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 @cindex eigenvalues and eigenvectors
 This chapter describes functions for computing eigenvalues and
 eigenvectors of matrices.  There are routines for real symmetric,
 real nonsymmetric, and complex hermitian matrices. Eigenvalues can
 be computed with or without eigenvectors. The hermitian matrix algorithms
 used are symmetric bidiagonalization followed by QR reduction. The
 nonsymmetric algorithm is the Francis QR double-shift.

 @cindex LAPACK, recommended for linear algebra
 These routines are intended for ``small'' systems where simple algorithms are
 acceptable.  Anyone interested in finding eigenvalues and eigenvectors of
 large matrices will want to use the sophisticated routines found in
 @sc{lapack}. The Fortran version of @sc{lapack} is recommended as the
 standard package for large-scale linear algebra.

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

 @menu
 * Real Symmetric Matrices::
 * Complex Hermitian Matrices::
 * Real Nonsymmetric Matrices::
 * Sorting Eigenvalues and Eigenvectors::
 * Eigenvalue and Eigenvector Examples::
 * Eigenvalue and Eigenvector References::
 @end menu

 @node Real Symmetric Matrices
 @section Real Symmetric Matrices
 @cindex symmetric matrix, real, eigensystem
 @cindex real symmetric matrix, eigensystem

 @deftypefun {gsl_eigen_symm_workspace *} gsl_eigen_symm_alloc (const size_t @var{n})
 This function allocates a workspace for computing eigenvalues of
 @var{n}-by-@var{n} real symmetric matrices.  The size of the workspace
 is @math{O(2n)}.
 @end deftypefun

 @deftypefun void gsl_eigen_symm_free (gsl_eigen_symm_workspace * @var{w})
 This function frees the memory associated with the workspace @var{w}.
 @end deftypefun

 @deftypefun int gsl_eigen_symm (gsl_matrix * @var{A}, gsl_vector * @var{eval}, gsl_eigen_symm_workspace * @var{w})
 This function computes the eigenvalues of the real symmetric matrix
 @var{A}.  Additional workspace of the appropriate size must be provided
 in @var{w}.  The diagonal and lower triangular part of @var{A} are
 destroyed during the computation, but the strict upper triangular part
 is not referenced.  The eigenvalues are stored in the vector @var{eval}
 and are unordered.
 @end deftypefun

 @deftypefun {gsl_eigen_symmv_workspace *} gsl_eigen_symmv_alloc (const size_t @var{n})
 This function allocates a workspace for computing eigenvalues and
 eigenvectors of @var{n}-by-@var{n} real symmetric matrices.  The size of
 the workspace is @math{O(4n)}.
 @end deftypefun

 @deftypefun void gsl_eigen_symmv_free (gsl_eigen_symmv_workspace * @var{w})
 This function frees the memory associated with the workspace @var{w}.
 @end deftypefun

 @deftypefun int gsl_eigen_symmv (gsl_matrix * @var{A}, gsl_vector * @var{eval}, gsl_matrix * @var{evec}, gsl_eigen_symmv_workspace * @var{w})
 This function computes the eigenvalues and eigenvectors of the real
 symmetric matrix @var{A}.  Additional workspace of the appropriate size
 must be provided in @var{w}.  The diagonal and lower triangular part of
 @var{A} are destroyed during the computation, but the strict upper
 triangular part is not referenced.  The eigenvalues are stored in the
 vector @var{eval} and are unordered.  The corresponding eigenvectors are
 stored in the columns of the matrix @var{evec}.  For example, the
 eigenvector in the first column corresponds to the first eigenvalue.
 The eigenvectors are guaranteed to be mutually orthogonal and normalised
 to unit magnitude.
 @end deftypefun

 @node Complex Hermitian Matrices
 @section Complex Hermitian Matrices

 @cindex hermitian matrix, complex, eigensystem
 @cindex complex hermitian matrix, eigensystem

 @deftypefun {gsl_eigen_herm_workspace *} gsl_eigen_herm_alloc (const size_t @var{n})
 This function allocates a workspace for computing eigenvalues of
 @var{n}-by-@var{n} complex hermitian matrices.  The size of the workspace
 is @math{O(3n)}.
 @end deftypefun

 @deftypefun void gsl_eigen_herm_free (gsl_eigen_herm_workspace * @var{w})
 This function frees the memory associated with the workspace @var{w}.
 @end deftypefun

 @deftypefun int gsl_eigen_herm (gsl_matrix_complex * @var{A}, gsl_vector * @var{eval}, gsl_eigen_herm_workspace * @var{w})
 This function computes the eigenvalues of the complex hermitian matrix
 @var{A}.  Additional workspace of the appropriate size must be provided
 in @var{w}.  The diagonal and lower triangular part of @var{A} are
 destroyed during the computation, but the strict upper triangular part
 is not referenced.  The imaginary parts of the diagonal are assumed to be
 zero and are not referenced. The eigenvalues are stored in the vector
 @var{eval} and are unordered.
 @end deftypefun

