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 square matrix A has an LU decomposition into upper and lower triangular matrices,

     P A = L U

where P is a permutation matrix, L is unit lower triangular matrix and U is upper triangular matrix. For square matrices this decomposition can be used to convert the linear system A x = b into a pair of triangular systems (L y = P b, U x = y), which can be solved by forward and back-substitution. Note that the LU decomposition is valid for singular matrices.

 -- Function: int gsl_linalg_LU_decomp (gsl_matrix * A, gsl_permutation * P, int * SIGNUM)
 -- Function: int gsl_linalg_complex_LU_decomp (gsl_matrix_complex * A, gsl_permutation * P, int * SIGNUM)
     These functions factorize the square matrix A into the LU decomposition PA = LU.  On output the diagonal and upper triangular part of the input matrix A contain the matrix U. The lower triangular part of the input matrix (excluding the diagonal) contains L.  The diagonal elements of L are unity, and are not stored.

     The permutation matrix P is encoded in the permutation P. The j-th column of the matrix P is given by the k-th column of the identity matrix, where k = p_j the j-th element of the permutation vector. The sign of the permutation is given by SIGNUM. It has the value (-1)^n, where n 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).

 -- Function: int gsl_linalg_LU_solve (const gsl_matrix * LU, const gsl_permutation * P, const gsl_vector * B, gsl_vector * X)
 -- Function: 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 A x = b using the LU decomposition of A into (LU, P) given by gsl_linalg_LU_decomp or gsl_linalg_complex_LU_decomp as input.

 -- Function: int gsl_linalg_LU_svx (const gsl_matrix * LU, const gsl_permutation * P, gsl_vector * X)
 -- Function: 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 A x = b in-place using the precomputed LU decomposition of A into (LU,P). On input X should contain the right-hand side b, which is replaced by the solution on output.

 -- Function: 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 * RESIDUAL)
 -- Function: 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 * RESIDUAL)
     These functions apply an iterative improvement to X, the solution of A x = b, from the precomputed LU decomposition of A into (LU,P). The initial residual r = A x - b is also computed and stored in RESIDUAL.

 -- Function: int gsl_linalg_LU_invert (const gsl_matrix * LU, const gsl_permutation * P, gsl_matrix * INVERSE)
 -- Function: 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 A from its LU decomposition (LU,P), storing the result in the matrix INVERSE. The inverse is computed by solving the system A x = b 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).

 -- Function: double gsl_linalg_LU_det (gsl_matrix * LU, int SIGNUM)
 -- Function: gsl_complex gsl_linalg_complex_LU_det (gsl_matrix_complex * LU, int SIGNUM)
     These functions compute the determinant of a matrix A from its LU decomposition, LU. The determinant is computed as the product of the diagonal elements of U and the sign of the row permutation SIGNUM.

 -- Function: double gsl_linalg_LU_lndet (gsl_matrix * LU)
 -- Function: 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 A, \ln|\det(A)|, from its LU decomposition, LU.  This function may be useful if the direct computation of the determinant would overflow or underflow.

 -- Function: int gsl_linalg_LU_sgndet (gsl_matrix * LU, int SIGNUM)
 -- Function: 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 A, \det(A)/|\det(A)|, from its LU decomposition, LU.

QR Decomposition
================

A general rectangular M-by-N matrix A has a QR decomposition into the product of an orthogonal M-by-M square matrix Q (where Q^T Q = I) and an M-by-N right-triangular matrix R,

     A = Q R

This decomposition can be used to convert the linear system A x = b into the triangular system R x = Q^T b, which can be solved by back-substitution. Another use of the QR decomposition is to compute an orthonormal basis for a set of vectors. The first N columns of Q form an orthonormal basis for the range of A, ran(A), when A has full column rank.

 -- Function: int gsl_linalg_QR_decomp (gsl_matrix * A, gsl_vector * TAU)
     This function factorizes the M-by-N matrix A into the QR decomposition A = Q R.  On output the diagonal and upper triangular part of the input matrix contain the matrix R. 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 k=\min(M,N). The matrix Q is related to these components by, Q = Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i v_i^T and v_i is the Householder vector v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)). 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).

