@cindex linear algebra
@cindex solution of linear systems, Ax=b
@cindex matrix factorization
@cindex factorization of matrices

This chapter describes functions for solving linear systems.  The
library provides linear algebra operations which operate directly on
the @code{gsl_vector} and @code{gsl_matrix} objects.  These routines
use the standard algorithms from Golub & Van Loan's @cite{Matrix
Computations}.

@cindex LAPACK, recommended for linear algebra
When dealing with very large systems the routines found in @sc{lapack}
should be considered.  These support specialized data representations
and other optimizations.

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


@menu
* LU Decomposition::            
* QR Decomposition::            
* QR Decomposition with Column Pivoting::  
* Singular Value Decomposition::  
* Cholesky Decomposition::      
* Tridiagonal Decomposition of Real Symmetric Matrices::  
* Tridiagonal Decomposition of Hermitian Matrices::  
* Hessenberg Decomposition of Real Matrices::
* Bidiagonalization::           
* Householder Transformations::  
* Householder solver for linear systems::  
* Tridiagonal Systems::         
* Balancing::
* Linear Algebra Examples::     
* Linear Algebra References and Further Reading::  
@end menu

@node LU Decomposition
@section LU Decomposition
@cindex LU decomposition

A general square matrix @math{A} has an @math{LU} decomposition into
upper and lower triangular matrices,
@tex
\beforedisplay
$$
P A = L U
$$
\afterdisplay
@end tex
@ifinfo

@example
P A = L U
@end example

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

@deftypefun int gsl_linalg_LU_decomp (gsl_matrix * @var{A}, gsl_permutation * @var{p}, int * @var{signum})
@deftypefunx int gsl_linalg_complex_LU_decomp (gsl_matrix_complex * @var{A}, gsl_permutation * @var{p}, int * @var{signum})
These functions factorize the square matrix @var{A} into the @math{LU}
decomposition @math{PA = LU}.  On output the diagonal and upper
triangular part of the input matrix @var{A} contain the matrix
@math{U}. The lower triangular part of the input matrix (excluding the
diagonal) contains @math{L}.  The diagonal elements of @math{L} are
unity, and are not stored.

The permutation matrix @math{P} is encoded in the permutation
@var{p}. The @math{j}-th column of the matrix @math{P} is given by the
@math{k}-th column of the identity matrix, where @math{k = p_j} the
@math{j}-th element of the permutation vector. The sign of the
permutation is given by @var{signum}. It has the value @math{(-1)^n},
where @math{n} is the number of interchanges in the permutation.

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

@cindex linear systems, solution of
@deftypefun int gsl_linalg_LU_solve (const gsl_matrix * @var{LU}, const gsl_permutation * @var{p}, const gsl_vector * @var{b}, gsl_vector * @var{x})
@deftypefunx int gsl_linalg_complex_LU_solve (const gsl_matrix_complex * @var{LU}, const gsl_permutation * @var{p}, const gsl_vector_complex * @var{b}, gsl_vector_complex * @var{x})
These functions solve the square system @math{A x = b} using the @math{LU}
decomposition of @math{A} into (@var{LU}, @var{p}) given by
@code{gsl_linalg_LU_decomp} or @code{gsl_linalg_complex_LU_decomp}.
@end deftypefun

@deftypefun int gsl_linalg_LU_svx (const gsl_matrix * @var{LU}, const gsl_permutation * @var{p}, gsl_vector * @var{x})
@deftypefunx int gsl_linalg_complex_LU_svx (const gsl_matrix_complex * @var{LU}, const gsl_permutation * @var{p}, gsl_vector_complex * @var{x})
These functions solve the square system @math{A x = b} in-place using the
@math{LU} decomposition of @math{A} into (@var{LU},@var{p}). On input
@var{x} should contain the right-hand side @math{b}, which is replaced
by the solution on output.
@end deftypefun

@cindex refinement of solutions in linear systems
@cindex iterative refinement of solutions in linear systems
@cindex linear systems, refinement of solutions
@deftypefun int gsl_linalg_LU_refine (const gsl_matrix * @var{A}, const gsl_matrix * @var{LU}, const gsl_permutation * @var{p}, const gsl_vector * @var{b}, gsl_vector * @var{x}, gsl_vector * @var{residual})
@deftypefunx int gsl_linalg_complex_LU_refine (const gsl_matrix_complex * @var{A}, const gsl_matrix_complex * @var{LU}, const gsl_permutation * @var{p}, const gsl_vector_complex * @var{b}, gsl_vector_complex * @var{x}, gsl_vector_complex * @var{residual})
These functions apply an iterative improvement to @var{x}, the solution
of @math{A x = b}, using the @math{LU} decomposition of @math{A} into
(@var{LU},@var{p}). The initial residual @math{r = A x - b} is also
computed and stored in @var{residual}.
@end deftypefun

@cindex inverse of a matrix, by LU decomposition
@cindex matrix inverse
@deftypefun int gsl_linalg_LU_invert (const gsl_matrix * @var{LU}, const gsl_permutation * @var{p}, gsl_matrix * @var{inverse})
@deftypefunx int gsl_linalg_complex_LU_invert (const gsl_matrix_complex * @var{LU}, const gsl_permutation * @var{p}, gsl_matrix_complex * @var{inverse})
These functions compute the inverse of a matrix @math{A} from its
@math{LU} decomposition (@var{LU},@var{p}), storing the result in the
matrix @var{inverse}. The inverse is computed by solving the system
@math{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).
@end deftypefun