 @deftypefun {gsl_eigen_hermv_workspace *} gsl_eigen_hermv_alloc (const size_t @var{n})
 This function allocates a workspace for computing eigenvalues and
 eigenvectors of @var{n}-by-@var{n} complex hermitian matrices.  The size of
 the workspace is @math{O(5n)}.
 @end deftypefun

 @deftypefun void gsl_eigen_hermv_free (gsl_eigen_hermv_workspace * @var{w})
 This function frees the memory associated with the workspace @var{w}.
 @end deftypefun

 @deftypefun int gsl_eigen_hermv (gsl_matrix_complex * @var{A}, gsl_vector * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_eigen_hermv_workspace * @var{w})
 This function computes the eigenvalues and eigenvectors of the complex
 hermitian matrix @var{A}.  Additional workspace of the appropriate size
 must be provided in @var{w}.  The diagonal and lower triangular part of
 @var{A} are destroyed during the computation, but the strict upper
 triangular part is not referenced. The imaginary parts of the diagonal
 are assumed to be zero and are not referenced.  The eigenvalues are
 stored in the vector @var{eval} and are unordered.  The corresponding
 complex eigenvectors are stored in the columns of the matrix @var{evec}.
 For example, the eigenvector in the first column corresponds to the
 first eigenvalue.  The eigenvectors are guaranteed to be mutually
 orthogonal and normalised to unit magnitude.
 @end deftypefun

 @node Real Nonsymmetric Matrices
 @section Real Nonsymmetric Matrices
 @cindex nonsymmetric matrix, real, eigensystem
 @cindex real nonsymmetric matrix, eigensystem

 The solution of the real nonsymmetric eigensystem problem for a
 matrix @math{A} involves computing the Schur decomposition
 @tex
 \beforedisplay
 $$
 A = Z T Z^T
 $$
 \afterdisplay
 @end tex
 @ifinfo

 @example
 A = Z T Z^T
 @end example

 @end ifinfo
 where @math{Z} is an orthogonal matrix of Schur vectors and @math{T},
 the Schur form, is quasi upper triangular with diagonal
 @math{1}-by-@math{1} blocks which are real eigenvalues of @math{A}, and
 diagonal @math{2}-by-@math{2} blocks whose eigenvalues are complex
 conjugate eigenvalues of @math{A}. The algorithm used is the double
 shift Francis method.

 @deftypefun {gsl_eigen_nonsymm_workspace *} gsl_eigen_nonsymm_alloc (const size_t @var{n})
 This function allocates a workspace for computing eigenvalues of
 @var{n}-by-@var{n} real nonsymmetric matrices. The size of the workspace
 is @math{O(2n)}.
 @end deftypefun

 @deftypefun void gsl_eigen_nonsymm_free (gsl_eigen_nonsymm_workspace * @var{w})
 This function frees the memory associated with the workspace @var{w}.
 @end deftypefun

 @deftypefun void gsl_eigen_nonsymm_params (const int @var{compute_t}, const int @var{balance}, gsl_eigen_nonsymm_workspace * @var{w})
 This function sets some parameters which determine how the eigenvalue
 problem is solved in subsequent calls to @code{gsl_eigen_nonsymm}.

 If @var{compute_t} is set to 1, the full Schur form @math{T} will be
 computed by @code{gsl_eigen_nonsymm}. If it is set to 0,
 @math{T} will not be computed (this is the default setting). Computing
 the full Schur form @math{T} requires approximately 1.5-2 times the
 number of flops.

 If @var{balance} is set to 1, a balancing transformation is applied
 to the matrix prior to computing eigenvalues. This transformation is
 designed to make the rows and columns of the matrix have comparable
 norms, and can result in more accurate eigenvalues for matrices
 whose entries vary widely in magnitude. See @ref{Balancing} for more
 information. Note that the balancing transformation does not preserve
 the orthogonality of the Schur vectors, so if you wish to compute the
 Schur vectors with @code{gsl_eigen_nonsymm_Z} you will obtain the Schur
 vectors of the balanced matrix instead of the original matrix. The
 relationship will be
 @tex
 \beforedisplay
 $$
 T = Q^t D^{-1} A D Q
 $$
 \afterdisplay
 @end tex
 @ifinfo

 @example
 T = Q^t D^(-1) A D Q
 @end example

 @end ifinfo
 @noindent
 where @var{Q} is the matrix of Schur vectors for the balanced matrix, and
 @var{D} is the balancing transformation. Then @code{gsl_eigen_nonsymm_Z}
 will compute a matrix @var{Z} which satisfies
 @tex
 \beforedisplay
 $$
 T = Z^{-1} A Z
 $$
 \afterdisplay
 @end tex
 @ifinfo