 -- Function: 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 A x = b using the QR decomposition of A 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.

 -- Function: int gsl_linalg_QR_svx (const gsl_matrix * QR, const gsl_vector * TAU, gsl_vector * X)
     This function solves the square system A x = b in-place using the QR decomposition of A held in (QR,TAU) which must have been computed previously by gsl_linalg_QR_decomp.  On input X should contain the right-hand side b, which is replaced by the solution on output.

 -- Function: 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 A x = b where the matrix A has more rows than columns.  The least squares solution minimizes the Euclidean norm of the residual, ||Ax - b||.The routine requires as input the QR decomposition of A 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.

 -- Function: int gsl_linalg_QR_QTvec (const gsl_matrix * QR, const gsl_vector * TAU, gsl_vector * V)
     This function applies the matrix Q^T encoded in the decomposition (QR,TAU) to the vector V, storing the result Q^T v in V.  The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q^T.

 -- Function: int gsl_linalg_QR_Qvec (const gsl_matrix * QR, const gsl_vector * TAU, gsl_vector * V)
     This function applies the matrix Q encoded in the decomposition (QR,TAU) to the vector V, storing the result Q v in V.  The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q.

 -- Function: int gsl_linalg_QR_QTmat (const gsl_matrix * QR, const gsl_vector * TAU, gsl_matrix * A)
     This function applies the matrix Q^T encoded in the decomposition (QR,TAU) to the matrix A, storing the result Q^T A in A.  The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q^T.

 -- Function: int gsl_linalg_QR_Rsolve (const gsl_matrix * QR, const gsl_vector * B, gsl_vector * X)
     This function solves the triangular system R x = b for X. It may be useful if the product b' = Q^T b has already been computed using gsl_linalg_QR_QTvec.

 -- Function: int gsl_linalg_QR_Rsvx (const gsl_matrix * QR, gsl_vector * X)
     This function solves the triangular system R x = b for X in-place. On input X should contain the right-hand side b and is replaced by the solution on output. This function may be useful if the product b' = Q^T b has already been computed using gsl_linalg_QR_QTvec.

 -- Function: int gsl_linalg_QR_unpack (const gsl_matrix * QR, const gsl_vector * TAU, gsl_matrix * Q, gsl_matrix * R)
     This function unpacks the encoded QR decomposition (QR,TAU) into the matrices Q and R, where Q is M-by-M and R is M-by-N.

 -- Function: int gsl_linalg_QR_QRsolve (gsl_matrix * Q, gsl_matrix * R, const gsl_vector * B, gsl_vector * X)
     This function solves the system R x = Q^T b for X. It can be used when the QR decomposition of a matrix is available in unpacked form as (Q, R).

 -- Function: 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 w v^T of the QR decomposition (Q, R). The update is given by Q'R' = Q (R + w v^T) where the output matrices Q' and R' are also orthogonal and right triangular. Note that W is destroyed by the update.

 -- Function: int gsl_linalg_R_solve (const gsl_matrix * R, const gsl_vector * B, gsl_vector * X)
     This function solves the triangular system R x = b for the N-by-N matrix R.

 -- Function: int gsl_linalg_R_svx (const gsl_matrix * R, gsl_vector * X)
     This function solves the triangular system R x = b in-place. On input X should contain the right-hand side b, which is replaced by the solution on output.

QR Decomposition with Column Pivoting
=====================================

The QR decomposition can be extended to the rank deficient case by introducing a column permutation P,

     A P = Q R

The first r columns of Q form an orthonormal basis for the range of A for a matrix with column rank r.  This decomposition can also be used to convert the linear system A x = b into the triangular system R y = Q^T b, x = P y, which can be solved by back-substitution and permutation.  We denote the QR decomposition with column pivoting by QRP^T since A = Q R P^T.