@cindex determinant of a matrix, by LU decomposition
@cindex matrix determinant
@deftypefun double gsl_linalg_LU_det (gsl_matrix * @var{LU}, int @var{signum})
@deftypefunx gsl_complex gsl_linalg_complex_LU_det (gsl_matrix_complex * @var{LU}, int @var{signum})
These functions compute the determinant of a matrix @math{A} from its
@math{LU} decomposition, @var{LU}. The determinant is computed as the
product of the diagonal elements of @math{U} and the sign of the row
permutation @var{signum}.
@end deftypefun

@cindex logarithm of the determinant of a matrix
@deftypefun double gsl_linalg_LU_lndet (gsl_matrix * @var{LU})
@deftypefunx double gsl_linalg_complex_LU_lndet (gsl_matrix_complex * @var{LU})
These functions compute the logarithm of the absolute value of the
determinant of a matrix @math{A}, @math{\ln|\det(A)|}, from its @math{LU}
decomposition, @var{LU}.  This function may be useful if the direct
computation of the determinant would overflow or underflow.
@end deftypefun

@cindex sign of the determinant of a matrix
@deftypefun int gsl_linalg_LU_sgndet (gsl_matrix * @var{LU}, int @var{signum})
@deftypefunx gsl_complex gsl_linalg_complex_LU_sgndet (gsl_matrix_complex * @var{LU}, int @var{signum})
These functions compute the sign or phase factor of the determinant of a
matrix @math{A}, @math{\det(A)/|\det(A)|}, from its @math{LU} decomposition,
@var{LU}.
@end deftypefun

@node QR Decomposition
@section QR Decomposition
@cindex QR decomposition

A general rectangular @math{M}-by-@math{N} matrix @math{A} has a
@math{QR} decomposition into the product of an orthogonal
@math{M}-by-@math{M} square matrix @math{Q} (where @math{Q^T Q = I}) and
an @math{M}-by-@math{N} right-triangular matrix @math{R},
@tex
\beforedisplay
$$
A = Q R
$$
\afterdisplay
@end tex
@ifinfo

@example
A = Q R
@end example

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

@deftypefun int gsl_linalg_QR_decomp (gsl_matrix * @var{A}, gsl_vector * @var{tau})
This function factorizes the @math{M}-by-@math{N} matrix @var{A} into
the @math{QR} decomposition @math{A = Q R}.  On output the diagonal and
upper triangular part of the input matrix contain the matrix
@math{R}. The vector @var{tau} and the columns of the lower triangular
part of the matrix @var{A} contain the Householder coefficients and
Householder vectors which encode the orthogonal matrix @var{Q}.  The
vector @var{tau} must be of length @math{k=\min(M,N)}. The matrix
@math{Q} is related to these components by, @math{Q = Q_k ... Q_2 Q_1}
where @math{Q_i = I - \tau_i v_i v_i^T} and @math{v_i} is the
Householder vector @math{v_i =
(0,...,1,A(i+1,i),A(i+2,i),...,A(m,i))}. This is the same storage scheme
as used by @sc{lapack}.

The algorithm used to perform the decomposition is Householder QR (Golub
& Van Loan, @cite{Matrix Computations}, Algorithm 5.2.1).
@end deftypefun

@deftypefun int gsl_linalg_QR_solve (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, const gsl_vector * @var{b}, gsl_vector * @var{x})
This function solves the square system @math{A x = b} using the @math{QR}
decomposition of @math{A} into (@var{QR}, @var{tau}) given by
@code{gsl_linalg_QR_decomp}. The least-squares solution for rectangular systems can
be found using @code{gsl_linalg_QR_lssolve}.
@end deftypefun

@deftypefun int gsl_linalg_QR_svx (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, gsl_vector * @var{x})
This function solves the square system @math{A x = b} in-place using the
@math{QR} decomposition of @math{A} into (@var{QR},@var{tau}) given by
@code{gsl_linalg_QR_decomp}. On input @var{x} should contain the
right-hand side @math{b}, which is replaced by the solution on output.
@end deftypefun

@deftypefun int gsl_linalg_QR_lssolve (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, const gsl_vector * @var{b}, gsl_vector * @var{x}, gsl_vector * @var{residual})
This function finds the least squares solution to the overdetermined
system @math{A x = b} where the matrix @var{A} has more rows than
columns.  The least squares solution minimizes the Euclidean norm of the
residual, @math{||Ax - b||}.The routine uses the @math{QR} decomposition
of @math{A} into (@var{QR}, @var{tau}) given by
@code{gsl_linalg_QR_decomp}.  The solution is returned in @var{x}.  The
residual is computed as a by-product and stored in @var{residual}.
@end deftypefun

@deftypefun int gsl_linalg_QR_QTvec (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, gsl_vector * @var{v})
This function applies the matrix @math{Q^T} encoded in the decomposition
(@var{QR},@var{tau}) to the vector @var{v}, storing the result @math{Q^T
v} in @var{v}.  The matrix multiplication is carried out directly using
the encoding of the Householder vectors without needing to form the full
matrix @math{Q^T}.
@end deftypefun

@deftypefun int gsl_linalg_QR_Qvec (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, gsl_vector * @var{v})
This function applies the matrix @math{Q} encoded in the decomposition
(@var{QR},@var{tau}) to the vector @var{v}, storing the result @math{Q
v} in @var{v}.  The matrix multiplication is carried out directly using
the encoding of the Householder vectors without needing to form the full
matrix @math{Q}.
@end deftypefun