 @example
 T = Z^(-1) A Z
 @end example

 @end ifinfo
 @noindent
 with @math{Z = D Q}. Note that @var{Z} will not be orthogonal. For
 this reason, balancing is not performed by default.
 @end deftypefun

 @deftypefun int gsl_eigen_nonsymm (gsl_matrix * @var{A}, gsl_vector_complex * @var{eval}, gsl_eigen_nonsymm_workspace * @var{w})
 This function computes the eigenvalues of the real nonsymmetric matrix
 @var{A} and stores them in the vector @var{eval}. If @math{T} is
 desired, it is stored in the upper portion of @var{A} on output.
 Otherwise, on output, the diagonal of @var{A} will contain the
 @math{1}-by-@math{1} real eigenvalues and @math{2}-by-@math{2}
 complex conjugate eigenvalue systems, and the rest of @var{A} is
 destroyed. In rare cases, this function will fail to find all
 eigenvalues. If this happens, an error code is returned
 and the number of converged eigenvalues is stored in @code{w->n_evals}.
 The converged eigenvalues are stored in the beginning of @var{eval}.
 @end deftypefun

 @deftypefun int gsl_eigen_nonsymm_Z (gsl_matrix * @var{A}, gsl_vector_complex * @var{eval}, gsl_matrix * @var{Z}, gsl_eigen_nonsymm_workspace * @var{w})
 This function is identical to @code{gsl_eigen_nonsymm} except it also
 computes the Schur vectors and stores them into @var{Z}.
 @end deftypefun

 @deftypefun {gsl_eigen_nonsymmv_workspace *} gsl_eigen_nonsymmv_alloc (const size_t @var{n})
 This function allocates a workspace for computing eigenvalues and
 eigenvectors of @var{n}-by-@var{n} real nonsymmetric matrices. The
 size of the workspace is @math{O(5n)}.
 @end deftypefun

 @deftypefun void gsl_eigen_nonsymmv_free (gsl_eigen_nonsymmv_workspace * @var{w})
 This function frees the memory associated with the workspace @var{w}.
 @end deftypefun

 @deftypefun int gsl_eigen_nonsymmv (gsl_matrix * @var{A}, gsl_vector_complex * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_eigen_nonsymmv_workspace * @var{w})
 This function computes eigenvalues and right eigenvectors of the
 @var{n}-by-@var{n} real nonsymmetric matrix @var{A}. It first calls
 @code{gsl_eigen_nonsymm} to compute the eigenvalues, Schur form @math{T}, and
 Schur vectors. Then it finds eigenvectors of @math{T} and backtransforms
 them using the Schur vectors. The Schur vectors are destroyed in the
 process, but can be saved by using @code{gsl_eigen_nonsymmv_Z}. The
 computed eigenvectors are normalized to have Euclidean norm 1. On
 output, the upper portion of @var{A} contains the Schur form
 @math{T}. If @code{gsl_eigen_nonsymm} fails, no eigenvectors are
 computed, and an error code is returned.
 @end deftypefun

 @deftypefun int gsl_eigen_nonsymmv_Z (gsl_matrix * @var{A}, gsl_vector_complex * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_matrix * @var{Z}, gsl_eigen_nonsymmv_workspace * @var{w})
 This function is identical to @code{gsl_eigen_nonsymmv} except it also saves
 the Schur vectors into @var{Z}.
 @end deftypefun

 @node Sorting Eigenvalues and Eigenvectors
 @section Sorting Eigenvalues and Eigenvectors
 @cindex sorting eigenvalues and eigenvectors

 @deftypefun int gsl_eigen_symmv_sort (gsl_vector * @var{eval}, gsl_matrix * @var{evec}, gsl_eigen_sort_t @var{sort_type})
 This function simultaneously sorts the eigenvalues stored in the vector
 @var{eval} and the corresponding real eigenvectors stored in the columns
 of the matrix @var{evec} into ascending or descending order according to
 the value of the parameter @var{sort_type},

 @table @code
 @item GSL_EIGEN_SORT_VAL_ASC
 ascending order in numerical value
 @item GSL_EIGEN_SORT_VAL_DESC
 descending order in numerical value
 @item GSL_EIGEN_SORT_ABS_ASC
 ascending order in magnitude
 @item GSL_EIGEN_SORT_ABS_DESC
 descending order in magnitude
 @end table

 @end deftypefun

 @deftypefun int gsl_eigen_hermv_sort (gsl_vector * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_eigen_sort_t @var{sort_type})
 This function simultaneously sorts the eigenvalues stored in the vector
 @var{eval} and the corresponding complex eigenvectors stored in the
 columns of the matrix @var{evec} into ascending or descending order
 according to the value of the parameter @var{sort_type} as shown above.
 @end deftypefun

