Linear Algebra

This chapter describes functions for solving linear systems. The library provides linear algebra operations which operate directly on the gsl_vector and gsl_matrix objects. These routines use the standard algorithms from Golub & Van Loan’s Matrix Computations with Level-1 and Level-2 BLAS calls for efficiency.

The functions described in this chapter are declared in the header file gsl_linalg.h.

LU Decomposition

A general

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-by-

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square matrix

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has an

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decomposition into upper and lower triangular matrices,

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where

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is a permutation matrix,

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is unit lower triangular matrix and

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is upper triangular matrix. For square matrices this decomposition can be used to convert the linear system

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into a pair of triangular systems (

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,

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), which can be solved by forward and back-substitution. Note that the

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decomposition is valid for singular matrices.

int gsl_linalg_LU_decomp(gsl_matrix * A, gsl_permutation * p, int * signum)

int gsl_linalg_complex_LU_decomp(gsl_matrix_complex * A, gsl_permutation * p, int * signum)

These functions factorize the square matrix A into the

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decomposition

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. On output the diagonal and upper triangular part of the input matrix A contain the matrix

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. The lower triangular part of the input matrix (excluding the diagonal) contains

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. The diagonal elements of

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are unity, and are not stored.

The permutation matrix

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is encoded in the permutation p on output. The

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-th column of the matrix

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is given by the

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-th column of the identity matrix, where

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the

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-th element of the permutation vector. The sign of the permutation is given by signum. It has the value

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, where

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is the number of interchanges in the permutation.

The algorithm used in the decomposition is Gaussian Elimination with partial pivoting (Golub & Van Loan, Matrix Computations, Algorithm 3.4.1).

int gsl_linalg_LU_solve(const gsl_matrix * LU, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x)

int gsl_linalg_complex_LU_solve(const gsl_matrix_complex * LU, const gsl_permutation * p, const gsl_vector_complex * b, gsl_vector_complex * x)

These functions solve the square system

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using the

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decomposition of

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into (LU, p) given by gsl_linalg_LU_decomp() or gsl_linalg_complex_LU_decomp() as input.

int gsl_linalg_LU_svx(const gsl_matrix * LU, const gsl_permutation * p, gsl_vector * x)

int gsl_linalg_complex_LU_svx(const gsl_matrix_complex * LU, const gsl_permutation * p, gsl_vector_complex * x)

These functions solve the square system

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in-place using the precomputed

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decomposition of

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into (LU, p). On input x should contain the right-hand side

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, which is replaced by the solution on output.

int gsl_linalg_LU_refine(const gsl_matrix * A, const gsl_matrix * LU, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x, gsl_vector * work)

int gsl_linalg_complex_LU_refine(const gsl_matrix_complex * A, const gsl_matrix_complex * LU, const gsl_permutation * p, const gsl_vector_complex * b, gsl_vector_complex * x, gsl_vector_complex * work)

These functions apply an iterative improvement to x, the solution of

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, from the precomputed

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decomposition of

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into (LU, p). Additional workspace of length N is required in work.

int gsl_linalg_LU_invert(const gsl_matrix * LU, const gsl_permutation * p, gsl_matrix * inverse)

int gsl_linalg_complex_LU_invert(const gsl_matrix_complex * LU, const gsl_permutation * p, gsl_matrix_complex * inverse)

These functions compute the inverse of a matrix

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from its

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decomposition (LU, p), storing the result in the matrix inverse. The inverse is computed by solving the system

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for each column of the identity matrix. It is preferable to avoid direct use of the inverse whenever possible, as the linear solver functions can obtain the same result more efficiently and reliably (consult any introductory textbook on numerical linear algebra for details).

double gsl_linalg_LU_det(gsl_matrix * LU, int signum)

gsl_complex gsl_linalg_complex_LU_det(gsl_matrix_complex * LU, int signum)

These functions compute the determinant of a matrix

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from its

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decomposition, LU. The determinant is computed as the product of the diagonal elements of

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and the sign of the row permutation signum.

double gsl_linalg_LU_lndet(gsl_matrix * LU)

double gsl_linalg_complex_LU_lndet(gsl_matrix_complex * LU)

These functions compute the logarithm of the absolute value of the determinant of a matrix

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,

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, from its

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decomposition, LU. This function may be useful if the direct computation of the determinant would overflow or underflow.

int gsl_linalg_LU_sgndet(gsl_matrix * LU, int signum)

gsl_complex gsl_linalg_complex_LU_sgndet(gsl_matrix_complex * LU, int signum)

These functions compute the sign or phase factor of the determinant of a matrix

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,

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, from its

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decomposition, LU.

QR Decomposition

A general rectangular

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-by-

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matrix

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has a

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decomposition into the product of an orthogonal

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-by-

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square matrix

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(where

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) and an

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-by-

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right-triangular matrix

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,

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This decomposition can be used to convert the linear system

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into the triangular system

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, which can be solved by back-substitution. Another use of the

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decomposition is to compute an orthonormal basis for a set of vectors. The first

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columns of

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form an orthonormal basis for the range of

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,

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, when

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has full column rank.

int gsl_linalg_QR_decomp(gsl_matrix * A, gsl_vector * tau)

This function factorizes the

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-by-

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matrix A into the

System Message: WARNING/2 (QR)

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decomposition

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. On output the diagonal and upper triangular part of the input matrix contain the matrix

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. The vector tau and the columns of the lower triangular part of the matrix A contain the Householder coefficients and Householder vectors which encode the orthogonal matrix Q. The vector tau must be of length

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. The matrix

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is related to these components by,

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where

System Message: WARNING/2 (Q_i = I - \tau_i v_i v_i^T)

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and

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is the Householder vector

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. This is the same storage scheme as used by LAPACK.

The algorithm used to perform the decomposition is Householder QR (Golub & Van Loan, “Matrix Computations”, Algorithm 5.2.1).

int gsl_linalg_QR_solve(const gsl_matrix * QR, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x)

This function solves the square system

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using the

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decomposition of

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held in (QR, tau) which must have been computed previously with gsl_linalg_QR_decomp(). The least-squares solution for rectangular systems can be found using gsl_linalg_QR_lssolve().

int gsl_linalg_QR_svx(const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * x)

This function solves the square system

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in-place using the

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decomposition of

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held in (QR, tau) which must have been computed previously by gsl_linalg_QR_decomp(). On input x should contain the right-hand side

System Message: WARNING/2 (b)

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, which is replaced by the solution on output.

int gsl_linalg_QR_lssolve(const gsl_matrix * QR, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x, gsl_vector * residual)

This function finds the least squares solution to the overdetermined system

System Message: WARNING/2 (A x = b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

where the matrix A has more rows than columns. The least squares solution minimizes the Euclidean norm of the residual,

System Message: WARNING/2 (||Ax - b||)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

.The routine requires as input the

System Message: WARNING/2 (QR)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

decomposition of

System Message: WARNING/2 (A)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

into (QR, tau) given by gsl_linalg_QR_decomp(). The solution is returned in x. The residual is computed as a by-product and stored in residual.

int gsl_linalg_QR_QTvec(const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * v)

This function applies the matrix

System Message: WARNING/2 (Q^T)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

encoded in the decomposition (QR, tau) to the vector v, storing the result

System Message: WARNING/2 (Q^T v)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

in v. The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix

System Message: WARNING/2 (Q^T)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

.

int gsl_linalg_QR_Qvec(const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * v)

This function applies the matrix

System Message: WARNING/2 (Q)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

encoded in the decomposition (QR, tau) to the vector v, storing the result

System Message: WARNING/2 (Q v)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

in v. The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix

System Message: WARNING/2 (Q)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

.

int gsl_linalg_QR_QTmat(const gsl_matrix * QR, const gsl_vector * tau, gsl_matrix * A)

This function applies the matrix

System Message: WARNING/2 (Q^T)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

encoded in the decomposition (QR, tau) to the matrix A, storing the result

System Message: WARNING/2 (Q^T A)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

in A. The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix

System Message: WARNING/2 (Q^T)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

.

int gsl_linalg_QR_Rsolve(const gsl_matrix * QR, const gsl_vector * b, gsl_vector * x)

This function solves the triangular system

System Message: WARNING/2 (R x = b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

for x. It may be useful if the product

System Message: WARNING/2 (b' = Q^T b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

has already been computed using gsl_linalg_QR_QTvec().

int gsl_linalg_QR_Rsvx(const gsl_matrix * QR, gsl_vector * x)