@deftypefun int gsl_linalg_QR_Rsolve (const gsl_matrix * @var{QR}, const gsl_vector * @var{b}, gsl_vector * @var{x})
This function solves the triangular system @math{R x = b} for
@var{x}. It may be useful if the product @math{b' = Q^T b} has already
been computed using @code{gsl_linalg_QR_QTvec}.
@end deftypefun

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

@deftypefun int gsl_linalg_QR_unpack (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, gsl_matrix * @var{Q}, gsl_matrix * @var{R})
This function unpacks the encoded @math{QR} decomposition
(@var{QR},@var{tau}) into the matrices @var{Q} and @var{R}, where
@var{Q} is @math{M}-by-@math{M} and @var{R} is @math{M}-by-@math{N}. 
@end deftypefun

@deftypefun int gsl_linalg_QR_QRsolve (gsl_matrix * @var{Q}, gsl_matrix * @var{R}, const gsl_vector * @var{b}, gsl_vector * @var{x})
This function solves the system @math{R x = Q^T b} for @var{x}. It can
be used when the @math{QR} decomposition of a matrix is available in
unpacked form as (@var{Q}, @var{R}).
@end deftypefun

@deftypefun int gsl_linalg_QR_update (gsl_matrix * @var{Q}, gsl_matrix * @var{R}, gsl_vector * @var{w}, const gsl_vector * @var{v})
This function performs a rank-1 update @math{w v^T} of the @math{QR}
decomposition (@var{Q}, @var{R}). The update is given by @math{Q'R' = Q
R + w v^T} where the output matrices @math{Q'} and @math{R'} are also
orthogonal and right triangular. Note that @var{w} is destroyed by the
update.
@end deftypefun

@deftypefun int gsl_linalg_R_solve (const gsl_matrix * @var{R}, const gsl_vector * @var{b}, gsl_vector * @var{x})
This function solves the triangular system @math{R x = b} for the
@math{N}-by-@math{N} matrix @var{R}.
@end deftypefun

@deftypefun int gsl_linalg_R_svx (const gsl_matrix * @var{R}, gsl_vector * @var{x})
This function solves the triangular system @math{R x = b} in-place. On
input @var{x} should contain the right-hand side @math{b}, which is
replaced by the solution on output.
@end deftypefun

@node QR Decomposition with Column Pivoting
@section QR Decomposition with Column Pivoting
@cindex QR decomposition with column pivoting

The @math{QR} decomposition can be extended to the rank deficient case
by introducing a column permutation @math{P},
@tex
\beforedisplay
$$
A P = Q R
$$
\afterdisplay
@end tex
@ifinfo

@example
A P = Q R
@end example

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

@deftypefun int gsl_linalg_QRPT_decomp (gsl_matrix * @var{A}, gsl_vector * @var{tau}, gsl_permutation * @var{p}, int * @var{signum}, gsl_vector * @var{norm})
This function factorizes the @math{M}-by-@math{N} matrix @var{A} into
the @math{QRP^T} decomposition @math{A = Q R P^T}.  On output the
diagonal and upper triangular part of the input matrix contain the
matrix @math{R}. The permutation matrix @math{P} is stored in the
permutation @var{p}.  The sign of the permutation is given by
@var{signum}. It has the value @math{(-1)^n}, where @math{n} is the
number of interchanges in the permutation. The vector @var{tau} and the
columns of the lower triangular part of the matrix @var{A} contain the
Householder coefficients and vectors which encode the orthogonal matrix
@var{Q}.  The vector @var{tau} must be of length @math{k=\min(M,N)}. The
matrix @math{Q} is related to these components by, @math{Q = Q_k ... Q_2
Q_1} where @math{Q_i = I - \tau_i v_i v_i^T} and @math{v_i} is the
Householder vector @math{v_i =
(0,...,1,A(i+1,i),A(i+2,i),...,A(m,i))}. This is the same storage scheme
as used by @sc{lapack}.  The vector @var{norm} is a workspace of length
@var{N} used for column pivoting.

The algorithm used to perform the decomposition is Householder QR with
column pivoting (Golub & Van Loan, @cite{Matrix Computations}, Algorithm
5.4.1).
@end deftypefun

@deftypefun int gsl_linalg_QRPT_decomp2 (const gsl_matrix * @var{A}, gsl_matrix * @var{q}, gsl_matrix * @var{r}, gsl_vector * @var{tau}, gsl_permutation * @var{p}, int * @var{signum}, gsl_vector * @var{norm})
This function factorizes the matrix @var{A} into the decomposition
@math{A = Q R P^T} without modifying @var{A} itself and storing the
output in the separate matrices @var{q} and @var{r}.
@end deftypefun

@deftypefun int gsl_linalg_QRPT_solve (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, const gsl_permutation * @var{p}, const gsl_vector * @var{b}, gsl_vector * @var{x})
This function solves the square system @math{A x = b} using the @math{QRP^T}
decomposition of @math{A} into (@var{QR}, @var{tau}, @var{p}) given by
@code{gsl_linalg_QRPT_decomp}.
@end deftypefun

@deftypefun int gsl_linalg_QRPT_svx (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, const gsl_permutation * @var{p}, gsl_vector * @var{x})
This function solves the square system @math{A x = b} in-place using the
@math{QRP^T} decomposition of @math{A} into
(@var{QR},@var{tau},@var{p}). On input @var{x} should contain the
right-hand side @math{b}, which is replaced by the solution on output.
@end deftypefun

