NAG Toolbox

• 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