This function solves the triangular system

System Message: WARNING/2 (R x = b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

for x in-place. On input x should contain the right-hand side

System Message: WARNING/2 (b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

and is replaced by the solution on output. This function may be useful if the product

System Message: WARNING/2 (b' = Q^T b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

has already been computed using gsl_linalg_QR_QTvec().

int gsl_linalg_QR_unpack(const gsl_matrix * QR, const gsl_vector * tau, gsl_matrix * Q, gsl_matrix * R)

This function unpacks the encoded

System Message: WARNING/2 (QR)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

decomposition (QR, tau) into the matrices Q and R, where Q is

System Message: WARNING/2 (M)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

-by-

System Message: WARNING/2 (M)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

and R is

System Message: WARNING/2 (M)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

-by-

System Message: WARNING/2 (N)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

.

int gsl_linalg_QR_QRsolve(gsl_matrix * Q, gsl_matrix * R, const gsl_vector * b, gsl_vector * x)

This function solves the system

System Message: WARNING/2 (R x = Q^T b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

for x. It can be used when the

System Message: WARNING/2 (QR)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

decomposition of a matrix is available in unpacked form as (Q, R).

int gsl_linalg_QR_update(gsl_matrix * Q, gsl_matrix * R, gsl_vector * w, const gsl_vector * v)

This function performs a rank-1 update

System Message: WARNING/2 (w v^T)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

of the

System Message: WARNING/2 (QR)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

decomposition (Q, R). The update is given by

System Message: WARNING/2 (Q'R' = Q (R + w v^T))

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where the output matrices

System Message: WARNING/2 (Q)

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and

System Message: WARNING/2 (R)

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are also orthogonal and right triangular. Note that w is destroyed by the update.

int gsl_linalg_R_solve(const gsl_matrix * R, const gsl_vector * b, gsl_vector * x)

This function solves the triangular system

System Message: WARNING/2 (R x = b)

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for the

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-by-

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matrix R.

int gsl_linalg_R_svx(const gsl_matrix * R, gsl_vector * x)

This function solves the triangular system

System Message: WARNING/2 (R x = b)

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in-place. On input x should contain the right-hand side

System Message: WARNING/2 (b)

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, which is replaced by the solution on output.

QR Decomposition with Column Pivoting

The

System Message: WARNING/2 (QR)

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decomposition of an

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-by-

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matrix

System Message: WARNING/2 (A)

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can be extended to the rank deficient case by introducing a column permutation

System Message: WARNING/2 (P)

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,

System Message: WARNING/2 (A P = Q R )

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The first

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columns of

System Message: WARNING/2 (Q)

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form an orthonormal basis for the range of

System Message: WARNING/2 (A)

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for a matrix with column rank

System Message: WARNING/2 (r)

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. This decomposition can also be used to convert the linear system

System Message: WARNING/2 (A x = b)

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into the triangular system

System Message: WARNING/2 (R y = Q^T b, x = P y)

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, which can be solved by back-substitution and permutation. We denote the

System Message: WARNING/2 (QR)

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decomposition with column pivoting by

System Message: WARNING/2 (QRP^T)

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since

System Message: WARNING/2 (A = Q R P^T)

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. When

System Message: WARNING/2 (A)

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is rank deficient with

System Message: WARNING/2 (r = {\rm rank}(A))

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, the matrix

System Message: WARNING/2 (R)

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can be partitioned as

System Message: WARNING/2 (R = \left( \begin{matrix} R_{11} & R_{12} \\ 0 & R_{22} \end{matrix} \right) \approx \left( \begin{matrix} R_{11} & R_{12} \\ 0 & 0 \end{matrix} \right))

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where

System Message: WARNING/2 (R_{11})

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is

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-by-

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and nonsingular. In this case, a basic least squares solution for the overdetermined system

System Message: WARNING/2 (A x = b)

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can be obtained as

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where

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consists of the first

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elements of

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. This basic solution is not guaranteed to be the minimum norm solution unless

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(see Complete Orthogonal Decomposition).

int gsl_linalg_QRPT_decomp(gsl_matrix * A, gsl_vector * tau, gsl_permutation * p, int * signum, gsl_vector * norm)

This function factorizes the

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-by-

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matrix A into the

System Message: WARNING/2 (QRP^T)

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decomposition

System Message: WARNING/2 (A = Q R P^T)

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. On output the diagonal and upper triangular part of the input matrix contain the matrix

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. The permutation matrix

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is stored in the permutation p. The sign of the permutation is given by signum. It has the value

System Message: WARNING/2 ((-1)^n)

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, where

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is the number of interchanges in the permutation. The vector tau and the columns of the lower triangular part of the matrix A contain the Householder coefficients and vectors which encode the orthogonal matrix Q. The vector tau must be of length

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. The matrix

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is related to these components by,

System Message: WARNING/2 (Q = Q_k ... Q_2 Q_1)

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where

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and

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is the Householder vector

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This is the same storage scheme as used by LAPACK. The vector norm is a workspace of length N used for column pivoting.

The algorithm used to perform the decomposition is Householder QR with column pivoting (Golub & Van Loan, “Matrix Computations”, Algorithm 5.4.1).

int gsl_linalg_QRPT_decomp2(const gsl_matrix * A, gsl_matrix * q, gsl_matrix * r, gsl_vector * tau, gsl_permutation * p, int * signum, gsl_vector * norm)

This function factorizes the matrix A into the decomposition

System Message: WARNING/2 (A = Q R P^T)

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without modifying A itself and storing the output in the separate matrices q and r.

int gsl_linalg_QRPT_solve(const gsl_matrix * QR, const gsl_vector * tau, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x)

This function solves the square system

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using the

System Message: WARNING/2 (QRP^T)

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decomposition of

System Message: WARNING/2 (A)

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held in (QR, tau, p) which must have been computed previously by gsl_linalg_QRPT_decomp().

int gsl_linalg_QRPT_svx(const gsl_matrix * QR, const gsl_vector * tau, const gsl_permutation * p, gsl_vector * x)

This function solves the square system

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in-place using the

System Message: WARNING/2 (QRP^T)

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decomposition of

System Message: WARNING/2 (A)

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held in (QR, tau, p). On input x should contain the right-hand side

System Message: WARNING/2 (b)

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, which is replaced by the solution on output.

int gsl_linalg_QRPT_lssolve(const gsl_matrix * QR, const gsl_vector * tau, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x, gsl_vector * residual)

This function finds the least squares solution to the overdetermined system

System Message: WARNING/2 (A x = b)

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where the matrix A has more rows than columns and is assumed to have full rank. The least squares solution minimizes the Euclidean norm of the residual,

System Message: WARNING/2 (||b - A x||)

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. The routine requires as input the

System Message: WARNING/2 (QR)

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decomposition of

System Message: WARNING/2 (A)

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into (QR, tau, p) given by gsl_linalg_QRPT_decomp(). The solution is returned in x. The residual is computed as a by-product and stored in residual. For rank deficient matrices, gsl_linalg_QRPT_lssolve2() should be used instead.

int gsl_linalg_QRPT_lssolve2(const gsl_matrix * QR, const gsl_vector * tau, const gsl_permutation * p, const gsl_vector * b, const size_t rank, gsl_vector * x, gsl_vector * residual)

This function finds the least squares solution to the overdetermined system

System Message: WARNING/2 (A x = b)

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where the matrix A has more rows than columns and has rank given by the input rank. If the user does not know the rank of

System Message: WARNING/2 (A)

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, the routine gsl_linalg_QRPT_rank() can be called to estimate it. The least squares solution is the so-called “basic” solution discussed above and may not be the minimum norm solution. The routine requires as input the

System Message: WARNING/2 (QR)

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decomposition of

System Message: WARNING/2 (A)

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into (QR, tau, p) given by gsl_linalg_QRPT_decomp(). The solution is returned in x. The residual is computed as a by-product and stored in residual.

int gsl_linalg_QRPT_QRsolve(const gsl_matrix * Q, const gsl_matrix * R, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x)

This function solves the square system

System Message: WARNING/2 (R P^T x = Q^T b)

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for x. It can be used when the

System Message: WARNING/2 (QR)

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decomposition of a matrix is available in unpacked form as (Q, R).

int gsl_linalg_QRPT_update(gsl_matrix * Q, gsl_matrix * R, const gsl_permutation * p, gsl_vector * w, const gsl_vector * v)

This function performs a rank-1 update

System Message: WARNING/2 (w v^T)

