blas.texi

开放gsl矩阵运算

matrix @var{A} is symmetric.  When @var{Uplo} is @code{CblasUpper} then
the upper triangle and diagonal of @var{A} are used, and when @var{Uplo}
is @code{CblasLower} then the lower triangle and diagonal of @var{A} are
used.
@end deftypefun

@deftypefun int gsl_blas_chemm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, const gsl_matrix_complex_float * @var{B}, const gsl_complex_float @var{beta}, gsl_matrix_complex_float * @var{C})
@deftypefunx int gsl_blas_zhemm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, const gsl_matrix_complex * @var{B}, const gsl_complex @var{beta}, gsl_matrix_complex * @var{C})
@cindex HEMM, Level-3 BLAS
These functions compute the matrix-matrix product and sum @math{C =
\alpha A B + \beta C} for @var{Side} is @code{CblasLeft} and @math{C =
\alpha B A + \beta C} for @var{Side} is @code{CblasRight}, where the
matrix @var{A} is hermitian.  When @var{Uplo} is @code{CblasUpper} then
the upper triangle and diagonal of @var{A} are used, and when @var{Uplo}
is @code{CblasLower} then the lower triangle and diagonal of @var{A} are
used.  The imaginary elements of the diagonal are automatically set to
zero.
@end deftypefun

@deftypefun int gsl_blas_strmm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, float @var{alpha}, const gsl_matrix_float * @var{A}, gsl_matrix_float * @var{B})
@deftypefunx int gsl_blas_dtrmm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, double @var{alpha}, const gsl_matrix * @var{A}, gsl_matrix * @var{B})
@deftypefunx int gsl_blas_ctrmm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, gsl_matrix_complex_float * @var{B})
@deftypefunx int gsl_blas_ztrmm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, gsl_matrix_complex * @var{B})
@cindex TRMM, Level-3 BLAS
These functions compute the matrix-matrix product @math{B = \alpha op(A)
B} for @var{Side} is @code{CblasLeft} and @math{B = \alpha B op(A)} for
@var{Side} is @code{CblasRight}.  The matrix @var{A} is triangular and
@math{op(A) = A}, @math{A^T}, @math{A^H} for @var{TransA} =
@code{CblasNoTrans}, @code{CblasTrans}, @code{CblasConjTrans} When
@var{Uplo} is @code{CblasUpper} then the upper triangle of @var{A} is
used, and when @var{Uplo} is @code{CblasLower} then the lower triangle
of @var{A} is used.  If @var{Diag} is @code{CblasNonUnit} then the
diagonal of @var{A} is used, but if @var{Diag} is @code{CblasUnit} then
the diagonal elements of the matrix @var{A} are taken as unity and are
not referenced.
@end deftypefun


@deftypefun int gsl_blas_strsm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, float @var{alpha}, const gsl_matrix_float * @var{A}, gsl_matrix_float * @var{B})
@deftypefunx int gsl_blas_dtrsm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, double @var{alpha}, const gsl_matrix * @var{A}, gsl_matrix * @var{B})
@deftypefunx int gsl_blas_ctrsm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, gsl_matrix_complex_float * @var{B})
@deftypefunx int gsl_blas_ztrsm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, gsl_matrix_complex * @var{B})
@cindex TRSM, Level-3 BLAS
These functions compute the matrix-matrix product @math{B = \alpha op(inv(A))
B} for @var{Side} is @code{CblasLeft} and @math{B = \alpha B op(inv(A))} for
@var{Side} is @code{CblasRight}.  The matrix @var{A} is triangular and
@math{op(A) = A}, @math{A^T}, @math{A^H} for @var{TransA} =
@code{CblasNoTrans}, @code{CblasTrans}, @code{CblasConjTrans} When
@var{Uplo} is @code{CblasUpper} then the upper triangle of @var{A} is
used, and when @var{Uplo} is @code{CblasLower} then the lower triangle
of @var{A} is used.  If @var{Diag} is @code{CblasNonUnit} then the
diagonal of @var{A} is used, but if @var{Diag} is @code{CblasUnit} then
the diagonal elements of the matrix @var{A} are taken as unity and are
not referenced.
@end deftypefun

@deftypefun int gsl_blas_ssyrk (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, float @var{alpha}, const gsl_matrix_float * @var{A}, float @var{beta}, gsl_matrix_float * @var{C})
@deftypefunx int gsl_blas_dsyrk (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, double @var{alpha}, const gsl_matrix * @var{A}, double @var{beta}, gsl_matrix * @var{C})
@deftypefunx int gsl_blas_csyrk (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, const gsl_complex_float @var{beta}, gsl_matrix_complex_float * @var{C})
@deftypefunx int gsl_blas_zsyrk (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, const gsl_complex @var{beta}, gsl_matrix_complex * @var{C})
@cindex SYRK, Level-3 BLAS
These functions compute a rank-k update of the symmetric matrix @var{C},
@math{C = \alpha A A^T + \beta C} when @var{Trans} is
@code{CblasNoTrans} and @math{C = \alpha A^T A + \beta C} when
@var{Trans} is @code{CblasTrans}.  Since the matrix @var{C} is symmetric
only its upper half or lower half need to be stored.  When @var{Uplo} is
@code{CblasUpper} then the upper triangle and diagonal of @var{C} are
used, and when @var{Uplo} is @code{CblasLower} then the lower triangle
and diagonal of @var{C} are used.
@end deftypefun