 @deftypefun int gsl_eigen_nonsymmv_sort (gsl_vector_complex * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_eigen_sort_t @var{sort_type})
 This function simultaneously sorts the eigenvalues stored in the vector
 @var{eval} and the corresponding complex eigenvectors stored in the
 columns of the matrix @var{evec} into ascending or descending order
 according to the value of the parameter @var{sort_type} as shown above.
 Only GSL_EIGEN_SORT_ABS_ASC and GSL_EIGEN_SORT_ABS_DESC are supported
 due to the eigenvalues being complex.
 @end deftypefun


 @comment @deftypefun int gsl_eigen_jacobi (gsl_matrix * @var{matrix}, gsl_vector * @var{eval}, gsl_matrix * @var{evec}, unsigned int @var{max_rot}, unsigned int * @var{nrot})
 @comment This function finds the eigenvectors and eigenvalues of a real symmetric
 @comment matrix by Jacobi iteration. The data in the input matrix is destroyed.
 @comment @end deftypefun

 @comment @deftypefun int gsl_la_invert_jacobi (const gsl_matrix * @var{matrix}, gsl_matrix * @var{ainv}, unsigned int @var{max_rot})
 @comment Invert a matrix by Jacobi iteration.
 @comment @end deftypefun

 @comment @deftypefun int gsl_eigen_sort (gsl_vector * @var{eval}, gsl_matrix * @var{evec}, gsl_eigen_sort_t @var{sort_type})
 @comment This functions sorts the eigensystem results based on eigenvalues.
 @comment Sorts in order of increasing value or increasing
 @comment absolute value, depending on the value of
 @comment @var{sort_type}, which can be @code{GSL_EIGEN_SORT_VALUE}
 @comment or @code{GSL_EIGEN_SORT_ABSVALUE}.
 @comment @end deftypefun

 @node Eigenvalue and Eigenvector Examples
 @section Examples

 The following program computes the eigenvalues and eigenvectors of the 4-th order Hilbert matrix, @math{H(i,j) = 1/(i + j + 1)}.

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

 @noindent
 Here is the beginning of the output from the program,

 @example
 $ ./a.out
 eigenvalue = 9.67023e-05
 eigenvector =
 -0.0291933
 0.328712
 -0.791411
 0.514553
 ...
 @end example

 @noindent
 This can be compared with the corresponding output from @sc{gnu octave},

 @example
 octave> [v,d] = eig(hilb(4));
 octave> diag(d)
 ans =

    9.6702e-05
    6.7383e-03
    1.6914e-01
    1.5002e+00

 octave> v
 v =

    0.029193   0.179186  -0.582076   0.792608
   -0.328712  -0.741918   0.370502   0.451923
    0.791411   0.100228   0.509579   0.322416
   -0.514553   0.638283   0.514048   0.252161
 @end example

 @noindent
 Note that the eigenvectors can differ by a change of sign, since the
 sign of an eigenvector is arbitrary.

 The following program illustrates the use of the nonsymmetric
 eigensolver, by computing the eigenvalues and eigenvectors of
 the Vandermonde matrix @math{V(x;i,j) = x_i^{n - j}} with
 @math{x = (-1,-2,3,4)}.

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

 @noindent
 Here is the beginning of the output from the program,

 @example
 $ ./a.out
 eigenvalue = -6.41391 + 0i
 eigenvector =
 -0.0998822 + 0i
 -0.111251 + 0i
 0.292501 + 0i
 0.944505 + 0i
 eigenvalue = 5.54555 + 3.08545i
 eigenvector =
 -0.043487 + -0.0076308i
 0.0642377 + -0.142127i
 -0.515253 + 0.0405118i
 -0.840592 + -0.00148565i
 ...
 @end example

 @noindent
 This can be compared with the corresponding output from @sc{gnu octave},

 @example
 octave> [v,d] = eig(vander([-1 -2 3 4]));
 octave> diag(d)
 ans =

   -6.4139 + 0.0000i
    5.5456 + 3.0854i
    5.5456 - 3.0854i
    2.3228 + 0.0000i

 octave> v
 v =

  Columns 1 through 3:

   -0.09988 + 0.00000i  -0.04350 - 0.00755i  -0.04350 + 0.00755i
   -0.11125 + 0.00000i   0.06399 - 0.14224i   0.06399 + 0.14224i
    0.29250 + 0.00000i  -0.51518 + 0.04142i  -0.51518 - 0.04142i
    0.94451 + 0.00000i  -0.84059 + 0.00000i  -0.84059 - 0.00000i

  Column 4:

   -0.14493 + 0.00000i
    0.35660 + 0.00000i
    0.91937 + 0.00000i
    0.08118 + 0.00000i

 @end example
 Note that the eigenvectors corresponding to the eigenvalue
 @math{5.54555 + 3.08545i} are slightly different. This is because
 they differ by the multiplicative constant
 @math{0.9999984 + 0.0017674i} which has magnitude 1.