@deftypefun int gsl_linalg_QRPT_QRsolve (const gsl_matrix * @var{Q}, const gsl_matrix * @var{R}, const gsl_permutation * @var{p}, const gsl_vector * @var{b}, gsl_vector * @var{x})
This function solves the square system @math{R P^T x = Q^T b} for
@var{x}. It can be used when the @math{QR} decomposition of a matrix is
available in unpacked form as (@var{Q}, @var{R}).
@end deftypefun

@deftypefun int gsl_linalg_QRPT_update (gsl_matrix * @var{Q}, gsl_matrix * @var{R}, const gsl_permutation * @var{p}, gsl_vector * @var{u}, const gsl_vector * @var{v})
This function performs a rank-1 update @math{w v^T} of the @math{QRP^T}
decomposition (@var{Q}, @var{R}, @var{p}). The update is given by
@math{Q'R' = Q R + w v^T} where the output matrices @math{Q'} and
@math{R'} are also orthogonal and right triangular. Note that @var{w} is
destroyed by the update. The permutation @var{p} is not changed.
@end deftypefun

@deftypefun int gsl_linalg_QRPT_Rsolve (const gsl_matrix * @var{QR}, const gsl_permutation * @var{p}, const gsl_vector * @var{b}, gsl_vector * @var{x})
This function solves the triangular system @math{R P^T x = b} for the
@math{N}-by-@math{N} matrix @math{R} contained in @var{QR}.
@end deftypefun

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

@node Singular Value Decomposition
@section Singular Value Decomposition
@cindex SVD
@cindex singular value decomposition

A general rectangular @math{M}-by-@math{N} matrix @math{A} has a
singular value decomposition (@sc{svd}) into the product of an
@math{M}-by-@math{N} orthogonal matrix @math{U}, an @math{N}-by-@math{N}
diagonal matrix of singular values @math{S} and the transpose of an
@math{N}-by-@math{N} orthogonal square matrix @math{V},
@tex
\beforedisplay
$$
A = U S V^T
$$
\afterdisplay
@end tex
@ifinfo

@example
A = U S V^T
@end example

@end ifinfo
@noindent
The singular values
@c{$\sigma_i = S_{ii}$}
@math{\sigma_i = S_@{ii@}} are all non-negative and are
generally chosen to form a non-increasing sequence 
@c{$\sigma_1 \ge \sigma_2 \ge ... \ge \sigma_N \ge 0$}
@math{\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 @math{A} is given by
the columns of @math{V} corresponding to the zero singular values.
Similarly, the range of @math{A} is given by columns of @math{U}
corresponding to the non-zero singular values.

@deftypefun int gsl_linalg_SV_decomp (gsl_matrix * @var{A}, gsl_matrix * @var{V}, gsl_vector * @var{S}, gsl_vector * @var{work})
This function factorizes the @math{M}-by-@math{N} matrix @var{A} into
the singular value decomposition @math{A = U S V^T} for @c{$M \ge N$}
@math{M >= N}.  On output the matrix @var{A} is replaced by
@math{U}. The diagonal elements of the singular value matrix @math{S}
are stored in the vector @var{S}. The singular values are non-negative
and form a non-increasing sequence from @math{S_1} to @math{S_N}. The
matrix @var{V} contains the elements of @math{V} in untransposed
form. To form the product @math{U S V^T} it is necessary to take the
transpose of @var{V}.  A workspace of length @var{N} is required in
@var{work}.

This routine uses the Golub-Reinsch SVD algorithm.
@end deftypefun

@deftypefun int gsl_linalg_SV_decomp_mod (gsl_matrix * @var{A}, gsl_matrix * @var{X}, gsl_matrix * @var{V}, gsl_vector * @var{S}, gsl_vector * @var{work})
This function computes the SVD using the modified Golub-Reinsch
algorithm, which is faster for @c{$M \gg N$}
@math{M>>N}.  It requires the vector @var{work} of length @var{N} and the
@math{N}-by-@math{N} matrix @var{X} as additional working space.
@end deftypefun

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

@deftypefun int gsl_linalg_SV_solve (gsl_matrix * @var{U}, gsl_matrix * @var{V}, gsl_vector * @var{S}, const gsl_vector * @var{b}, gsl_vector * @var{x})
This function solves the system @math{A x = b} using the singular value
decomposition (@var{U}, @var{S}, @var{V}) of @math{A} given by
@code{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 @var{A} has more rows than columns the
system is solved in the least squares sense, returning the solution
@var{x} which minimizes @math{||A x - b||_2}.
@end deftypefun

@node Cholesky Decomposition
@section Cholesky Decomposition
@cindex Cholesky decomposition
@cindex square root of a matrix, Cholesky decomposition
@cindex matrix square root, Cholesky decomposition

A symmetric, positive definite square matrix @math{A} has a Cholesky
decomposition into a product of a lower triangular matrix @math{L} and
its transpose @math{L^T},
@tex
\beforedisplay
$$
A = L L^T
$$
\afterdisplay
@end tex
@ifinfo

