eigen.texi

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error code is returned.
@end deftypefun

@deftypefun int gsl_eigen_gen_QZ (gsl_matrix * @var{A}, gsl_matrix * @var{B}, gsl_vector_complex * @var{alpha}, gsl_vector * @var{beta}, gsl_matrix * @var{Q}, gsl_matrix * @var{Z}, gsl_eigen_gen_workspace * @var{w})
This function is identical to @code{gsl_eigen_gen} except it also
computes the left and right Schur vectors and stores them into @var{Q}
and @var{Z} respectively.
@end deftypefun

@deftypefun {gsl_eigen_genv_workspace *} gsl_eigen_genv_alloc (const size_t @var{n})
This function allocates a workspace for computing eigenvalues and
eigenvectors of @var{n}-by-@var{n} real generalized nonsymmetric
eigensystems. The size of the workspace is @math{O(7n)}.
@end deftypefun

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

@deftypefun int gsl_eigen_genv (gsl_matrix * @var{A}, gsl_matrix * @var{B}, gsl_vector_complex * @var{alpha}, gsl_vector * @var{beta}, gsl_matrix_complex * @var{evec}, gsl_eigen_genv_workspace * @var{w})
This function computes eigenvalues and right eigenvectors of the
@var{n}-by-@var{n} real generalized nonsymmetric matrix pair
(@var{A}, @var{B}). The eigenvalues are stored in (@var{alpha}, @var{beta})
and the eigenvectors are stored in @var{evec}. It first calls
@code{gsl_eigen_gen} to compute the eigenvalues, Schur forms, and
Schur vectors. Then it finds eigenvectors of the Schur forms and
backtransforms them using the Schur vectors. The Schur vectors are
destroyed in the process, but can be saved by using
@code{gsl_eigen_genv_QZ}. The computed eigenvectors are normalized
to have unit magnitude. On output, (@var{A}, @var{B}) contains
the generalized Schur form (@math{S}, @math{T}). If @code{gsl_eigen_gen}
fails, no eigenvectors are computed, and an error code is returned.
@end deftypefun

@deftypefun int gsl_eigen_genv_QZ (gsl_matrix * @var{A}, gsl_matrix * @var{B}, gsl_vector_complex * @var{alpha}, gsl_vector * @var{beta}, gsl_matrix_complex * @var{evec}, gsl_matrix * @var{Q}, gsl_matrix * @var{Z}, gsl_eigen_genv_workspace * @var{w})
This function is identical to @code{gsl_eigen_genv} except it also
computes the left and right Schur vectors and stores them into @var{Q}
and @var{Z} respectively.
@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 @code{GSL_EIGEN_SORT_ABS_ASC} and @code{GSL_EIGEN_SORT_ABS_DESC} are
supported due to the eigenvalues being complex.
@end deftypefun

@deftypefun int gsl_eigen_gensymmv_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} as shown above.
@end deftypefun

@deftypefun int gsl_eigen_genhermv_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_genv_sort (gsl_vector_complex * @var{alpha}, gsl_vector * @var{beta}, gsl_matrix_complex * @var{evec}, gsl_eigen_sort_t @var{sort_type})
This function simultaneously sorts the eigenvalues stored in the vectors
(@var{alpha}, @var{beta}) 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 @code{GSL_EIGEN_SORT_ABS_ASC} and
@code{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
@c{$V(x;i,j) = x_i^{n - j}$}
@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
Further information on the generalized eigensystems QZ algorithm
can be found in this paper,

@itemize @asis
@item
C. Moler, G. Stewart, "An Algorithm for Generalized Matrix Eigenvalue
Problems," SIAM J. Numer. Anal., Vol 10, No 2, 1973.
@end itemize

@noindent
@cindex LAPACK
Eigensystem routines for very large matrices can be found in the
Fortran library @sc{lapack}. 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.