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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 Level1 and Level2 BLAS calls for efficiency.
The functions described in this chapter are declared in the header file
`gsl_linalg.h'.
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 backsubstitution.
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 jth column of the matrix P is given by the
kth column of the identity matrix, where k = p_j the
jth 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 inplace using the
precomputed LU decomposition of A into (LU,p). On input
x should contain the righthand 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.
A general rectangular MbyN matrix A has a
QR decomposition into the product of an orthogonal
MbyM square matrix Q (where Q^T Q = I) and
an MbyN righttriangular 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
backsubstitution. 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 MbyN 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 leastsquares 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 inplace 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
righthand 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 byproduct 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
inplace. On input x should contain the righthand 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 MbyM and R is MbyN.
 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 rank1 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
NbyN matrix R.
 Function: int gsl_linalg_R_svx (const gsl_matrix * R, gsl_vector * x)

This function solves the triangular system R x = b inplace. On
input x should contain the righthand side b, which is
replaced by the solution on output.
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 backsubstitution 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 MbyN 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 inplace using the
QRP^T decomposition of A held in
(QR,tau,p). On input x should contain the
righthand 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 rank1 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
NbyN 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 inplace
for the NbyN matrix R contained in QR. On
input x should contain the righthand side b, which is
replaced by the solution on output.
A general rectangular MbyN matrix A has a
singular value decomposition (SVD) into the product of an
MbyN orthogonal matrix U, an NbyN
diagonal matrix of singular values S and the transpose of an
NbyN orthogonal square matrix V,
A = U S V^T
The singular values
\sigma_i = S_{ii} are all nonnegative and are
generally chosen to form a nonincreasing sequence
\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
nonzero singular values indicates the rank of the matrix. In practice
singular value decomposition of a rankdeficient 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 rankdeficient 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 nonzero singular values.
Note that the routines here compute the "thin" version of the SVD
with U as MbyN orthogonal matrix. This allows
inplace computation and is the most commonlyused form in practice.
Mathematically, the "full" SVD is defined with U as an
MbyM orthogonal matrix and S as an
MbyN 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 MbyN matrix A into
the singular value decomposition A = U S V^T for
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 nonnegative
and form a nonincreasing 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 GolubReinsch 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 GolubReinsch
algorithm, which is faster for
M>>N. It requires the vector work of length N and the
NbyN 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 MbyN matrix A
using onesided Jacobi orthogonalization for
M >= N. The Jacobi method can compute singular values to higher
relative accuracy than GolubReinsch 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 nonzero 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 overdetermined 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.
 Function: int gsl_linalg_SV_leverage (const gsl_matrix * U, gsl_vector * h)

This function computes the statistical leverage values h_i of a matrix A
using its singular value decomposition (U, S, V) previously computed
with
gsl_linalg_SV_decomp
. h_i are the diagonal values of the matrix
A (A^T A)^{1 A^T} and depend only on the matrix U which is the input to
this function.
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 squareroot 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
backsubstitution.
 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, positivedefinite square matrix
A into the Cholesky decomposition A = L L^T (or
A = L L^H
for the complex case). On input, the values from the diagonal and lowertriangular 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
positivedefinite then the decomposition will fail, returning the
error code
GSL_EDOM
.
When testing whether a matrix is positivedefinite, 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 inplace 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 righthand 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 inplace in cholesky.
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.
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.
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
.
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
HessenbergTriangular 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 HessenbergTriangular 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.
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
NbyN bidiagonal matrix with nonzero entries only on the
diagonal and superdiagonal. The size of U is MbyN
and the size of V is NbyN.
 Function: int gsl_linalg_bidiag_decomp (gsl_matrix * A, gsl_vector * tau_U, gsl_vector * tau_V)

This function factorizes the MbyN 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 MbyN 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 inplace 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.
A Householder transformation is a rank1 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
rank1 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 lefthand 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 righthand 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.
 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 inplace using
Householder transformations. On input x should contain the
righthand side b, which is replaced by the solution on output. The
matrix A is destroyed by the Householder transformations.
The functions described in this section efficiently solve symmetric,
nonsymmetric 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 nonpositive 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 NbyN system A x =
b where A is tridiagonal (
N >= 2). The superdiagonal and
subdiagonal vectors e and f must be one element shorter
than the diagonal vector diag. The form of A for the 4by4
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 NbyN system A x =
b where A is symmetric tridiagonal (
N >= 2). The offdiagonal vector
e must be one element shorter than the diagonal vector diag.
The form of A for the 4by4 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 NbyN system A x =
b where A is cyclic tridiagonal (
N >= 3). The cyclic superdiagonal and
subdiagonal vectors e and f must have the same number of
elements as the diagonal vector diag. The form of A for the
4by4 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 NbyN system A x =
b where A is symmetric cyclic tridiagonal (
N >= 3). The cyclic
offdiagonal vector e must have the same number of elements as the
diagonal vector diag. The form of A for the 4by4 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 )
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.
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.
#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 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 righthand side vector, b, in
accordance with the equation A x = b.
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 0801854148.
The LAPACK library is described in the following manual,
The LAPACK source code can be found at the website above, along
with an online copy of the users guide.
The Modified GolubReinsch 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 7283.
The Jacobi algorithm for singular value decomposition is described in
the following papers,

J.C. Nash, "A onesided transformation method for the singular value
decomposition and algebraic eigenproblem", Computer Journal,
Volume 18, Number 1 (1975), p 7476

J.C. Nash and S. Shlien "Simple algorithms for the partial singular
value decomposition", Computer Journal, Volume 30 (1987), p
268275.

James Demmel, Kre@v{s}imir Veseli'c, "Jacobi's Method is more accurate than
QR", Lapack Working Note 15 (LAWN15), October 1989. Available
from netlib, http://www.netlib.org/lapack/ in the
lawns
or
lawnspdf
directories.
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