@example
A = L L^T
@end example

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

@deftypefun int gsl_linalg_cholesky_decomp (gsl_matrix * @var{A})
This function factorizes the positive-definite symmetric square matrix
@var{A} into the Cholesky decomposition @math{A = L L^T}. On input
only the diagonal and lower-triangular part of the matrix @var{A} are
needed.  On output the diagonal and lower triangular part of the input
matrix @var{A} contain the matrix @math{L}. The upper triangular part
of the input matrix contains @math{L^T}, the diagonal terms being
identical for both @math{L} and @math{L^T}.  If the matrix is not
positive-definite then the decomposition will fail, returning the
error code @code{GSL_EDOM}.
@end deftypefun

@deftypefun int gsl_linalg_cholesky_solve (const gsl_matrix * @var{cholesky}, const gsl_vector * @var{b}, gsl_vector * @var{x})
This function solves the system @math{A x = b} using the Cholesky
decomposition of @math{A} into the matrix @var{cholesky} given by
@code{gsl_linalg_cholesky_decomp}.
@end deftypefun

@deftypefun int gsl_linalg_cholesky_svx (const gsl_matrix * @var{cholesky}, gsl_vector * @var{x})
This function solves the system @math{A x = b} in-place using the
Cholesky decomposition of @math{A} into the matrix @var{cholesky} given
by @code{gsl_linalg_cholesky_decomp}. On input @var{x} should contain
the right-hand side @math{b}, which is replaced by the solution on
output.
@end deftypefun

@node Tridiagonal Decomposition of Real Symmetric Matrices
@section Tridiagonal Decomposition of Real Symmetric Matrices
@cindex  tridiagonal decomposition

A symmetric matrix @math{A} can be factorized by similarity
transformations into the form,
@tex
\beforedisplay
$$
A = Q T Q^T
$$
\afterdisplay
@end tex
@ifinfo

@example
A = Q T Q^T
@end example

@end ifinfo
@noindent
where @math{Q} is an orthogonal matrix and @math{T} is a symmetric
tridiagonal matrix.

@deftypefun int gsl_linalg_symmtd_decomp (gsl_matrix * @var{A}, gsl_vector * @var{tau})
This function factorizes the symmetric square matrix @var{A} into the
symmetric tridiagonal decomposition @math{Q T Q^T}.  On output the
diagonal and subdiagonal part of the input matrix @var{A} contain the
tridiagonal matrix @math{T}.  The remaining lower triangular part of the
input matrix contains the Householder vectors which, together with the
Householder coefficients @var{tau}, encode the orthogonal matrix
@math{Q}. This storage scheme is the same as used by @sc{lapack}.  The
upper triangular part of @var{A} is not referenced.
@end deftypefun

@deftypefun int gsl_linalg_symmtd_unpack (const gsl_matrix * @var{A}, const gsl_vector * @var{tau}, gsl_matrix * @var{Q}, gsl_vector * @var{diag}, gsl_vector * @var{subdiag})
This function unpacks the encoded symmetric tridiagonal decomposition
(@var{A}, @var{tau}) obtained from @code{gsl_linalg_symmtd_decomp} into
the orthogonal matrix @var{Q}, the vector of diagonal elements @var{diag}
and the vector of subdiagonal elements @var{subdiag}.  
@end deftypefun

@deftypefun int gsl_linalg_symmtd_unpack_T (const gsl_matrix * @var{A}, gsl_vector * @var{diag}, gsl_vector * @var{subdiag})
This function unpacks the diagonal and subdiagonal of the encoded
symmetric tridiagonal decomposition (@var{A}, @var{tau}) obtained from
@code{gsl_linalg_symmtd_decomp} into the vectors @var{diag} and @var{subdiag}.
@end deftypefun

@node Tridiagonal Decomposition of Hermitian Matrices
@section Tridiagonal Decomposition of Hermitian Matrices
@cindex tridiagonal decomposition

A hermitian matrix @math{A} can be factorized by similarity
transformations into the form,
@tex
\beforedisplay
$$
A = U T U^T
$$
\afterdisplay
@end tex
@ifinfo

@example
A = U T U^T
@end example

@end ifinfo
@noindent
where @math{U} is a unitary matrix and @math{T} is a real symmetric
tridiagonal matrix.


@deftypefun int gsl_linalg_hermtd_decomp (gsl_matrix_complex * @var{A}, gsl_vector_complex * @var{tau})
This function factorizes the hermitian matrix @var{A} into the symmetric
tridiagonal decomposition @math{U T U^T}.  On output the real parts of
the diagonal and subdiagonal part of the input matrix @var{A} contain
the tridiagonal matrix @math{T}.  The remaining lower triangular part of
the input matrix contains the Householder vectors which, together with
the Householder coefficients @var{tau}, encode the orthogonal matrix
@math{Q}. This storage scheme is the same as used by @sc{lapack}.  The
upper triangular part of @var{A} and imaginary parts of the diagonal are
not referenced.
@end deftypefun

@deftypefun int gsl_linalg_hermtd_unpack (const gsl_matrix_complex * @var{A}, const gsl_vector_complex * @var{tau}, gsl_matrix_complex * @var{Q}, gsl_vector * @var{diag}, gsl_vector * @var{subdiag})
This function unpacks the encoded tridiagonal decomposition (@var{A},
@var{tau}) obtained from @code{gsl_linalg_hermtd_decomp} into the
unitary matrix @var{U}, the real vector of diagonal elements @var{diag} and
the real vector of subdiagonal elements @var{subdiag}. 
@end deftypefun

@deftypefun int gsl_linalg_hermtd_unpack_T (const gsl_matrix_complex * @var{A}, gsl_vector * @var{diag}, gsl_vector * @var{subdiag})
This function unpacks the diagonal and subdiagonal of the encoded
tridiagonal decomposition (@var{A}, @var{tau}) obtained from the
@code{gsl_linalg_hermtd_decomp} into the real vectors
@var{diag} and @var{subdiag}.
@end deftypefun

@node Hessenberg Decomposition of Real Matrices
@section Hessenberg Decomposition of Real Matrices
@cindex hessenberg decomposition

A general matrix @math{A} can be decomposed by orthogonal
similarity transformations into the form
@tex
\beforedisplay
$$
A = U H U^T
$$
\afterdisplay
@end tex
@ifinfo

@example
A = U H U^T
@end example

@end ifinfo
where @math{U} is orthogonal and @math{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.

