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This chapter describes functions for solving linear systems. The
library provides simple linear algebra operations which operate directly
on the gsl_vector
and gsl_matrix
objects. These are meant
for "small" systems where simple algorithms are acceptable.
Anyone interested in large systems will want to use the sophisticated
routines found in LAPACK. The Fortran version of LAPACK is
recommended as the standard package for linear algebra. It supports
blocked algorithms, specialized data representations and other
optimizations.
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 R 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.
- Function: int gsl_linalg_LU_decomp (gsl_matrix * A, gsl_permutation * p, int *signum)
-
This function factorizes 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
R. 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)
-
This function solves the system A x = b using the LU
decomposition of A into (LU, p) given by
gsl_linalg_LU_decomp
.
- Function: int gsl_linalg_LU_svx (const gsl_matrix * LU, const gsl_permutation * p, gsl_vector * x)
-
This function solves the system A x = b in-place using the
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)
-
This function applies an iterative improvement to x, the solution
of A x = b, using the 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)
-
This function computes 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.
- Function: double gsl_linalg_LU_det (gsl_matrix * LU, int signum)
-
This function computes 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)
-
This function computes 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)
-
This function computes the sign of the determinant of a matrix A,
sign(det(A)), from its LU decomposition, LU.
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 system A x = b using the QR
decomposition of A into (QR, tau) given by
gsl_linalg_QR_decomp
.
- Function: int gsl_linalg_QR_svx (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * x)
-
This function solves the system A x = b in-place using the
QR decomposition of A into (QR,tau). On input
x should contain the right-hand side b, which is replaced by
the solution on output.
- 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_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.
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 this 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)
-
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 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_solve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x)
-
This function solves the system A x = b using the QRP^T
decomposition of A into (QR, tau, p) given 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 system A x = b in-place using the
QRP^T decomposition of A into
(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 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 * u, 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 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.
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
M-by-M orthogonal square matrix Q,
A = U S Q^T
The singular values
\sigma_i = S_{ii} are all non-negative and are
generally chosen to form a non-increasing 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
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 chosing a suitable
tolerance.
- Function: int gsl_linalg_SV_decomp (gsl_matrix * A, gsl_matrix * Q, gsl_vector * S)
-
This function factorizes the M-by-N matrix A into
the singular value decomposition A = U S Q^T. 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 Q contains the elements of
Q in untransposed form. To form the product U S Q^T it is
necessary to take the transpose of Q.
The algorithm used in the decomposition is One-Sided Jacobi
orthogonalization (see references for details).
- Function: int gsl_linalg_SV_solve (gsl_matrix * U, gsl_matrix * Q, gsl_vector * S, const gsl_vector * b, gsl_vector * x)
-
This function solves the system A x = b using the singular value
decomposition of A given by
gsl_linalg_SV_decomp
,
(U, S, Q).
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.
A symmetric, positive definite square matrix A has a Cholesky
decomposition into a product of a lower triagular 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)
-
This function factorizes the positive-definite square matrix A
into the Cholesky decomposition A = L L^T. On output the diagonal
and lower triangular part of the input matrix A contain the matrix
L. The upper triangular part of the input matrix contains
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
.
- Function: int gsl_linalg_cholesky_solve (const gsl_matrix * cholesky, const gsl_vector * b, gsl_vector * x)
-
This function solves the system A x = b using the Cholesky
decomposition of A into the matrix cholesky given by
gsl_linalg_cholesky_decomp
.
- Function: int gsl_linalg_cholesky_svx (const gsl_matrix * cholesky, gsl_vector * x)
-
This function solves the system A x = b in-place using the
Cholesky decomposition of A into the matrix cholesky. On
input x should contain the right-hand side b, which is
replaced by the solution on output.
- 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.
- Function: int gsl_linalg_solve_symm_tridiag (const gsl_vector * diag, const gsl_vector * offdiag, const gsl_vector * b, gsl_vector * x)
-
- Function: int gsl_linalg_solve_symm_cyc_tridiag (const gsl_vector * diag, const gsl_vector * offdiag, const gsl_vector * b, gsl_vector * x)
-
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,
-
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 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 (1973), p 74--76
-
James Demmel, Kresimir 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.
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