 -- Function: int gsl_linalg_QRPT_decomp (gsl_matrix * A, gsl_vector * TAU, gsl_permutation * P, int * SIGNUM, gsl_vector * NORM)
     This function factorizes the M-by-N matrix A into the QRP^T decomposition A = Q R P^T.  On output the diagonal and upper triangular part of the input matrix contain the matrix R. The permutation matrix P is stored in the permutation P.  The sign of the permutation is given by SIGNUM. It has the value (-1)^n, where n 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 k=\min(M,N). The matrix Q is related to these components by, Q = Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i v_i^T and v_i is the Householder vector v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)). 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).

 -- Function: 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 A = Q R P^T without modifying A itself and storing the output in the separate matrices Q and R.

 -- Function: 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 A x = b using the QRP^T decomposition of A held in (QR, TAU, P) which must have been computed previously by gsl_linalg_QRPT_decomp.

 -- Function: 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 A x = b in-place using the QRP^T decomposition of A held in (QR,TAU,P). On input X should contain the right-hand side b, which is replaced by the solution on output.

 -- Function: 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 R P^T x = Q^T b for X. It can be used when the QR decomposition of a matrix is available in unpacked form as (Q, R).

 -- Function: 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 w v^T of the QRP^T decomposition (Q, R, P). The update is given by Q'R' = Q (R + w v^T P) where the output matrices Q' and R' are also orthogonal and right triangular. Note that W is destroyed by the update. The permutation P is not changed.

 -- Function: 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 R P^T x = b for the N-by-N matrix R contained in QR.

 -- Function: int gsl_linalg_QRPT_Rsvx (const gsl_matrix * QR, const gsl_permutation * P, gsl_vector * X)
     This function solves the triangular system R P^T x = b in-place for the N-by-N matrix R contained in QR. On input X should contain the right-hand side b, which is replaced by the solution on output.

Singular Value Decomposition
============================

A general rectangular M-by-N matrix A has a singular value decomposition (SVD) into the product of an M-by-N orthogonal matrix U, an N-by-N diagonal matrix of singular values S and the transpose of an N-by-N orthogonal square matrix V,

     A = U S V^T

The singular values $\sigma_i = S_ii$ \sigma_i = S_{ii} are all non-negative and are generally chosen to form a non-increasing sequence $\sigma_1 \ge \sigma_2 \ge ... \ge \sigma_N \ge 0$ \sigma_1 >= \sigma_2 >= ... >= \sigma_N >= 0.

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 A is given by the columns of V corresponding to the zero singular values. Similarly, the range of A is given by columns of U corresponding to the non-zero singular values.

Note that the routines here compute the ``thin'' version of the SVD with U as M-by-N orthogonal matrix. This allows in-place computation and is the most commonly-used form in practice. Mathematically, the ``full'' SVD is defined with U as an M-by-M orthogonal matrix and S as an M-by-N diagonal matrix (with additional rows of zeros).

 -- Function: int gsl_linalg_SV_decomp (gsl_matrix * A, gsl_matrix * V, gsl_vector * S, gsl_vector * WORK)
     This function factorizes the M-by-N matrix A into the singular value decomposition A = U S V^T for $M \ge N$ M >= N.  On output the matrix A is replaced by U. The diagonal elements of the singular value matrix S are stored in the vector S. The singular values are non-negative and form a non-increasing sequence from S_1 to S_N. The matrix V contains the elements of V in untransposed form. To form the product U S V^T 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.

 -- Function: 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 $M \gg N$ M>>N.  It requires the vector WORK of length N and the N-by-N matrix X as additional working space.

 -- Function: int gsl_linalg_SV_decomp_jacobi (gsl_matrix * A, gsl_matrix * V, gsl_vector * S)
     This function computes the SVD of the M-by-N matrix A using one-sided Jacobi orthogonalization for $M \ge N$ M >= N.  The Jacobi method can compute singular values to higher relative accuracy than Golub-Reinsch algorithms (see references for details).

 -- Function: 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 A x = b using the singular value decomposition (U, S, V) of A 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 ||A x - b||_2.

Cholesky Decomposition
======================

A symmetric, positive definite square matrix A has a Cholesky decomposition into a product of a lower triangular matrix L and its transpose L^T,

     A = L L^T

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 A x = b into a pair of triangular systems (L y = b, L^T x = y), which can be solved by forward and back-substitution.