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of the

System Message: WARNING/2 (QRP^T)

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decomposition (Q, R, p). The update is given by

System Message: WARNING/2 (Q'R' = Q (R + w v^T P))

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where the output matrices

System Message: WARNING/2 (Q')

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and

System Message: WARNING/2 (R')

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are also orthogonal and right triangular. Note that w is destroyed by the update. The permutation p is not changed.

int gsl_linalg_QRPT_Rsolve(const gsl_matrix * QR, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x)

This function solves the triangular system

System Message: WARNING/2 (R P^T x = b)

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for the

System Message: WARNING/2 (N)

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-by-

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matrix

System Message: WARNING/2 (R)

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contained in QR.

int gsl_linalg_QRPT_Rsvx(const gsl_matrix * QR, const gsl_permutation * p, gsl_vector * x)

This function solves the triangular system

System Message: WARNING/2 (R P^T x = b)

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in-place for the

System Message: WARNING/2 (N)

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-by-

System Message: WARNING/2 (N)

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matrix

System Message: WARNING/2 (R)

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contained in QR. On input x should contain the right-hand side

System Message: WARNING/2 (b)

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, which is replaced by the solution on output.

size_t gsl_linalg_QRPT_rank(const gsl_matrix * QR, const double tol)

This function estimates the rank of the triangular matrix

System Message: WARNING/2 (R)

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contained in QR. The algorithm simply counts the number of diagonal elements of

System Message: WARNING/2 (R)

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whose absolute value is greater than the specified tolerance tol. If the input tol is negative, a default value of

System Message: WARNING/2 (20 (M + N) eps(max(|diag(R)|)))

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is used.

int gsl_linalg_QRPT_rcond(const gsl_matrix * QR, double * rcond, gsl_vector * work)

This function estimates the reciprocal condition number (using the 1-norm) of the

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factor, stored in the upper triangle of QR. The reciprocal condition number estimate, defined as

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, is stored in rcond. Additional workspace of size

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is required in work.

Complete Orthogonal Decomposition

The complete orthogonal decomposition of a

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-by-

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matrix

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is a generalization of the QR decomposition with column pivoting, given by

System Message: WARNING/2 (A P = Q \left( \begin{matrix} R_{11} & 0 \\ 0 & 0 \end{matrix} \right) Z^T)

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where

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is a

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-by-

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permutation matrix,

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is

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-by-

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orthogonal,

System Message: WARNING/2 (R_{11})

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is

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-by-

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upper triangular, with

System Message: WARNING/2 (r = {\rm rank}(A))

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, and

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is

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-by-

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orthogonal. If

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has full rank, then

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,

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and this reduces to the QR decomposition with column pivoting.

For a rank deficient least squares problem,

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, the solution vector

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is not unique. However if we further require that

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is minimized, then the complete orthogonal decomposition gives us the ability to compute the unique minimum norm solution, which is given by

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and the vector

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is the first

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elements of

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.

The COD also enables a straightforward solution of regularized least squares problems in Tikhonov standard form, written as

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where

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is a regularization parameter which represents a tradeoff between minimizing the residual norm

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and the solution norm

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. For this system, the solution is given by

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where

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is a vector of length

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which is found by solving

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and

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is defined above. The equation above can be solved efficiently for different values of

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using QR factorizations of the left hand side matrix.

int gsl_linalg_COD_decomp(gsl_matrix * A, gsl_vector * tau_Q, gsl_vector * tau_Z, gsl_permutation * p, size_t * rank, gsl_vector * work)

int gsl_linalg_COD_decomp_e(gsl_matrix * A, gsl_vector * tau_Q, gsl_vector * tau_Z, gsl_permutation * p, double tol, size_t * rank, gsl_vector * work)

These functions factor the

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-by-

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matrix A into the decomposition

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. The rank of A is computed as the number of diagonal elements of

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greater than the tolerance tol and output in rank. If tol is not specified, a default value is used (see gsl_linalg_QRPT_rank()). On output, the permutation matrix

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is stored in p. The matrix

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is stored in the upper rank-by-rank block of A. The matrices

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and

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are encoded in packed storage in A on output. The vectors tau_Q and tau_Z contain the Householder scalars corresponding to the matrices

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and

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respectively and must be of length

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. The vector work is additional workspace of length

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.

int gsl_linalg_COD_lssolve(const gsl_matrix * QRZT, const gsl_vector * tau_Q, const gsl_vector * tau_Z, const gsl_permutation * p, const size_t rank, const gsl_vector * b, gsl_vector * x, gsl_vector * residual)

This function finds the unique minimum norm least squares solution to the overdetermined system

System Message: WARNING/2 (A x = b)

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where the matrix A has more rows than columns. The least squares solution minimizes the Euclidean norm of the residual,

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as well as the norm of the solution

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. The routine requires as input the

System Message: WARNING/2 (QRZT)

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decomposition of

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into (QRZT, tau_Q, tau_Z, p, rank) given by gsl_linalg_COD_decomp(). The solution is returned in x. The residual,

System Message: WARNING/2 (b - Ax)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

, is computed as a by-product and stored in residual.

int gsl_linalg_COD_lssolve2(const double lambda, const gsl_matrix * QRZT, const gsl_vector * tau_Q, const gsl_vector * tau_Z, const gsl_permutation * p, const size_t rank, const gsl_vector * b, gsl_vector * x, gsl_vector * residual, gsl_matrix * S, gsl_vector * work)

This function finds the solution to the regularized least squares problem in Tikhonov standard form,

System Message: WARNING/2 (\min_x ||b - Ax||^2 + \lambda^2 ||x||^2)

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. The routine requires as input the

System Message: WARNING/2 (QRZT)

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decomposition of

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into (QRZT, tau_Q, tau_Z, p, rank) given by gsl_linalg_COD_decomp(). The parameter

System Message: WARNING/2 (\lambda)

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is supplied in lambda. The solution is returned in x. The residual,

System Message: WARNING/2 (b - Ax)

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, is stored in residual on output. S is additional workspace of size rank-by-rank. work is additional workspace of length rank.

int gsl_linalg_COD_unpack(const gsl_matrix * QRZT, const gsl_vector * tau_Q, const gsl_vector * tau_Z, const size_t rank, gsl_matrix * Q, gsl_matrix * R, gsl_matrix * Z)

This function unpacks the encoded

System Message: WARNING/2 (QRZT)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

decomposition (QRZT, tau_Q, tau_Z, rank) into the matrices Q, R, and Z, where Q is

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-by-

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, R is

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-by-

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, and Z is

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-by-

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latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

.

int gsl_linalg_COD_matZ(const gsl_matrix * QRZT, const gsl_vector * tau_Z, const size_t rank, gsl_matrix * A, gsl_vector * work)

This function multiplies the input matrix A on the right by Z,

System Message: WARNING/2 (A' = A Z)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

using the encoded

System Message: WARNING/2 (QRZT)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

decomposition (QRZT, tau_Z, rank). A must have

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columns but may have any number of rows. Additional workspace of length

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is provided in work.

Singular Value Decomposition

A general rectangular

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-by-

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matrix

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has a singular value decomposition (SVD) into the product of an

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-by-

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orthogonal matrix

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, an

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-by-

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diagonal matrix of singular values

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and the transpose of an

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-by-

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orthogonal square matrix

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,

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The singular values

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are all non-negative and are generally chosen to form a non-increasing sequence

System Message: WARNING/2 (\sigma_1 \ge \sigma_2 \ge ... \ge \sigma_N \ge 0 )

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The singular value decomposition of a matrix has many practical uses. The condition number of the matrix is given by the ratio of the largest singular value to the smallest singular value. The presence of a zero singular value indicates that the matrix is singular. The number of non-zero singular values indicates the rank of the matrix. In practice singular value decomposition of a rank-deficient matrix will not produce exact zeroes for singular values, due to finite numerical precision. Small singular values should be edited by choosing a suitable tolerance.

For a rank-deficient matrix, the null space of

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is given by the columns of

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corresponding to the zero singular values. Similarly, the range of

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is given by columns of

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corresponding to the non-zero singular values.