 @node Eigenvalue and Eigenvector References
 @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,

 @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.
	@cindex eigenvalues and eigenvectors
	This chapter describes functions for computing eigenvalues and
	eigenvectors of matrices. There are routines for real symmetric,
	real nonsymmetric, and complex hermitian matrices. Eigenvalues can
	be computed with or without eigenvectors. The hermitian matrix algorithms
	used are symmetric bidiagonalization followed by QR reduction. The
	nonsymmetric algorithm is the Francis QR double-shift.

	@cindex LAPACK, recommended for linear algebra
	These routines are intended for ``small'' systems where simple algorithms are
	acceptable. Anyone interested in finding eigenvalues and eigenvectors of
	large matrices will want to use the sophisticated routines found in
	@sc{lapack}. The Fortran version of @sc{lapack} is recommended as the
	standard package for large-scale linear algebra.

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

	@menu
	* Real Symmetric Matrices::
	* Complex Hermitian Matrices::
	* Real Nonsymmetric Matrices::
	* Sorting Eigenvalues and Eigenvectors::
	* Eigenvalue and Eigenvector Examples::
	* Eigenvalue and Eigenvector References::
	@end menu

	@node Real Symmetric Matrices
	@section Real Symmetric Matrices
	@cindex symmetric matrix, real, eigensystem
	@cindex real symmetric matrix, eigensystem

	@deftypefun {gsl_eigen_symm_workspace *} gsl_eigen_symm_alloc (const size_t @var{n})
	This function allocates a workspace for computing eigenvalues of
	@var{n}-by-@var{n} real symmetric matrices. The size of the workspace
	is @math{O(2n)}.
	@end deftypefun

	@deftypefun void gsl_eigen_symm_free (gsl_eigen_symm_workspace * @var{w})
	This function frees the memory associated with the workspace @var{w}.
	@end deftypefun

	@deftypefun int gsl_eigen_symm (gsl_matrix * @var{A}, gsl_vector * @var{eval}, gsl_eigen_symm_workspace * @var{w})
	This function computes the eigenvalues of the real symmetric matrix
	@var{A}. Additional workspace of the appropriate size must be provided
	in @var{w}. The diagonal and lower triangular part of @var{A} are
	destroyed during the computation, but the strict upper triangular part
	is not referenced. The eigenvalues are stored in the vector @var{eval}
	and are unordered.
	@end deftypefun

	@deftypefun {gsl_eigen_symmv_workspace *} gsl_eigen_symmv_alloc (const size_t @var{n})
	This function allocates a workspace for computing eigenvalues and
	eigenvectors of @var{n}-by-@var{n} real symmetric matrices. The size of
	the workspace is @math{O(4n)}.
	@end deftypefun

	@deftypefun void gsl_eigen_symmv_free (gsl_eigen_symmv_workspace * @var{w})
	This function frees the memory associated with the workspace @var{w}.
	@end deftypefun

	@deftypefun int gsl_eigen_symmv (gsl_matrix * @var{A}, gsl_vector * @var{eval}, gsl_matrix * @var{evec}, gsl_eigen_symmv_workspace * @var{w})
	This function computes the eigenvalues and eigenvectors of the real
	symmetric matrix @var{A}. Additional workspace of the appropriate size
	must be provided in @var{w}. The diagonal and lower triangular part of
	@var{A} are destroyed during the computation, but the strict upper
	triangular part is not referenced. The eigenvalues are stored in the
	vector @var{eval} and are unordered. The corresponding eigenvectors are
	stored in the columns of the matrix @var{evec}. For example, the
	eigenvector in the first column corresponds to the first eigenvalue.
	The eigenvectors are guaranteed to be mutually orthogonal and normalised
	to unit magnitude.
	@end deftypefun

	@node Complex Hermitian Matrices
	@section Complex Hermitian Matrices

	@cindex hermitian matrix, complex, eigensystem
	@cindex complex hermitian matrix, eigensystem

	@deftypefun {gsl_eigen_herm_workspace *} gsl_eigen_herm_alloc (const size_t @var{n})
	This function allocates a workspace for computing eigenvalues of
	@var{n}-by-@var{n} complex hermitian matrices. The size of the workspace
	is @math{O(3n)}.
	@end deftypefun

	@deftypefun void gsl_eigen_herm_free (gsl_eigen_herm_workspace * @var{w})
	This function frees the memory associated with the workspace @var{w}.
	@end deftypefun

	@deftypefun int gsl_eigen_herm (gsl_matrix_complex * @var{A}, gsl_vector * @var{eval}, gsl_eigen_herm_workspace * @var{w})
	This function computes the eigenvalues of the complex hermitian matrix
	@var{A}. Additional workspace of the appropriate size must be provided
	in @var{w}. The diagonal and lower triangular part of @var{A} are
	destroyed during the computation, but the strict upper triangular part
	is not referenced. The imaginary parts of the diagonal are assumed to be
	zero and are not referenced. The eigenvalues are stored in the vector
	@var{eval} and are unordered.
	@end deftypefun