 -- Function: int gsl_linalg_cholesky_decomp (gsl_matrix * A)
 -- Function: int gsl_linalg_complex_cholesky_decomp (gsl_matrix_complex * A)
     These functions factorize the symmetric, positive-definite square matrix A into the Cholesky decomposition A = L L^T (or $A = L L^\dagger$ A = L L^H 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 L, while the upper triangular part of the input matrix is overwritten with L^T (the diagonal terms being identical for both L and L^T).  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.

 -- Function: int gsl_linalg_cholesky_solve (const gsl_matrix * CHOLESKY, const gsl_vector * B, gsl_vector * X)
 -- Function: 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 A x = b using the Cholesky decomposition of A held in the matrix CHOLESKY which must have been previously computed by gsl_linalg_cholesky_decomp or gsl_linalg_complex_cholesky_decomp.

 -- Function: int gsl_linalg_cholesky_svx (const gsl_matrix * CHOLESKY, gsl_vector * X)
 -- Function: int gsl_linalg_complex_cholesky_svx (const gsl_matrix_complex * CHOLESKY, gsl_vector_complex * X)
     These functions solve the system A x = b in-place using the Cholesky decomposition of A 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 b, which is replaced by the solution on output.

 -- Function: int gsl_linalg_cholesky_invert (gsl_matrix * CHOLESKY)
 -- Function: 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.

Tridiagonal Decomposition of Real Symmetric Matrices
====================================================

A symmetric matrix A can be factorized by similarity transformations into the form,

     A = Q T Q^T

where Q is an orthogonal matrix and T is a symmetric tridiagonal matrix.

 -- Function: int gsl_linalg_symmtd_decomp (gsl_matrix * A, gsl_vector * TAU)
     This function factorizes the symmetric square matrix A into the symmetric tridiagonal decomposition Q T Q^T.  On output the diagonal and subdiagonal part of the input matrix A contain the tridiagonal matrix T.  The remaining lower triangular part of the input matrix contains the Householder vectors which, together with the Householder coefficients TAU, encode the orthogonal matrix Q. This storage scheme is the same as used by LAPACK.  The upper triangular part of A is not referenced.

 -- Function: 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.

 -- Function: 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 A can be factorized by similarity transformations into the form,

     A = U T U^T

where U is a unitary matrix and T is a real symmetric tridiagonal matrix.

 -- Function: 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 U T U^T.  On output the real parts of the diagonal and subdiagonal part of the input matrix A contain the tridiagonal matrix T.  The remaining lower triangular part of the input matrix contains the Householder vectors which, together with the Householder coefficients TAU, encode the unitary matrix U. 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.

 -- Function: 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.

 -- Function: 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 A can be decomposed by orthogonal similarity transformations into the form

     A = U H U^T

where U is orthogonal and H 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.

 -- Function: 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 H = U^T A U. On output, H is stored in the upper portion of A. The information required to construct the matrix U is stored in the lower triangular portion of A. U is a product of N - 2 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.

 -- Function: int gsl_linalg_hessenberg_unpack (gsl_matrix * H, gsl_vector * TAU, gsl_matrix * U)
     This function constructs the orthogonal matrix U from the information stored in the Hessenberg matrix H along with the vector TAU. H and TAU are outputs from gsl_linalg_hessenberg_decomp.

 -- Function: 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 V' = VU. 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.

 -- Function: 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 (A, B) can be decomposed by orthogonal similarity transformations into the form

     A = U H V^T
     B = U R V^T

where U and V are orthogonal, H is an upper Hessenberg matrix, and R is upper triangular. The Hessenberg-Triangular reduction is the first step in the generalized Schur decomposition for the generalized eigenvalue problem.

 -- Function: 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, H is stored in A, and R 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 N is needed in WORK.

Bidiagonalization
=================

A general matrix A can be factorized by similarity transformations into the form,

     A = U B V^T

where U and V are orthogonal matrices and B is a N-by-N bidiagonal matrix with non-zero entries only on the diagonal and superdiagonal.  The size of U is M-by-N and the size of V is N-by-N.