@deftypefun int gsl_linalg_hessenberg (gsl_matrix * @var{A}, gsl_vector * @var{tau})
This function computes the Hessenberg decomposition of the matrix
@var{A} by applying the similarity transformation @math{H = U^T A U}.
On output, @math{H} is stored in the upper portion of @var{A}. The
information required to construct the matrix @math{U} is stored in
the lower triangular portion of @var{A}. @math{U} is a product
of @math{N - 2} Householder matrices. The Householder vectors
are stored in the lower portion of @var{A} (below the subdiagonal)
and the Householder coefficients are stored in the vector @var{tau}.
@var{tau} must be of length @var{N}.
@end deftypefun

@deftypefun int gsl_linalg_hessenberg_unpack (gsl_matrix * @var{H}, gsl_vector * @var{tau}, gsl_matrix * @var{U})
This function constructs the orthogonal matrix @math{U} from the
information stored in the Hessenberg matrix @var{H} along with the
vector @var{tau}. @var{H} and @var{tau} are outputs from
@code{gsl_linalg_hessenberg}.
@end deftypefun

@deftypefun int gsl_linalg_hessenberg_unpack_accum (gsl_matrix * @var{H}, gsl_vector * @var{tau}, gsl_matrix * @var{V})
This function is similar to @code{gsl_linalg_hessenberg_unpack}, except
it accumulates the matrix @var{U} into @var{V}, so that @math{V' = VU}.
The matrix @var{V} must be initialized prior to calling this function.
Setting @var{V} to the identity matrix provides the same result as
@code{gsl_linalg_hessenberg_unpack}. If @var{H} is order @var{N}, then
@var{V} must have @var{N} columns but may have any number of rows.
@end deftypefun

@deftypefun void gsl_linalg_hessenberg_set_zero (gsl_matrix * @var{H})
This function sets the lower triangular portion of @var{H}, below
the subdiagonal, to zero. It is useful for clearing out the
Householder vectors after calling @code{gsl_linalg_hessenberg}.
@end deftypefun

@node Bidiagonalization
@section Bidiagonalization 
@cindex bidiagonalization of real matrices

A general matrix @math{A} can be factorized by similarity
transformations into the form,
@tex
\beforedisplay
$$
A = U B V^T
$$
\afterdisplay
@end tex
@ifinfo

@example
A = U B V^T
@end example

@end ifinfo
@noindent
where @math{U} and @math{V} are orthogonal matrices and @math{B} is a
@math{N}-by-@math{N} bidiagonal matrix with non-zero entries only on the
diagonal and superdiagonal.  The size of @var{U} is @math{M}-by-@math{N}
and the size of @var{V} is @math{N}-by-@math{N}.

@deftypefun int gsl_linalg_bidiag_decomp (gsl_matrix * @var{A}, gsl_vector * @var{tau_U}, gsl_vector * @var{tau_V})
This function factorizes the @math{M}-by-@math{N} matrix @var{A} into
bidiagonal form @math{U B V^T}.  The diagonal and superdiagonal of the
matrix @math{B} are stored in the diagonal and superdiagonal of @var{A}.
The orthogonal matrices @math{U} and @var{V} are stored as compressed
Householder vectors in the remaining elements of @var{A}.  The
Householder coefficients are stored in the vectors @var{tau_U} and
@var{tau_V}.  The length of @var{tau_U} must equal the number of
elements in the diagonal of @var{A} and the length of @var{tau_V} should
be one element shorter.
@end deftypefun

@deftypefun int gsl_linalg_bidiag_unpack (const gsl_matrix * @var{A}, const gsl_vector * @var{tau_U}, gsl_matrix * @var{U}, const gsl_vector * @var{tau_V}, gsl_matrix * @var{V}, gsl_vector * @var{diag}, gsl_vector * @var{superdiag})
This function unpacks the bidiagonal decomposition of @var{A} given by
@code{gsl_linalg_bidiag_decomp}, (@var{A}, @var{tau_U}, @var{tau_V})
into the separate orthogonal matrices @var{U}, @var{V} and the diagonal
vector @var{diag} and superdiagonal @var{superdiag}.  Note that @var{U}
is stored as a compact @math{M}-by-@math{N} orthogonal matrix satisfying
@math{U^T U = I} for efficiency.
@end deftypefun

@deftypefun int gsl_linalg_bidiag_unpack2 (gsl_matrix * @var{A}, gsl_vector * @var{tau_U}, gsl_vector * @var{tau_V}, gsl_matrix * @var{V})
This function unpacks the bidiagonal decomposition of @var{A} given by
@code{gsl_linalg_bidiag_decomp}, (@var{A}, @var{tau_U}, @var{tau_V})
into the separate orthogonal matrices @var{U}, @var{V} and the diagonal
vector @var{diag} and superdiagonal @var{superdiag}.  The matrix @var{U}
is stored in-place in @var{A}.
@end deftypefun