	@deftypefun {gsl_eigen_hermv_workspace *} gsl_eigen_hermv_alloc (const size_t @var{n})
	This function allocates a workspace for computing eigenvalues and
	eigenvectors of @var{n}-by-@var{n} complex hermitian matrices. The size of
	the workspace is @math{O(5n)}.
	@end deftypefun

	@deftypefun void gsl_eigen_hermv_free (gsl_eigen_hermv_workspace * @var{w})
	This function frees the memory associated with the workspace @var{w}.
	@end deftypefun

	@deftypefun int gsl_eigen_hermv (gsl_matrix_complex * @var{A}, gsl_vector * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_eigen_hermv_workspace * @var{w})
	This function computes the eigenvalues and eigenvectors of the complex
	hermitian matrix @var{A}. Additional workspace of the appropriate size
	must be provided in @var{w}. The diagonal and lower triangular part of
	@var{A} are destroyed during the computation, but the strict upper
	triangular part is not referenced. The imaginary parts of the diagonal
	are assumed to be zero and are not referenced. The eigenvalues are
	stored in the vector @var{eval} and are unordered. The corresponding
	complex eigenvectors are stored in the columns of the matrix @var{evec}.
	For example, the eigenvector in the first column corresponds to the
	first eigenvalue. The eigenvectors are guaranteed to be mutually
	orthogonal and normalised to unit magnitude.
	@end deftypefun

	@node Real Nonsymmetric Matrices
	@section Real Nonsymmetric Matrices
	@cindex nonsymmetric matrix, real, eigensystem
	@cindex real nonsymmetric matrix, eigensystem

	The solution of the real nonsymmetric eigensystem problem for a
	matrix @math{A} involves computing the Schur decomposition
	@tex
	\beforedisplay
	$$
	A = Z T Z^T
	$$
	\afterdisplay
	@end tex
	@ifinfo

	@example
	A = Z T Z^T
	@end example

	@end ifinfo
	where @math{Z} is an orthogonal matrix of Schur vectors and @math{T},
	the Schur form, is quasi upper triangular with diagonal
	@math{1}-by-@math{1} blocks which are real eigenvalues of @math{A}, and
	diagonal @math{2}-by-@math{2} blocks whose eigenvalues are complex
	conjugate eigenvalues of @math{A}. The algorithm used is the double
	shift Francis method.

	@deftypefun {gsl_eigen_nonsymm_workspace *} gsl_eigen_nonsymm_alloc (const size_t @var{n})
	This function allocates a workspace for computing eigenvalues of
	@var{n}-by-@var{n} real nonsymmetric matrices. The size of the workspace
	is @math{O(2n)}.
	@end deftypefun

	@deftypefun void gsl_eigen_nonsymm_free (gsl_eigen_nonsymm_workspace * @var{w})
	This function frees the memory associated with the workspace @var{w}.
	@end deftypefun

	@deftypefun void gsl_eigen_nonsymm_params (const int @var{compute_t}, const int @var{balance}, gsl_eigen_nonsymm_workspace * @var{w})
	This function sets some parameters which determine how the eigenvalue
	problem is solved in subsequent calls to @code{gsl_eigen_nonsymm}.

	If @var{compute_t} is set to 1, the full Schur form @math{T} will be
	computed by @code{gsl_eigen_nonsymm}. If it is set to 0,
	@math{T} will not be computed (this is the default setting). Computing
	the full Schur form @math{T} requires approximately 1.5-2 times the
	number of flops.

	If @var{balance} is set to 1, a balancing transformation is applied
	to the matrix prior to computing eigenvalues. This transformation is
	designed to make the rows and columns of the matrix have comparable
	norms, and can result in more accurate eigenvalues for matrices
	whose entries vary widely in magnitude. See @ref{Balancing} for more
	information. Note that the balancing transformation does not preserve
	the orthogonality of the Schur vectors, so if you wish to compute the
	Schur vectors with @code{gsl_eigen_nonsymm_Z} you will obtain the Schur
	vectors of the balanced matrix instead of the original matrix. The
	relationship will be
	@tex
	\beforedisplay
	$$
	T = Q^t D^{-1} A D Q
	$$
	\afterdisplay
	@end tex
	@ifinfo

	@example
	T = Q^t D^(-1) A D Q
	@end example

	@end ifinfo
	@noindent
	where @var{Q} is the matrix of Schur vectors for the balanced matrix, and
	@var{D} is the balancing transformation. Then @code{gsl_eigen_nonsymm_Z}
	will compute a matrix @var{Z} which satisfies
	@tex
	\beforedisplay
	$$
	T = Z^{-1} A Z
	$$
	\afterdisplay
	@end tex
	@ifinfo

