DTGEVC nagf_lapack_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_lapack_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_lapack_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_lapack_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_lapack_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_lapack_ztgsyl Solves the complex generalized Sylvester equation f08yxf (ztgevc) Example Text Example Data 20 ZTGEVC nagf_lapack_ztgevc Computes left and right eigenvectors of a pair of complex upper triangular matrices f08yyf (ztgsna) Example Text Example Data 22 ZTGSNA nagf_lapack_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_lapack_dgglse Solves the real linear equality-constrained least squares (LSE) problem f08zbf (dggglm) Example Text Example Data 21 DGGGLM nagf_lapack_dggglm Solves a real general Gauss–Markov linear model (GLM) problem f08zef (dggqrf) Example Text Example Data 22 DGGQRF nagf_lapack_dggqrf Computes a generalized $QR$ factorization of a real matrix pair f08zff (dggrqf) Example Text Example Data 22 DGGRQF nagf_lapack_dggrqf Computes a generalized $RQ$ factorization of a real matrix pair f08znf (zgglse) Example Text Example Data 21 ZGGLSE nagf_lapack_zgglse Solves the complex linear equality-constrained least squares (LSE) problem f08zpf (zggglm) Example Text Example Data 21 ZGGGLM nagf_lapack_zggglm Solves a complex general Gauss–Markov linear model (GLM) problem f08zsf (zggqrf) Example Text Example Data 22 ZGGQRF nagf_lapack_zggqrf Computes a generalized $QR$ factorization of a complex matrix pair f08ztf (zggrqf) Example Text Example Data 22 ZGGRQF nagf_lapack_zggrqf Computes a generalized $RQ$ factorization of a complex matrix pair
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017