linalg.texi

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(@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 real 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_decomp (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_decomp}.
@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 int 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_decomp}.
@end deftypefun

@node Hessenberg-Triangular Decomposition of Real Matrices
@section Hessenberg-Triangular Decomposition of Real Matrices
@cindex hessenberg triangular decomposition

A general real matrix pair (@math{A}, @math{B}) can be decomposed by
orthogonal similarity transformations into the form
@tex
\beforedisplay
$$
A = U H V^T
$$
$$
B = U R V^T
$$
\afterdisplay
@end tex
@ifinfo

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

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

@deftypefun int gsl_linalg_hesstri_decomp (gsl_matrix * @var{A}, gsl_matrix * @var{B}, gsl_matrix * @var{U}, gsl_matrix * @var{V}, gsl_vector * @var{work})
This function computes the Hessenberg-Triangular decomposition of the
matrix pair (@var{A}, @var{B}). On output, @math{H} is stored in @var{A},
and @math{R} is stored in @var{B}. If @var{U} and @var{V} are provided
(they may be null), the similarity transformations are stored in them.
Additional workspace of length @math{N} is needed in @var{work}.
@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} produced 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} produced 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} from @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})
@deftypefunx gsl_complex gsl_linalg_complex_householder_transform (gsl_vector_complex * @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)
@deftypefunx int gsl_linalg_complex_householder_hm (gsl_complex tau, const gsl_vector_complex * v, gsl_matrix_complex * 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)
@deftypefunx int gsl_linalg_complex_householder_mh (gsl_complex tau, const gsl_vector_complex * v, gsl_matrix_complex * 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)
@deftypefunx int gsl_linalg_complex_householder_hv (gsl_complex tau, const gsl_vector_complex * v, gsl_vector_complex * 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.