	@example
	T = Z^(-1) A Z
	@end example

	@end ifinfo
	@noindent
	with @math{Z = D Q}. Note that @var{Z} will not be orthogonal. For
	this reason, balancing is not performed by default.
	@end deftypefun

	@deftypefun int gsl_eigen_nonsymm (gsl_matrix * @var{A}, gsl_vector_complex * @var{eval}, gsl_eigen_nonsymm_workspace * @var{w})
	This function computes the eigenvalues of the real nonsymmetric matrix
	@var{A} and stores them in the vector @var{eval}. If @math{T} is
	desired, it is stored in the upper portion of @var{A} on output.
	Otherwise, on output, the diagonal of @var{A} will contain the
	@math{1}-by-@math{1} real eigenvalues and @math{2}-by-@math{2}
	complex conjugate eigenvalue systems, and the rest of @var{A} is
	destroyed. In rare cases, this function will fail to find all
	eigenvalues. If this happens, an error code is returned
	and the number of converged eigenvalues is stored in @code{w->n_evals}.
	The converged eigenvalues are stored in the beginning of @var{eval}.
	@end deftypefun

	@deftypefun int gsl_eigen_nonsymm_Z (gsl_matrix * @var{A}, gsl_vector_complex * @var{eval}, gsl_matrix * @var{Z}, gsl_eigen_nonsymm_workspace * @var{w})
	This function is identical to @code{gsl_eigen_nonsymm} except it also
	computes the Schur vectors and stores them into @var{Z}.
	@end deftypefun

	@deftypefun {gsl_eigen_nonsymmv_workspace *} gsl_eigen_nonsymmv_alloc (const size_t @var{n})
	This function allocates a workspace for computing eigenvalues and
	eigenvectors of @var{n}-by-@var{n} real nonsymmetric matrices. The
	size of the workspace is @math{O(5n)}.
	@end deftypefun

	@deftypefun void gsl_eigen_nonsymmv_free (gsl_eigen_nonsymmv_workspace * @var{w})
	This function frees the memory associated with the workspace @var{w}.
	@end deftypefun

	@deftypefun int gsl_eigen_nonsymmv (gsl_matrix * @var{A}, gsl_vector_complex * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_eigen_nonsymmv_workspace * @var{w})
	This function computes eigenvalues and right eigenvectors of the
	@var{n}-by-@var{n} real nonsymmetric matrix @var{A}. It first calls
	@code{gsl_eigen_nonsymm} to compute the eigenvalues, Schur form @math{T}, and
	Schur vectors. Then it finds eigenvectors of @math{T} and backtransforms
	them using the Schur vectors. The Schur vectors are destroyed in the
	process, but can be saved by using @code{gsl_eigen_nonsymmv_Z}. The
	computed eigenvectors are normalized to have Euclidean norm 1. On
	output, the upper portion of @var{A} contains the Schur form
	@math{T}. If @code{gsl_eigen_nonsymm} fails, no eigenvectors are
	computed, and an error code is returned.
	@end deftypefun

	@deftypefun int gsl_eigen_nonsymmv_Z (gsl_matrix * @var{A}, gsl_vector_complex * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_matrix * @var{Z}, gsl_eigen_nonsymmv_workspace * @var{w})
	This function is identical to @code{gsl_eigen_nonsymmv} except it also saves
	the Schur vectors into @var{Z}.
	@end deftypefun

	@node Sorting Eigenvalues and Eigenvectors
	@section Sorting Eigenvalues and Eigenvectors
	@cindex sorting eigenvalues and eigenvectors

	@deftypefun int gsl_eigen_symmv_sort (gsl_vector * @var{eval}, gsl_matrix * @var{evec}, gsl_eigen_sort_t @var{sort_type})
	This function simultaneously sorts the eigenvalues stored in the vector
	@var{eval} and the corresponding real eigenvectors stored in the columns
	of the matrix @var{evec} into ascending or descending order according to
	the value of the parameter @var{sort_type},

	@table @code
	@item GSL_EIGEN_SORT_VAL_ASC
	ascending order in numerical value
	@item GSL_EIGEN_SORT_VAL_DESC
	descending order in numerical value
	@item GSL_EIGEN_SORT_ABS_ASC
	ascending order in magnitude
	@item GSL_EIGEN_SORT_ABS_DESC
	descending order in magnitude
	@end table

	@end deftypefun

	@deftypefun int gsl_eigen_hermv_sort (gsl_vector * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_eigen_sort_t @var{sort_type})
	This function simultaneously sorts the eigenvalues stored in the vector
	@var{eval} and the corresponding complex eigenvectors stored in the
	columns of the matrix @var{evec} into ascending or descending order
	according to the value of the parameter @var{sort_type} as shown above.
	@end deftypefun