Note that the routines here compute the “thin” version of the SVD with

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as

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-by-

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orthogonal matrix. This allows in-place computation and is the most commonly-used form in practice. Mathematically, the “full” SVD is defined with

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as an

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-by-

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orthogonal matrix and

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as an

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-by-

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diagonal matrix (with additional rows of zeros).

int gsl_linalg_SV_decomp(gsl_matrix * A, gsl_matrix * V, gsl_vector * S, gsl_vector * work)

This function factorizes the

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-by-

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matrix A into the singular value decomposition

System Message: WARNING/2 (A = U S V^T)

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for

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. On output the matrix A is replaced by

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. The diagonal elements of the singular value matrix

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are stored in the vector S. The singular values are non-negative and form a non-increasing sequence from

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to

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. The matrix V contains the elements of

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in untransposed form. To form the product

System Message: WARNING/2 (U S V^T)

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it is necessary to take the transpose of V. A workspace of length N is required in work.

This routine uses the Golub-Reinsch SVD algorithm.

int gsl_linalg_SV_decomp_mod(gsl_matrix * A, gsl_matrix * X, gsl_matrix * V, gsl_vector * S, gsl_vector * work)

This function computes the SVD using the modified Golub-Reinsch algorithm, which is faster for

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. It requires the vector work of length N and the

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-by-

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matrix X as additional working space.

int gsl_linalg_SV_decomp_jacobi(gsl_matrix * A, gsl_matrix * V, gsl_vector * S)

This function computes the SVD of the

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-by-

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matrix A using one-sided Jacobi orthogonalization for

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. The Jacobi method can compute singular values to higher relative accuracy than Golub-Reinsch algorithms (see references for details).

int gsl_linalg_SV_solve(const gsl_matrix * U, const gsl_matrix * V, const gsl_vector * S, const gsl_vector * b, gsl_vector * x)

This function solves the system

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using the singular value decomposition (U, S, V) of

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which must have been computed previously with gsl_linalg_SV_decomp().

Only non-zero singular values are used in computing the solution. The parts of the solution corresponding to singular values of zero are ignored. Other singular values can be edited out by setting them to zero before calling this function.

In the over-determined case where A has more rows than columns the system is solved in the least squares sense, returning the solution x which minimizes

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.

int gsl_linalg_SV_leverage(const gsl_matrix * U, gsl_vector * h)

This function computes the statistical leverage values

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of a matrix

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using its singular value decomposition (U, S, V) previously computed with gsl_linalg_SV_decomp().

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are the diagonal values of the matrix

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and depend only on the matrix U which is the input to this function.

Cholesky Decomposition

A symmetric, positive definite square matrix

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has a Cholesky decomposition into a product of a lower triangular matrix

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and its transpose

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,

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This is sometimes referred to as taking the square-root of a matrix. The Cholesky decomposition can only be carried out when all the eigenvalues of the matrix are positive. This decomposition can be used to convert the linear system

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into a pair of triangular systems (

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,

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), which can be solved by forward and back-substitution.

If the matrix

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is near singular, it is sometimes possible to reduce the condition number and recover a more accurate solution vector

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by scaling as

System Message: WARNING/2 (\left( S A S \right) \left( S^{-1} x \right) = S b )

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where

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is a diagonal matrix whose elements are given by

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. This scaling is also known as Jacobi preconditioning. There are routines below to solve both the scaled and unscaled systems.

int gsl_linalg_cholesky_decomp1(gsl_matrix * A)

int gsl_linalg_complex_cholesky_decomp(gsl_matrix_complex * A)

These functions factorize the symmetric, positive-definite square matrix A into the Cholesky decomposition

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(or

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for the complex case). On input, the values from the diagonal and lower-triangular part of the matrix A are used (the upper triangular part is ignored). On output the diagonal and lower triangular part of the input matrix A contain the matrix

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, while the upper triangular part is unmodified. If the matrix is not positive-definite then the decomposition will fail, returning the error code GSL_EDOM.

When testing whether a matrix is positive-definite, disable the error handler first to avoid triggering an error.

int gsl_linalg_cholesky_decomp(gsl_matrix * A)

This function is now deprecated and is provided only for backward compatibility.

int gsl_linalg_cholesky_solve(const gsl_matrix * cholesky, const gsl_vector * b, gsl_vector * x)

int gsl_linalg_complex_cholesky_solve(const gsl_matrix_complex * cholesky, const gsl_vector_complex * b, gsl_vector_complex * x)

These functions solve the system

System Message: WARNING/2 (A x = b)

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using the Cholesky decomposition of

System Message: WARNING/2 (A)

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held in the matrix cholesky which must have been previously computed by gsl_linalg_cholesky_decomp() or gsl_linalg_complex_cholesky_decomp().

int gsl_linalg_cholesky_svx(const gsl_matrix * cholesky, gsl_vector * x)

int gsl_linalg_complex_cholesky_svx(const gsl_matrix_complex * cholesky, gsl_vector_complex * x)

These functions solve the system

System Message: WARNING/2 (A x = b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

in-place using the Cholesky decomposition of

System Message: WARNING/2 (A)

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held in the matrix cholesky which must have been previously computed by gsl_linalg_cholesky_decomp() or gsl_linalg_complex_cholesky_decomp(). On input x should contain the right-hand side

System Message: WARNING/2 (b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

, which is replaced by the solution on output.

int gsl_linalg_cholesky_invert(gsl_matrix * cholesky)

int gsl_linalg_complex_cholesky_invert(gsl_matrix_complex * cholesky)

These functions compute the inverse of a matrix from its Cholesky decomposition cholesky, which must have been previously computed by gsl_linalg_cholesky_decomp() or gsl_linalg_complex_cholesky_decomp(). On output, the inverse is stored in-place in cholesky.

int gsl_linalg_cholesky_decomp2(gsl_matrix * A, gsl_vector * S)

This function calculates a diagonal scaling transformation

System Message: WARNING/2 (S)

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for the symmetric, positive-definite square matrix A, and then computes the Cholesky decomposition

System Message: WARNING/2 (S A S = L L^T)

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. On input, the values from the diagonal and lower-triangular part of the matrix A are used (the upper triangular part is ignored). On output the diagonal and lower triangular part of the input matrix A contain the matrix

System Message: WARNING/2 (L)

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, while the upper triangular part of the input matrix is overwritten with

System Message: WARNING/2 (L^T)

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(the diagonal terms being identical for both

System Message: WARNING/2 (L)

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and

System Message: WARNING/2 (L^T)

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). If the matrix is not positive-definite then the decomposition will fail, returning the error code GSL_EDOM. The diagonal scale factors are stored in S on output.

When testing whether a matrix is positive-definite, disable the error handler first to avoid triggering an error.

int gsl_linalg_cholesky_solve2(const gsl_matrix * cholesky, const gsl_vector * S, const gsl_vector * b, gsl_vector * x)

This function solves the system

System Message: WARNING/2 ((S A S) (S^{-1} x) = S b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

using the Cholesky decomposition of

System Message: WARNING/2 (S A S)

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held in the matrix cholesky which must have been previously computed by gsl_linalg_cholesky_decomp2().

int gsl_linalg_cholesky_svx2(const gsl_matrix * cholesky, const gsl_vector * S, gsl_vector * x)

This function solves the system

System Message: WARNING/2 ((S A S) (S^{-1} x) = S b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

in-place using the Cholesky decomposition of

System Message: WARNING/2 (S A S)

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held in the matrix cholesky which must have been previously computed by gsl_linalg_cholesky_decomp2(). On input x should contain the right-hand side

System Message: WARNING/2 (b)

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, which is replaced by the solution on output.

int gsl_linalg_cholesky_scale(const gsl_matrix * A, gsl_vector * S)

This function calculates a diagonal scaling transformation of the symmetric, positive definite matrix A, such that

System Message: WARNING/2 (S A S)

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has a condition number within a factor of

System Message: WARNING/2 (N)

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of the matrix of smallest possible condition number over all possible diagonal scalings. On output, S contains the scale factors, given by

System Message: WARNING/2 (S_i = 1/\sqrt{A_{ii}})

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. For any

System Message: WARNING/2 (A_{ii} \le 0)

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, the corresponding scale factor

System Message: WARNING/2 (S_i)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

is set to

System Message: WARNING/2 (1)

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.

int gsl_linalg_cholesky_scale_apply(gsl_matrix * A, const gsl_vector * S)

This function applies the scaling transformation S to the matrix A. On output, A is replaced by

System Message: WARNING/2 (S A S)

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.

int gsl_linalg_cholesky_rcond(const gsl_matrix * cholesky, double * rcond, gsl_vector * work)

This function estimates the reciprocal condition number (using the 1-norm) of the symmetric positive definite matrix

System Message: WARNING/2 (A)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

, using its Cholesky decomposition provided in cholesky. The reciprocal condition number estimate, defined as

System Message: WARNING/2 (1 / (||A||_1 \cdot ||A^{-1}||_1))

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, is stored in rcond. Additional workspace of size

System Message: WARNING/2 (3 N)

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is required in work.