@deftypefun int gsl_linalg_bidiag_unpack_B (const gsl_matrix * @var{A}, gsl_vector * @var{diag}, gsl_vector * @var{superdiag})
This function unpacks the diagonal and superdiagonal of the bidiagonal
decomposition of @var{A} given by @code{gsl_linalg_bidiag_decomp}, into
the diagonal vector @var{diag} and superdiagonal vector @var{superdiag}.
@end deftypefun

@node Householder Transformations
@section Householder Transformations
@cindex Householder matrix
@cindex Householder transformation
@cindex transformation, Householder

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 @math{P} takes the form,
@tex
\beforedisplay
$$
P = I - \tau v v^T
$$
\afterdisplay
@end tex
@ifinfo

@example
P = I - \tau v v^T
@end example

@end ifinfo
@noindent
where @math{v} is a vector (called the @dfn{Householder vector}) and
@math{\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.

@deftypefun double gsl_linalg_householder_transform (gsl_vector * @var{v})
This function prepares a Householder transformation @math{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 @var{v}
and the scalar @math{\tau} is returned.
@end deftypefun

@deftypefun int gsl_linalg_householder_hm (double tau, const gsl_vector * v, gsl_matrix * A)
This function applies the Householder matrix @math{P} defined by the
scalar @var{tau} and the vector @var{v} to the left-hand side of the
matrix @var{A}. On output the result @math{P A} is stored in @var{A}.
@end deftypefun

@deftypefun int gsl_linalg_householder_mh (double tau, const gsl_vector * v, gsl_matrix * A)
This function applies the Householder matrix @math{P} defined by the
scalar @var{tau} and the vector @var{v} to the right-hand side of the
matrix @var{A}. On output the result @math{A P} is stored in @var{A}.
@end deftypefun

@deftypefun int gsl_linalg_householder_hv (double tau, const gsl_vector * v, gsl_vector * w)
This function applies the Householder transformation @math{P} defined by
the scalar @var{tau} and the vector @var{v} to the vector @var{w}.  On
output the result @math{P w} is stored in @var{w}.
@end deftypefun

@comment @deftypefun int gsl_linalg_householder_hm1 (double tau, gsl_matrix * A)
@comment This function applies the Householder transform, defined by the scalar
@comment @var{tau} and the vector @var{v}, to a matrix being build up from the
@comment identity matrix, using the first column of @var{A} as a householder vector.
@comment @end deftypefun

@node Householder solver for linear systems
@section Householder solver for linear systems
@cindex solution of linear system by Householder transformations
@cindex Householder linear solver

@deftypefun int gsl_linalg_HH_solve (gsl_matrix * @var{A}, const gsl_vector * @var{b}, gsl_vector * @var{x})
This function solves the system @math{A x = b} directly using
Householder transformations. On output the solution is stored in @var{x}
and @var{b} is not modified. The matrix @var{A} is destroyed by the
Householder transformations.
@end deftypefun

@deftypefun int gsl_linalg_HH_svx (gsl_matrix * @var{A}, gsl_vector * @var{x})
This function solves the system @math{A x = b} in-place using
Householder transformations.  On input @var{x} should contain the
right-hand side @math{b}, which is replaced by the solution on output.  The
matrix @var{A} is destroyed by the Householder transformations.
@end deftypefun

@node Tridiagonal Systems
@section Tridiagonal Systems
@cindex tridiagonal systems

@deftypefun int gsl_linalg_solve_tridiag (const gsl_vector * @var{diag}, const gsl_vector * @var{e}, const gsl_vector * @var{f}, const gsl_vector * @var{b}, gsl_vector * @var{x})
This function solves the general @math{N}-by-@math{N} system @math{A x =
b} where @var{A} is tridiagonal (@c{$N\geq 2$}
@math{N >= 2}). The super-diagonal and
sub-diagonal vectors @var{e} and @var{f} must be one element shorter
than the diagonal vector @var{diag}.  The form of @var{A} for the 4-by-4
case is shown below,
@tex
\beforedisplay
$$
A = \pmatrix{d_0&e_0&  0& 0\cr
             f_0&d_1&e_1& 0\cr
             0  &f_1&d_2&e_2\cr 
             0  &0  &f_2&d_3\cr}
$$
\afterdisplay
@end tex
@ifinfo

@example
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 )
@end example
@end ifinfo
@end deftypefun

@deftypefun int gsl_linalg_solve_symm_tridiag (const gsl_vector * @var{diag}, const gsl_vector * @var{e}, const gsl_vector * @var{b}, gsl_vector * @var{x})
This function solves the general @math{N}-by-@math{N} system @math{A x =
b} where @var{A} is symmetric tridiagonal (@c{$N\geq 2$}
@math{N >= 2}).  The off-diagonal vector
@var{e} must be one element shorter than the diagonal vector @var{diag}.
The form of @var{A} for the 4-by-4 case is shown below,
@tex
\beforedisplay
$$
A = \pmatrix{d_0&e_0&  0& 0\cr
             e_0&d_1&e_1& 0\cr
             0  &e_1&d_2&e_2\cr 
             0  &0  &e_2&d_3\cr}
$$
\afterdisplay
@end tex
@ifinfo