	@deftypefun int gsl_eigen_nonsymmv_sort (gsl_vector_complex * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_eigen_sort_t @var{sort_type})
	This function simultaneously sorts the eigenvalues stored in the vector
	@var{eval} and the corresponding complex eigenvectors stored in the
	columns of the matrix @var{evec} into ascending or descending order
	according to the value of the parameter @var{sort_type} as shown above.
	Only GSL_EIGEN_SORT_ABS_ASC and GSL_EIGEN_SORT_ABS_DESC are supported
	due to the eigenvalues being complex.
	@end deftypefun


	@comment @deftypefun int gsl_eigen_jacobi (gsl_matrix * @var{matrix}, gsl_vector * @var{eval}, gsl_matrix * @var{evec}, unsigned int @var{max_rot}, unsigned int * @var{nrot})
	@comment This function finds the eigenvectors and eigenvalues of a real symmetric
	@comment matrix by Jacobi iteration. The data in the input matrix is destroyed.
	@comment @end deftypefun

	@comment @deftypefun int gsl_la_invert_jacobi (const gsl_matrix * @var{matrix}, gsl_matrix * @var{ainv}, unsigned int @var{max_rot})
	@comment Invert a matrix by Jacobi iteration.
	@comment @end deftypefun

	@comment @deftypefun int gsl_eigen_sort (gsl_vector * @var{eval}, gsl_matrix * @var{evec}, gsl_eigen_sort_t @var{sort_type})
	@comment This functions sorts the eigensystem results based on eigenvalues.
	@comment Sorts in order of increasing value or increasing
	@comment absolute value, depending on the value of
	@comment @var{sort_type}, which can be @code{GSL_EIGEN_SORT_VALUE}
	@comment or @code{GSL_EIGEN_SORT_ABSVALUE}.
	@comment @end deftypefun

	@node Eigenvalue and Eigenvector Examples
	@section Examples

	The following program computes the eigenvalues and eigenvectors of the 4-th order Hilbert matrix, @math{H(i,j) = 1/(i + j + 1)}.

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

	@noindent
	Here is the beginning of the output from the program,

	@example
	$ ./a.out
	eigenvalue = 9.67023e-05
	eigenvector =
	-0.0291933
	0.328712
	-0.791411
	0.514553
	...
	@end example

	@noindent
	This can be compared with the corresponding output from @sc{gnu octave},

	@example
	octave> [v,d] = eig(hilb(4));
	octave> diag(d)
	ans =

	9.6702e-05
	6.7383e-03
	1.6914e-01
	1.5002e+00

	octave> v
	v =

	0.029193 0.179186 -0.582076 0.792608
	-0.328712 -0.741918 0.370502 0.451923
	0.791411 0.100228 0.509579 0.322416
	-0.514553 0.638283 0.514048 0.252161
	@end example

	@noindent
	Note that the eigenvectors can differ by a change of sign, since the
	sign of an eigenvector is arbitrary.

	The following program illustrates the use of the nonsymmetric
	eigensolver, by computing the eigenvalues and eigenvectors of
	the Vandermonde matrix @math{V(x;i,j) = x_i^{n - j}} with
	@math{x = (-1,-2,3,4)}.

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

	@noindent
	Here is the beginning of the output from the program,

	@example
	$ ./a.out
	eigenvalue = -6.41391 + 0i
	eigenvector =
	-0.0998822 + 0i
	-0.111251 + 0i
	0.292501 + 0i
	0.944505 + 0i
	eigenvalue = 5.54555 + 3.08545i
	eigenvector =
	-0.043487 + -0.0076308i
	0.0642377 + -0.142127i
	-0.515253 + 0.0405118i
	-0.840592 + -0.00148565i
	...
	@end example

	@noindent
	This can be compared with the corresponding output from @sc{gnu octave},

	@example
	octave> [v,d] = eig(vander([-1 -2 3 4]));
	octave> diag(d)
	ans =

	-6.4139 + 0.0000i
	5.5456 + 3.0854i
	5.5456 - 3.0854i
	2.3228 + 0.0000i

	octave> v
	v =

	Columns 1 through 3:

	-0.09988 + 0.00000i -0.04350 - 0.00755i -0.04350 + 0.00755i
	-0.11125 + 0.00000i 0.06399 - 0.14224i 0.06399 + 0.14224i
	0.29250 + 0.00000i -0.51518 + 0.04142i -0.51518 - 0.04142i
	0.94451 + 0.00000i -0.84059 + 0.00000i -0.84059 - 0.00000i

	Column 4:

	-0.14493 + 0.00000i
	0.35660 + 0.00000i
	0.91937 + 0.00000i
	0.08118 + 0.00000i

	@end example
	Note that the eigenvectors corresponding to the eigenvalue
	@math{5.54555 + 3.08545i} are slightly different. This is because
	they differ by the multiplicative constant
	@math{0.9999984 + 0.0017674i} which has magnitude 1.

	@node Eigenvalue and Eigenvector References
	@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,

	@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.