Pivoted Cholesky Decomposition

A symmetric, positive definite square matrix

System Message: WARNING/2 (A)

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has an alternate Cholesky decomposition into a product of a lower unit triangular matrix

System Message: WARNING/2 (L)

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, a diagonal matrix

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and

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, given by

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. This is equivalent to the Cholesky formulation discussed above, with the standard Cholesky lower triangular factor given by

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. For ill-conditioned matrices, it can help to use a pivoting strategy to prevent the entries of

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and

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from growing too large, and also ensure

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, where

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are the diagonal entries of

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. The final decomposition is given by

System Message: WARNING/2 (P A P^T = L D L^T )

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where

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is a permutation matrix.

int gsl_linalg_pcholesky_decomp(gsl_matrix * A, gsl_permutation * p)

This function factors the symmetric, positive-definite square matrix A into the Pivoted Cholesky decomposition

System Message: WARNING/2 (P A P^T = L D L^T)

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. On input, the values from the diagonal and lower-triangular part of the matrix A are used to construct the factorization. On output the diagonal of the input matrix A stores the diagonal elements of

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, and the lower triangular portion of A contains the matrix

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. Since

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has ones on its diagonal these do not need to be explicitely stored. The upper triangular portion of A is unmodified. The permutation matrix

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is stored in p on output.

int gsl_linalg_pcholesky_solve(const gsl_matrix * LDLT, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x)

This function solves the system

System Message: WARNING/2 (A x = b)

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using the Pivoted Cholesky decomposition of

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held in the matrix LDLT and permutation p which must have been previously computed by gsl_linalg_pcholesky_decomp().

int gsl_linalg_pcholesky_svx(const gsl_matrix * LDLT, const gsl_permutation * p, gsl_vector * x)

This function solves the system

System Message: WARNING/2 (A x = b)

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in-place using the Pivoted Cholesky decomposition of

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held in the matrix LDLT and permutation p which must have been previously computed by gsl_linalg_pcholesky_decomp(). On input, x contains the right hand side vector

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which is replaced by the solution vector on output.

int gsl_linalg_pcholesky_decomp2(gsl_matrix * A, gsl_permutation * p, gsl_vector * S)

This function computes the pivoted Cholesky factorization of the matrix

System Message: WARNING/2 (S A S)

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, where the input matrix A is symmetric and positive definite, and the diagonal scaling matrix S is computed to reduce the condition number of A as much as possible. See Cholesky Decomposition for more information on the matrix S. The Pivoted Cholesky decomposition satisfies

System Message: WARNING/2 (P S A S P^T = L D L^T)

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. On input, the values from the diagonal and lower-triangular part of the matrix A are used to construct the factorization. On output the diagonal of the input matrix A stores the diagonal elements of

System Message: WARNING/2 (D)

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, and the lower triangular portion of A contains the matrix

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. Since

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has ones on its diagonal these do not need to be explicitely stored. The upper triangular portion of A is unmodified. The permutation matrix

System Message: WARNING/2 (P)

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is stored in p on output. The diagonal scaling transformation is stored in S on output.

int gsl_linalg_pcholesky_solve2(const gsl_matrix * LDLT, const gsl_permutation * p, const gsl_vector * S, const gsl_vector * b, gsl_vector * x)

This function solves the system

System Message: WARNING/2 ((S A S) (S^{-1} x) = S b)

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using the Pivoted Cholesky decomposition of

System Message: WARNING/2 (S A S)

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held in the matrix LDLT, permutation p, and vector S, which must have been previously computed by gsl_linalg_pcholesky_decomp2().

int gsl_linalg_pcholesky_svx2(const gsl_matrix * LDLT, const gsl_permutation * p, const gsl_vector * S, gsl_vector * x)

This function solves the system

System Message: WARNING/2 ((S A S) (S^{-1} x) = S b)

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in-place using the Pivoted Cholesky decomposition of

System Message: WARNING/2 (S A S)

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held in the matrix LDLT, permutation p and vector S, which must have been previously computed by gsl_linalg_pcholesky_decomp2(). On input, x contains the right hand side vector

System Message: WARNING/2 (b)

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which is replaced by the solution vector on output.

int gsl_linalg_pcholesky_invert(const gsl_matrix * LDLT, const gsl_permutation * p, gsl_matrix * Ainv)

This function computes the inverse of the matrix

System Message: WARNING/2 (A)

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, using the Pivoted Cholesky decomposition stored in LDLT and p. On output, the matrix Ainv contains

System Message: WARNING/2 (A^{-1})

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.

int gsl_linalg_pcholesky_rcond(const gsl_matrix * LDLT, const gsl_permutation * p, double * rcond, gsl_vector * work)

This function estimates the reciprocal condition number (using the 1-norm) of the symmetric positive definite matrix

System Message: WARNING/2 (A)

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, using its pivoted Cholesky decomposition provided in LDLT. The reciprocal condition number estimate, defined as

System Message: WARNING/2 (1 / (||A||_1 \cdot ||A^{-1}||_1))

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, is stored in rcond. Additional workspace of size

System Message: WARNING/2 (3 N)

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is required in work.

Modified Cholesky Decomposition

The modified Cholesky decomposition is suitable for solving systems

System Message: WARNING/2 (A x = b)

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where

System Message: WARNING/2 (A)

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is a symmetric indefinite matrix. Such matrices arise in nonlinear optimization algorithms. The standard Cholesky decomposition requires a positive definite matrix and would fail in this case. Instead of resorting to a method like QR or SVD, which do not take into account the symmetry of the matrix, we can instead introduce a small perturbation to the matrix

System Message: WARNING/2 (A)

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to make it positive definite, and then use a Cholesky decomposition on the perturbed matrix. The resulting decomposition satisfies

System Message: WARNING/2 (P (A + E) P^T = L D L^T )

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where

System Message: WARNING/2 (P)

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is a permutation matrix,

System Message: WARNING/2 (E)

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is a diagonal perturbation matrix,

System Message: WARNING/2 (L)

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is unit lower triangular, and

System Message: WARNING/2 (D)

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is diagonal. If

System Message: WARNING/2 (A)

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is sufficiently positive definite, then the perturbation matrix

System Message: WARNING/2 (E)

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will be zero and this method is equivalent to the pivoted Cholesky algorithm. For indefinite matrices, the perturbation matrix

System Message: WARNING/2 (E)

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is computed to ensure that

System Message: WARNING/2 (A + E)

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is positive definite and well conditioned.

int gsl_linalg_mcholesky_decomp(gsl_matrix * A, gsl_permutation * p, gsl_vector * E)

This function factors the symmetric, indefinite square matrix A into the Modified Cholesky decomposition

System Message: WARNING/2 (P (A + E) P^T = L D L^T)

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. On input, the values from the diagonal and lower-triangular part of the matrix A are used to construct the factorization. On output the diagonal of the input matrix A stores the diagonal elements of

System Message: WARNING/2 (D)

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, and the lower triangular portion of A contains the matrix

System Message: WARNING/2 (L)

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. Since

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has ones on its diagonal these do not need to be explicitely stored. The upper triangular portion of A is unmodified. The permutation matrix

System Message: WARNING/2 (P)

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is stored in p on output. The diagonal perturbation matrix is stored in E on output. The parameter E may be set to NULL if it is not required.

int gsl_linalg_mcholesky_solve(const gsl_matrix * LDLT, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x)

This function solves the perturbed system

System Message: WARNING/2 ((A + E) x = b)

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using the Cholesky decomposition of

System Message: WARNING/2 (A + E)

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held in the matrix LDLT and permutation p which must have been previously computed by gsl_linalg_mcholesky_decomp().

int gsl_linalg_mcholesky_svx(const gsl_matrix * LDLT, const gsl_permutation * p, gsl_vector * x)

This function solves the perturbed system

System Message: WARNING/2 ((A + E) x = b)

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in-place using the Cholesky decomposition of

System Message: WARNING/2 (A + E)

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held in the matrix LDLT and permutation p which must have been previously computed by gsl_linalg_mcholesky_decomp(). On input, x contains the right hand side vector

System Message: WARNING/2 (b)

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which is replaced by the solution vector on output.

int gsl_linalg_mcholesky_rcond(const gsl_matrix * LDLT, const gsl_permutation * p, double * rcond, gsl_vector * work)

This function estimates the reciprocal condition number (using the 1-norm) of the perturbed matrix

System Message: WARNING/2 (A + E)

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, using its pivoted Cholesky decomposition provided in LDLT. The reciprocal condition number estimate, defined as

System Message: WARNING/2 (1 / (||A + E||_1 \cdot ||(A + E)^{-1}||_1))

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, is stored in rcond. Additional workspace of size

System Message: WARNING/2 (3 N)

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is required in work.