 -- Function: int gsl_linalg_bidiag_decomp (gsl_matrix * A, gsl_vector * TAU_U, gsl_vector * TAU_V)
     This function factorizes the M-by-N matrix A into bidiagonal form U B V^T.  The diagonal and superdiagonal of the matrix B are stored in the diagonal and superdiagonal of A. The orthogonal matrices U 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.

 -- Function: 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 M-by-N orthogonal matrix satisfying U^T U = I for efficiency.

 -- Function: 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.

 -- Function: 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.

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 P takes the form,

     P = I - \tau v v^T

where v is a vector (called the Householder vector) and \tau = 2/(v^T v).  The functions described in this section use the rank-1 structure of the Householder matrix to create and apply Householder transformations efficiently.

 -- Function: double gsl_linalg_householder_transform (gsl_vector * V)
 -- Function: gsl_complex gsl_linalg_complex_householder_transform (gsl_vector_complex * V)
     This function prepares a Householder transformation P = I - \tau v v^T which can be used to zero all the elements of the input vector except the first.  On output the transformation is stored in the vector V and the scalar \tau is returned.

 -- Function: int gsl_linalg_householder_hm (double TAU, const gsl_vector * V, gsl_matrix * A)
 -- Function: int gsl_linalg_complex_householder_hm (gsl_complex TAU, const gsl_vector_complex * V, gsl_matrix_complex * A)
     This function applies the Householder matrix P defined by the scalar TAU and the vector V to the left-hand side of the matrix A. On output the result P A is stored in A.

 -- Function: int gsl_linalg_householder_mh (double TAU, const gsl_vector * V, gsl_matrix * A)
 -- Function: int gsl_linalg_complex_householder_mh (gsl_complex TAU, const gsl_vector_complex * V, gsl_matrix_complex * A)
     This function applies the Householder matrix P defined by the scalar TAU and the vector V to the right-hand side of the matrix A. On output the result A P is stored in A.

 -- Function: int gsl_linalg_householder_hv (double TAU, const gsl_vector * V, gsl_vector * W)
 -- Function: int gsl_linalg_complex_householder_hv (gsl_complex TAU, const gsl_vector_complex * V, gsl_vector_complex * W)
     This function applies the Householder transformation P defined by the scalar TAU and the vector V to the vector W.  On output the result P w is stored in W.

Householder solver for linear systems
=====================================

 -- Function: int gsl_linalg_HH_solve (gsl_matrix * A, const gsl_vector * B, gsl_vector * X)
     This function solves the system A x = b 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.

 -- Function: int gsl_linalg_HH_svx (gsl_matrix * A, gsl_vector * X)
     This function solves the system A x = b in-place using Householder transformations.  On input X should contain the right-hand side b, 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.

 -- Function: 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 N-by-N system A x = b where A is tridiagonal ($N\geq 2$ N >= 2). 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,

          A = ( 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 )

 -- Function: 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 N-by-N system A x = b where A is symmetric tridiagonal ($N\geq 2$ N >= 2).  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,

          A = ( 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 )

 -- Function: 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 N-by-N system A x = b where A is cyclic tridiagonal ($N\geq 3$ N >= 3).  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,

          A = ( 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 )

 -- Function: 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 N-by-N system A x = b where A is symmetric cyclic tridiagonal ($N\geq 3$ N >= 3).  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,

          A = ( 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 )

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 A consists of replacing A with a similar matrix

     A' = D^(-1) A D

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

 -- Function: 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 A x = b. The system to be solved is,

     [ 0.18 0.60 0.57 0.96 ] [x0]   [1.0]
     [ 0.41 0.24 0.99 0.58 ] [x1] = [2.0]
     [ 0.14 0.30 0.97 0.66 ] [x2]   [3.0]
     [ 0.51 0.13 0.19 0.85 ] [x3]   [4.0]

and the solution is found using LU decomposition of the matrix A.

          [file linalglu.c not included]

Here is the output from the program,

          [file linalglu.out not included]

This can be verified by multiplying the solution x by the original matrix A 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, b, in accordance with the equation A x = b.

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.

     http://www.netlib.org/lapack

The LAPACK source code can be found at the website above, 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.

  James Demmel, Krešimir Veselić, ``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.