@example
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 )
@end example
@end ifinfo
The current implementation uses a variant of Cholesky decomposition
which can cause division by zero if the matrix is not positive definite.
@end deftypefun

@deftypefun int gsl_linalg_solve_cyc_tridiag (const gsl_vector * @var{diag}, const gsl_vector * @var{e}, const gsl_vector * @var{f}, const gsl_vector * @var{b}, gsl_vector * @var{x})
This function solves the general @math{N}-by-@math{N} system @math{A x =
b} where @var{A} is cyclic tridiagonal (@c{$N\geq 3$}
@math{N >= 3}).  The cyclic super-diagonal and
sub-diagonal vectors @var{e} and @var{f} must have the same number of
elements as the diagonal vector @var{diag}.  The form of @var{A} for the
4-by-4 case is shown below,
@tex
\beforedisplay
$$
A = \pmatrix{d_0&e_0& 0 &f_3\cr
             f_0&d_1&e_1& 0 \cr
              0 &f_1&d_2&e_2\cr 
             e_3& 0 &f_2&d_3\cr}
$$
\afterdisplay
@end tex
@ifinfo

@example
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 )
@end example
@end ifinfo
@end deftypefun


@deftypefun int gsl_linalg_solve_symm_cyc_tridiag (const gsl_vector * @var{diag}, const gsl_vector * @var{e}, const gsl_vector * @var{b}, gsl_vector * @var{x})
This function solves the general @math{N}-by-@math{N} system @math{A x =
b} where @var{A} is symmetric cyclic tridiagonal (@c{$N\geq 3$}
@math{N >= 3}).  The cyclic
off-diagonal vector @var{e} must have the same number of elements as the
diagonal vector @var{diag}.  The form of @var{A} for the 4-by-4 case is
shown below,
@tex
\beforedisplay
$$
A = \pmatrix{d_0&e_0& 0 &e_3\cr
             e_0&d_1&e_1& 0 \cr
              0 &e_1&d_2&e_2\cr 
             e_3& 0 &e_2&d_3\cr}
$$
\afterdisplay
@end tex
@ifinfo

@example
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 )
@end example
@end ifinfo
@end deftypefun

@node Balancing
@section Balancing
@cindex balancing matrices

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 @math{A} consists
of replacing @math{A} with a similar matrix
@tex
\beforedisplay
$$
A' = D^{-1} A D
$$
\afterdisplay
@end tex
@ifinfo

@example
A' = D^(-1) A D
@end example

@end ifinfo
where @math{D} is a diagonal matrix whose entries are powers
of the floating point radix.

@deftypefun int gsl_linalg_balance_matrix (gsl_matrix * @var{A}, gsl_vector * @var{D})
This function replaces the matrix @var{A} with its balanced counterpart
and stores the diagonal elements of the similarity transformation
into the vector @var{D}.
@end deftypefun

@node Linear Algebra Examples
@section Examples

The following program solves the linear system @math{A x = b}. The
system to be solved is,
@tex
\beforedisplay
$$
\left(
\matrix{0.18& 0.60& 0.57& 0.96\cr
0.41& 0.24& 0.99& 0.58\cr
0.14& 0.30& 0.97& 0.66\cr
0.51& 0.13& 0.19& 0.85}
\right)
\left(
\matrix{x_0\cr
x_1\cr
x_2\cr
x_3}
\right)
=
\left(
\matrix{1.0\cr
2.0\cr
3.0\cr
4.0}
\right)
$$
\afterdisplay
@end tex
@ifinfo

@example
[ 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]
@end example

@end ifinfo
@noindent
and the solution is found using LU decomposition of the matrix @math{A}.

@example
@verbatiminclude examples/linalglu.c
@end example

@noindent
Here is the output from the program,

@example
@verbatiminclude examples/linalglu.out
@end example

@noindent
This can be verified by multiplying the solution @math{x} by the
original matrix @math{A} using @sc{gnu octave},

@example
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
@end example

@noindent
This reproduces the original right-hand side vector, @math{b}, in
accordance with the equation @math{A x = b}.

@node Linear Algebra References and Further Reading
@section References and Further Reading

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

@itemize @asis
@item
G. H. Golub, C. F. Van Loan, @cite{Matrix Computations} (3rd Ed, 1996),
Johns Hopkins University Press, ISBN 0-8018-5414-8.
@end itemize

@noindent
The @sc{lapack} library is described in the following manual,

@itemize @asis
@item
@cite{LAPACK Users' Guide} (Third Edition, 1999), Published by SIAM,
ISBN 0-89871-447-8.

@uref{http://www.netlib.org/lapack} 
@end itemize

@noindent
The @sc{lapack} source code can be found at the website above, along
with an online copy of the users guide.

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

@itemize @asis
@item
T.F. Chan, ``An Improved Algorithm for Computing the Singular Value
Decomposition'', @cite{ACM Transactions on Mathematical Software}, 8
(1982), pp 72--83.
@end itemize

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

@itemize @asis
@item
J.C. Nash, ``A one-sided transformation method for the singular value
decomposition and algebraic eigenproblem'', @cite{Computer Journal},
Volume 18, Number 1 (1973), p 74--76

@item
James Demmel, Kresimir Veselic, ``Jacobi's Method is more accurate than
QR'', @cite{Lapack Working Note 15} (LAWN-15), October 1989. Available
from netlib, @uref{http://www.netlib.org/lapack/} in the @code{lawns} or
@code{lawnspdf} directories.
@end itemize