Tridiagonal Decomposition of Real Symmetric Matrices

A symmetric matrix

System Message: WARNING/2 (A)

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can be factorized by similarity transformations into the form,

System Message: WARNING/2 (A = Q T Q^T )

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where

System Message: WARNING/2 (Q)

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is an orthogonal matrix and

System Message: WARNING/2 (T)

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is a symmetric tridiagonal matrix.

int gsl_linalg_symmtd_decomp(gsl_matrix * A, gsl_vector * tau)

This function factorizes the symmetric square matrix A into the symmetric tridiagonal decomposition

System Message: WARNING/2 (Q T Q^T)

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. On output the diagonal and subdiagonal part of the input matrix A contain the tridiagonal matrix

System Message: WARNING/2 (T)

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. The remaining lower triangular part of the input matrix contains the Householder vectors which, together with the Householder coefficients tau, encode the orthogonal matrix

System Message: WARNING/2 (Q)

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. This storage scheme is the same as used by LAPACK. The upper triangular part of A is not referenced.

int gsl_linalg_symmtd_unpack(const gsl_matrix * A, const gsl_vector * tau, gsl_matrix * Q, gsl_vector * diag, gsl_vector * subdiag)

This function unpacks the encoded symmetric tridiagonal decomposition (A, tau) obtained from gsl_linalg_symmtd_decomp() into the orthogonal matrix Q, the vector of diagonal elements diag and the vector of subdiagonal elements subdiag.

int gsl_linalg_symmtd_unpack_T(const gsl_matrix * A, gsl_vector * diag, gsl_vector * subdiag)

This function unpacks the diagonal and subdiagonal of the encoded symmetric tridiagonal decomposition (A, tau) obtained from gsl_linalg_symmtd_decomp() into the vectors diag and subdiag.

Tridiagonal Decomposition of Hermitian Matrices

A hermitian matrix

System Message: WARNING/2 (A)

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can be factorized by similarity transformations into the form,

System Message: WARNING/2 (A = U T U^T )

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where

System Message: WARNING/2 (U)

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is a unitary matrix and

System Message: WARNING/2 (T)

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is a real symmetric tridiagonal matrix.

int gsl_linalg_hermtd_decomp(gsl_matrix_complex * A, gsl_vector_complex * tau)

This function factorizes the hermitian matrix A into the symmetric tridiagonal decomposition

System Message: WARNING/2 (U T U^T)

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. On output the real parts of the diagonal and subdiagonal part of the input matrix A contain the tridiagonal matrix

System Message: WARNING/2 (T)

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. The remaining lower triangular part of the input matrix contains the Householder vectors which, together with the Householder coefficients tau, encode the unitary matrix

System Message: WARNING/2 (U)

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. This storage scheme is the same as used by LAPACK. The upper triangular part of A and imaginary parts of the diagonal are not referenced.

int gsl_linalg_hermtd_unpack(const gsl_matrix_complex * A, const gsl_vector_complex * tau, gsl_matrix_complex * U, gsl_vector * diag, gsl_vector * subdiag)

This function unpacks the encoded tridiagonal decomposition (A, tau) obtained from gsl_linalg_hermtd_decomp() into the unitary matrix U, the real vector of diagonal elements diag and the real vector of subdiagonal elements subdiag.

int gsl_linalg_hermtd_unpack_T(const gsl_matrix_complex * A, gsl_vector * diag, gsl_vector * subdiag)

This function unpacks the diagonal and subdiagonal of the encoded tridiagonal decomposition (A, tau) obtained from the gsl_linalg_hermtd_decomp() into the real vectors diag and subdiag.

Hessenberg Decomposition of Real Matrices

A general real matrix

System Message: WARNING/2 (A)

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can be decomposed by orthogonal similarity transformations into the form

System Message: WARNING/2 (A = U H U^T )

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where

System Message: WARNING/2 (U)

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is orthogonal and

System Message: WARNING/2 (H)

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is an upper Hessenberg matrix, meaning that it has zeros below the first subdiagonal. The Hessenberg reduction is the first step in the Schur decomposition for the nonsymmetric eigenvalue problem, but has applications in other areas as well.

int gsl_linalg_hessenberg_decomp(gsl_matrix * A, gsl_vector * tau)

This function computes the Hessenberg decomposition of the matrix A by applying the similarity transformation

System Message: WARNING/2 (H = U^T A U)

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. On output,

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is stored in the upper portion of A. The information required to construct the matrix

System Message: WARNING/2 (U)

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is stored in the lower triangular portion of A.

System Message: WARNING/2 (U)

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is a product of

System Message: WARNING/2 (N - 2)

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Householder matrices. The Householder vectors are stored in the lower portion of A (below the subdiagonal) and the Householder coefficients are stored in the vector tau. tau must be of length N.

int gsl_linalg_hessenberg_unpack(gsl_matrix * H, gsl_vector * tau, gsl_matrix * U)

This function constructs the orthogonal matrix

System Message: WARNING/2 (U)

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from the information stored in the Hessenberg matrix H along with the vector tau. H and tau are outputs from gsl_linalg_hessenberg_decomp().

int gsl_linalg_hessenberg_unpack_accum(gsl_matrix * H, gsl_vector * tau, gsl_matrix * V)

This function is similar to gsl_linalg_hessenberg_unpack(), except it accumulates the matrix U into V, so that

System Message: WARNING/2 (V' = VU)

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. The matrix V must be initialized prior to calling this function. Setting V to the identity matrix provides the same result as gsl_linalg_hessenberg_unpack(). If H is order N, then V must have N columns but may have any number of rows.

int gsl_linalg_hessenberg_set_zero(gsl_matrix * H)

This function sets the lower triangular portion of H, below the subdiagonal, to zero. It is useful for clearing out the Householder vectors after calling gsl_linalg_hessenberg_decomp().

Hessenberg-Triangular Decomposition of Real Matrices

A general real matrix pair (

System Message: WARNING/2 (A)

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,

System Message: WARNING/2 (B)

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) can be decomposed by orthogonal similarity transformations into the form

System Message: WARNING/2 (A &= U H V^T \\ B &= U R V^T)

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where

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and

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are orthogonal,

System Message: WARNING/2 (H)

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is an upper Hessenberg matrix, and

System Message: WARNING/2 (R)

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is upper triangular. The Hessenberg-Triangular reduction is the first step in the generalized Schur decomposition for the generalized eigenvalue problem.

int gsl_linalg_hesstri_decomp(gsl_matrix * A, gsl_matrix * B, gsl_matrix * U, gsl_matrix * V, gsl_vector * work)

This function computes the Hessenberg-Triangular decomposition of the matrix pair (A, B). On output,

System Message: WARNING/2 (H)

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is stored in A, and

System Message: WARNING/2 (R)

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is stored in B. If U and V are provided (they may be null), the similarity transformations are stored in them. Additional workspace of length

System Message: WARNING/2 (N)

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is needed in work.

Bidiagonalization

A general matrix

System Message: WARNING/2 (A)

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can be factorized by similarity transformations into the form,

System Message: WARNING/2 (A = U B V^T )

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where

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and

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are orthogonal matrices and

System Message: WARNING/2 (B)

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is a

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-by-

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bidiagonal matrix with non-zero entries only on the diagonal and superdiagonal. The size of U is

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-by-

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and the size of V is

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-by-

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.

int gsl_linalg_bidiag_decomp(gsl_matrix * A, gsl_vector * tau_U, gsl_vector * tau_V)

This function factorizes the

System Message: WARNING/2 (M)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

-by-

System Message: WARNING/2 (N)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

matrix A into bidiagonal form

System Message: WARNING/2 (U B V^T)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

. The diagonal and superdiagonal of the matrix

System Message: WARNING/2 (B)

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are stored in the diagonal and superdiagonal of A. The orthogonal matrices

System Message: WARNING/2 (U)

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and V are stored as compressed Householder vectors in the remaining elements of A. The Householder coefficients are stored in the vectors tau_U and tau_V. The length of tau_U must equal the number of elements in the diagonal of A and the length of tau_V should be one element shorter.

int gsl_linalg_bidiag_unpack(const gsl_matrix * A, const gsl_vector * tau_U, gsl_matrix * U, const gsl_vector * tau_V, gsl_matrix * V, gsl_vector * diag, gsl_vector * superdiag)

This function unpacks the bidiagonal decomposition of A produced by gsl_linalg_bidiag_decomp(), (A, tau_U, tau_V) into the separate orthogonal matrices U, V and the diagonal vector diag and superdiagonal superdiag. Note that U is stored as a compact

System Message: WARNING/2 (M)

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-by-

System Message: WARNING/2 (N)

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orthogonal matrix satisfying

System Message: WARNING/2 (U^T U = I)

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for efficiency.

int gsl_linalg_bidiag_unpack2(gsl_matrix * A, gsl_vector * tau_U, gsl_vector * tau_V, gsl_matrix * V)

This function unpacks the bidiagonal decomposition of A produced by gsl_linalg_bidiag_decomp(), (A, tau_U, tau_V) into the separate orthogonal matrices U, V and the diagonal vector diag and superdiagonal superdiag. The matrix U is stored in-place in A.

int gsl_linalg_bidiag_unpack_B(const gsl_matrix * A, gsl_vector * diag, gsl_vector * superdiag)

This function unpacks the diagonal and superdiagonal of the bidiagonal decomposition of A from gsl_linalg_bidiag_decomp(), into the diagonal vector diag and superdiagonal vector superdiag.

Givens Rotations

A Givens rotation is a rotation in the plane acting on two elements of a given vector. It can be represented in matrix form as

System Message: WARNING/2 (G(i,j,\theta) = \left( \begin{matrix} 1 & \ldots & 0 & \ldots & 0 & \ldots & 0 \\ \vdots & \ddots & \vdots & & \vdots & & \vdots \\ 0 & \ldots & \cos{\theta} & \ldots & -\sin{\theta} & \ldots & 0 \\ \vdots & & \vdots & \ddots & \vdots & & \vdots \\ 0 & \ldots & \sin{\theta} & \ldots & \cos{\theta} & \ldots & 0 \\ \vdots & & \vdots & & \vdots & \ddots & \vdots \\ 0 & \ldots & 0 & \ldots & 0 & \ldots & 1 \end{matrix} \right))

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

where the

System Message: WARNING/2 (\cos{\theta})

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and

System Message: WARNING/2 (\sin{\theta})

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appear at the intersection of the

System Message: WARNING/2 (i)

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-th and

System Message: WARNING/2 (j)

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-th rows and columns. When acting on a vector

System Message: WARNING/2 (x)

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,

System Message: WARNING/2 (G(i,j,\theta) x)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

performs a rotation of the

System Message: WARNING/2 ((i,j))

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elements of

System Message: WARNING/2 (x)

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. Givens rotations are typically used to introduce zeros in vectors, such as during the QR decomposition of a matrix. In this case, it is typically desired to find

System Message: WARNING/2 (c)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

and

System Message: WARNING/2 (s)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

such that

System Message: WARNING/2 (\left( \begin{matrix} c & -s \\ s & c \end{matrix} \right) \left( \begin{matrix} a \\ b \end{matrix} \right) = \left( \begin{matrix} r \\ 0 \end{matrix} \right))

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

with

System Message: WARNING/2 (r = \sqrt{a^2 + b^2})

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.

void gsl_linalg_givens(const double a, const double b, double * c, double * s)

This function computes

System Message: WARNING/2 (c = \cos{\theta})

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and

System Message: WARNING/2 (s = \sin{\theta})

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so that the Givens matrix

System Message: WARNING/2 (G(\theta))

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

acting on the vector

System Message: WARNING/2 ((a,b))

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

produces

System Message: WARNING/2 ((r, 0))

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, with

System Message: WARNING/2 (r = \sqrt{a^2 + b^2})

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.

void gsl_linalg_givens_gv(gsl_vector * v, const size_t i, const size_t j, const double c, const double s)

This function applies the Givens rotation defined by

System Message: WARNING/2 (c = \cos{\theta})

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and

System Message: WARNING/2 (s = \sin{\theta})

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to the i and j elements of v. On output,

System Message: WARNING/2 ((v(i),v(j)) \leftarrow G(\theta) (v(i),v(j)))

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.

Householder Transformations

A Householder transformation is a rank-1 modification of the identity matrix which can be used to zero out selected elements of a vector. A Householder matrix

System Message: WARNING/2 (P)

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takes the form,

System Message: WARNING/2 (P = I - \tau v v^T )

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where

System Message: WARNING/2 (v)

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is a vector (called the Householder vector) and

System Message: WARNING/2 (\tau = 2/(v^T v))

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. The functions described in this section use the rank-1 structure of the Householder matrix to create and apply Householder transformations efficiently.

double gsl_linalg_householder_transform(gsl_vector * w)

gsl_complex gsl_linalg_complex_householder_transform(gsl_vector_complex * w)

This function prepares a Householder transformation

System Message: WARNING/2 (P = I - \tau v v^T)

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which can be used to zero all the elements of the input vector w except the first. On output the Householder vector v is stored in w and the scalar

System Message: WARNING/2 (\tau)

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is returned. The householder vector v is normalized so that v[0] = 1, however this 1 is not stored in the output vector. Instead, w[0] is set to the first element of the transformed vector, so that if

System Message: WARNING/2 (u = P w)

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, w[0] = u[0] on output and the remainder of

System Message: WARNING/2 (u)

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is zero.

int gsl_linalg_householder_hm(double tau, const gsl_vector * v, gsl_matrix * A)

int gsl_linalg_complex_householder_hm(gsl_complex tau, const gsl_vector_complex * v, gsl_matrix_complex * A)

This function applies the Householder matrix

System Message: WARNING/2 (P)

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defined by the scalar tau and the vector v to the left-hand side of the matrix A. On output the result

System Message: WARNING/2 (P A)

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is stored in A.

int gsl_linalg_householder_mh(double tau, const gsl_vector * v, gsl_matrix * A)

int gsl_linalg_complex_householder_mh(gsl_complex tau, const gsl_vector_complex * v, gsl_matrix_complex * A)

This function applies the Householder matrix

System Message: WARNING/2 (P)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

defined by the scalar tau and the vector v to the right-hand side of the matrix A. On output the result

System Message: WARNING/2 (A P)

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is stored in A.

int gsl_linalg_householder_hv(double tau, const gsl_vector * v, gsl_vector * w)

int gsl_linalg_complex_householder_hv(gsl_complex tau, const gsl_vector_complex * v, gsl_vector_complex * w)

This function applies the Householder transformation

System Message: WARNING/2 (P)

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defined by the scalar tau and the vector v to the vector w. On output the result

System Message: WARNING/2 (P w)

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is stored in w.

Householder solver for linear systems

int gsl_linalg_HH_solve(gsl_matrix * A, const gsl_vector * b, gsl_vector * x)

This function solves the system

System Message: WARNING/2 (A x = b)

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directly using Householder transformations. On output the solution is stored in x and b is not modified. The matrix A is destroyed by the Householder transformations.

int gsl_linalg_HH_svx(gsl_matrix * A, gsl_vector * x)

This function solves the system

System Message: WARNING/2 (A x = b)

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in-place using Householder transformations. On input x should contain the right-hand side

System Message: WARNING/2 (b)

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, which is replaced by the solution on output. The matrix A is destroyed by the Householder transformations.

Tridiagonal Systems

The functions described in this section efficiently solve symmetric, non-symmetric and cyclic tridiagonal systems with minimal storage. Note that the current implementations of these functions use a variant of Cholesky decomposition, so the tridiagonal matrix must be positive definite. For non-positive definite matrices, the functions return the error code GSL_ESING.

int gsl_linalg_solve_tridiag(const gsl_vector * diag, const gsl_vector * e, const gsl_vector * f, const gsl_vector * b, gsl_vector * x)

This function solves the general

System Message: WARNING/2 (N)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

-by-

System Message: WARNING/2 (N)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

system

System Message: WARNING/2 (A x = b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

where A is tridiagonal (

System Message: WARNING/2 (N \geq 2)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

). The super-diagonal and sub-diagonal vectors e and f must be one element shorter than the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

System Message: WARNING/2 (A = \left( \begin{matrix} d_0&e_0& 0& 0\\ f_0&d_1&e_1& 0\\ 0 &f_1&d_2&e_2\\ 0 &0 &f_2&d_3 \end{matrix} \right))

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

int gsl_linalg_solve_symm_tridiag(const gsl_vector * diag, const gsl_vector * e, const gsl_vector * b, gsl_vector * x)

This function solves the general

System Message: WARNING/2 (N)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

-by-

System Message: WARNING/2 (N)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

system

System Message: WARNING/2 (A x = b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

where A is symmetric tridiagonal (

System Message: WARNING/2 (N \geq 2)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

). The off-diagonal vector e must be one element shorter than the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

System Message: WARNING/2 (A = \left( \begin{matrix} d_0&e_0& 0& 0\\ e_0&d_1&e_1& 0\\ 0 &e_1&d_2&e_2\\ 0 &0 &e_2&d_3 \end{matrix} \right))

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

int gsl_linalg_solve_cyc_tridiag(const gsl_vector * diag, const gsl_vector * e, const gsl_vector * f, const gsl_vector * b, gsl_vector * x)

This function solves the general

System Message: WARNING/2 (N)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

-by-

System Message: WARNING/2 (N)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

system

System Message: WARNING/2 (A x = b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

where A is cyclic tridiagonal (

System Message: WARNING/2 (N \geq 3)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

). The cyclic super-diagonal and sub-diagonal vectors e and f must have the same number of elements as the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

System Message: WARNING/2 (A = \left( \begin{matrix} d_0&e_0& 0 &f_3\\ f_0&d_1&e_1& 0 \\ 0 &f_1&d_2&e_2\\ e_3& 0 &f_2&d_3 \end{matrix} \right))

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

int gsl_linalg_solve_symm_cyc_tridiag(const gsl_vector * diag, const gsl_vector * e, const gsl_vector * b, gsl_vector * x)

This function solves the general

System Message: WARNING/2 (N)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

-by-

System Message: WARNING/2 (N)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

system

System Message: WARNING/2 (A x = b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

where A is symmetric cyclic tridiagonal (

System Message: WARNING/2 (N \geq 3)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

). The cyclic off-diagonal vector e must have the same number of elements as the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

System Message: WARNING/2 (A = \left( \begin{matrix} d_0&e_0& 0 &e_3\\ e_0&d_1&e_1& 0 \\ 0 &e_1&d_2&e_2\\ e_3& 0 &e_2&d_3 \end{matrix} \right))

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

Triangular Systems

int gsl_linalg_tri_upper_invert(gsl_matrix * T)

int gsl_linalg_tri_lower_invert(gsl_matrix * T)

int gsl_linalg_tri_upper_unit_invert(gsl_matrix * T)

int gsl_linalg_tri_lower_unit_invert(gsl_matrix * T)

These functions calculate the in-place inverse of the triangular matrix T. When the upper prefix is specified, then the upper triangle of T is used, and when the lower prefix is specified, the lower triangle is used. If the unit prefix is specified, then the diagonal elements of the matrix T are taken as unity and are not referenced. Otherwise the diagonal elements are used in the inversion.

int gsl_linalg_tri_upper_rcond(const gsl_matrix * T, double * rcond, gsl_vector * work)

int gsl_linalg_tri_lower_rcond(const gsl_matrix * T, double * rcond, gsl_vector * work)

These functions estimate the reciprocal condition number, in the 1-norm, of the upper or lower

System Message: WARNING/2 (N)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

-by-

System Message: WARNING/2 (N)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

triangular matrix T. The reciprocal condition number is stored in rcond on output, and is defined by

System Message: WARNING/2 (1 / (||T||_1 \cdot ||T^{-1}||_1))

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

. Additional workspace of size

System Message: WARNING/2 (3 N)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

is required in work.

Balancing

The process of balancing a matrix applies similarity transformations to make the rows and columns have comparable norms. This is useful, for example, to reduce roundoff errors in the solution of eigenvalue problems. Balancing a matrix

System Message: WARNING/2 (A)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

consists of replacing

System Message: WARNING/2 (A)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

with a similar matrix

System Message: WARNING/2 (A' = D^{-1} A D )

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

where

System Message: WARNING/2 (D)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

is a diagonal matrix whose entries are powers of the floating point radix.

int gsl_linalg_balance_matrix(gsl_matrix * A, gsl_vector * D)

This function replaces the matrix A with its balanced counterpart and stores the diagonal elements of the similarity transformation into the vector D.

Examples

The following program solves the linear system

System Message: WARNING/2 (A x = b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

. The system to be solved is,

System Message: WARNING/2 (\left( \begin{matrix} 0.18& 0.60& 0.57& 0.96\\ 0.41& 0.24& 0.99& 0.58\\ 0.14& 0.30& 0.97& 0.66\\ 0.51& 0.13& 0.19& 0.85 \end{matrix} \right) \left( \begin{matrix} x_0\\ x_1\\ x_2\\ x_3 \end{matrix} \right) = \left( \begin{matrix} 1.0\\ 2.0\\ 3.0\\ 4.0 \end{matrix} \right))

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

and the solution is found using LU decomposition of the matrix

System Message: WARNING/2 (A)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

.

#include <stdio.h>
#include <gsl/gsl_linalg.h>

int
main (void)
{
  double a_data[] = { 0.18, 0.60, 0.57, 0.96,
                      0.41, 0.24, 0.99, 0.58,
                      0.14, 0.30, 0.97, 0.66,
                      0.51, 0.13, 0.19, 0.85 };

  double b_data[] = { 1.0, 2.0, 3.0, 4.0 };

  gsl_matrix_view m
    = gsl_matrix_view_array (a_data, 4, 4);

  gsl_vector_view b
    = gsl_vector_view_array (b_data, 4);

  gsl_vector *x = gsl_vector_alloc (4);

  int s;

  gsl_permutation * p = gsl_permutation_alloc (4);

  gsl_linalg_LU_decomp (&m.matrix, p, &s);

  gsl_linalg_LU_solve (&m.matrix, p, &b.vector, x);

  printf ("x = \n");
  gsl_vector_fprintf (stdout, x, "%g");

  gsl_permutation_free (p);
  gsl_vector_free (x);
  return 0;
}

Here is the output from the program,

x =
-4.05205
-12.6056
1.66091
8.69377

This can be verified by multiplying the solution

System Message: WARNING/2 (x)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

by the original matrix

System Message: WARNING/2 (A)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

using GNU octave,

octave> A = [ 0.18, 0.60, 0.57, 0.96;
              0.41, 0.24, 0.99, 0.58;
              0.14, 0.30, 0.97, 0.66;
              0.51, 0.13, 0.19, 0.85 ];

octave> x = [ -4.05205; -12.6056; 1.66091; 8.69377];

octave> A * x
ans =
  1.0000
  2.0000
  3.0000
  4.0000

This reproduces the original right-hand side vector,

System Message: WARNING/2 (b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

, in accordance with the equation

System Message: WARNING/2 (A x = b)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Mageia) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.10> and hyphenation patterns for 84 language(s) loaded. (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.158 \endinput ^^M No pages of output. Transcript written on math.log.

.

References and Further Reading

Further information on the algorithms described in this section can be found in the following book,

• G. H. Golub, C. F. Van Loan, “Matrix Computations” (3rd Ed, 1996), Johns Hopkins University Press, ISBN 0-8018-5414-8.

The LAPACK library is described in the following manual,

• LAPACK Users’ Guide (Third Edition, 1999), Published by SIAM, ISBN 0-89871-447-8

The LAPACK source code can be found at http://www.netlib.org/lapack, along with an online copy of the users guide.

The Modified Golub-Reinsch algorithm is described in the following paper,

• T.F. Chan, “An Improved Algorithm for Computing the Singular Value Decomposition”, ACM Transactions on Mathematical Software, 8 (1982), pp 72–83.

The Jacobi algorithm for singular value decomposition is described in the following papers,

• J.C. Nash, “A one-sided transformation method for the singular value decomposition and algebraic eigenproblem”, Computer Journal, Volume 18, Number 1 (1975), p 74–76
• J.C. Nash and S. Shlien “Simple algorithms for the partial singular value decomposition”, Computer Journal, Volume 30 (1987), p 268–275.
• J. Demmel, K. Veselic, “Jacobi’s Method is more accurate than QR”, Lapack Working Note 15 (LAWN-15), October 1989. Available from netlib, http://www.netlib.org/lapack/ in the lawns or lawnspdf directories.

The algorithm for estimating a matrix condition number is described in the following paper,

• N. J. Higham, “FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation”, ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.