@deftypefun int gsl_blas_cherk (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, float @var{alpha}, const gsl_matrix_complex_float * @var{A}, float @var{beta}, gsl_matrix_complex_float * @var{C})
@deftypefunx int gsl_blas_zherk (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, double @var{alpha}, const gsl_matrix_complex * @var{A}, double @var{beta}, gsl_matrix_complex * @var{C})
@cindex HERK, Level-3 BLAS
These functions compute a rank-k update of the hermitian matrix @var{C},
@math{C = \alpha A A^H + \beta C} when @var{Trans} is
@code{CblasNoTrans} and @math{C = \alpha A^H A + \beta C} when
@var{Trans} is @code{CblasTrans}.  Since the matrix @var{C} is hermitian
only its upper half or lower half need to be stored.  When @var{Uplo} is
@code{CblasUpper} then the upper triangle and diagonal of @var{C} are
used, and when @var{Uplo} is @code{CblasLower} then the lower triangle
and diagonal of @var{C} are used.  The imaginary elements of the
diagonal are automatically set to zero.
@end deftypefun

@deftypefun int gsl_blas_ssyr2k (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, float @var{alpha}, const gsl_matrix_float * @var{A}, const gsl_matrix_float * @var{B}, float @var{beta}, gsl_matrix_float * @var{C})
@deftypefunx int gsl_blas_dsyr2k (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, double @var{alpha}, const gsl_matrix * @var{A}, const gsl_matrix * @var{B}, double @var{beta}, gsl_matrix * @var{C})
@deftypefunx int gsl_blas_csyr2k (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, const gsl_matrix_complex_float * @var{B}, const gsl_complex_float @var{beta}, gsl_matrix_complex_float * @var{C})
@deftypefunx int gsl_blas_zsyr2k (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, const gsl_matrix_complex * @var{B}, const gsl_complex @var{beta}, gsl_matrix_complex *@var{C})
@cindex SYR2K, Level-3 BLAS
These functions compute a rank-2k update of the symmetric matrix @var{C},
@math{C = \alpha A B^T + \alpha B A^T + \beta C} when @var{Trans} is
@code{CblasNoTrans} and @math{C = \alpha A^T B + \alpha B^T A + \beta C} when
@var{Trans} is @code{CblasTrans}.  Since the matrix @var{C} is symmetric
only its upper half or lower half need to be stored.  When @var{Uplo} is
@code{CblasUpper} then the upper triangle and diagonal of @var{C} are
used, and when @var{Uplo} is @code{CblasLower} then the lower triangle
and diagonal of @var{C} are used.
@end deftypefun

@deftypefun int gsl_blas_cher2k (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, const gsl_matrix_complex_float * @var{B}, float @var{beta}, gsl_matrix_complex_float * @var{C})
@deftypefunx int gsl_blas_zher2k (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, const gsl_matrix_complex * @var{B}, double @var{beta}, gsl_matrix_complex * @var{C})
@cindex HER2K, Level-3 BLAS
These functions compute a rank-2k update of the hermitian matrix @var{C},
@math{C = \alpha A B^H + \alpha^* B A^H + \beta C} when @var{Trans} is
@code{CblasNoTrans} and @math{C = \alpha A^H B + \alpha^* B^H A + \beta C} when
@var{Trans} is @code{CblasTrans}.  Since the matrix @var{C} is hermitian
only its upper half or lower half need to be stored.  When @var{Uplo} is
@code{CblasUpper} then the upper triangle and diagonal of @var{C} are
used, and when @var{Uplo} is @code{CblasLower} then the lower triangle
and diagonal of @var{C} are used.  The imaginary elements of the
diagonal are automatically set to zero.
@end deftypefun

@node BLAS Examples
@section Examples

The following program computes the product of two matrices using the
Level-3 @sc{blas} function @sc{dgemm},

@tex
\beforedisplay
$$
\left(
\matrix{0.11&0.12&0.13\cr
0.21&0.22&0.23\cr}
\right)
\left(
\matrix{1011&1012\cr
1021&1022\cr
1031&1031\cr}
\right)
=
\left(
\matrix{367.76&368.12\cr
674.06&674.72\cr}
\right)
$$
\afterdisplay
@end tex
@ifinfo
@example
[ 0.11 0.12 0.13 ]  [ 1011 1012 ]     [ 367.76 368.12 ]
[ 0.21 0.22 0.23 ]  [ 1021 1022 ]  =  [ 674.06 674.72 ]
                    [ 1031 1032 ]
@end example
@end ifinfo
@noindent
The matrices are stored in row major order, according to the C convention 
for arrays.

@example
#include <stdio.h>
#include <gsl/gsl_blas.h>

int
main (void)
@{
  double a[] = @{ 0.11, 0.12, 0.13,
                 0.21, 0.22, 0.23 @};

  double b[] = @{ 1011, 1012,
                 1021, 1022,
                 1031, 1032 @};

  double c[] = @{ 0.00, 0.00,
                 0.00, 0.00 @};

  gsl_matrix_view A = gsl_matrix_view_array(a, 2, 3);
  gsl_matrix_view B = gsl_matrix_view_array(b, 3, 2);
  gsl_matrix_view C = gsl_matrix_view_array(c, 2, 2);

  /* Compute C = A B */

  gsl_blas_dgemm (CblasNoTrans, CblasNoTrans,
                  1.0, &A.matrix, &B.matrix,
                  0.0, &C.matrix);

  printf("[ %g, %g\n", c[0], c[1]);
  printf("  %g, %g ]\n", c[2], c[3]);

  return 0;  
@}
@end example
@noindent
Here is the output from the program,
@example
$ ./a.out
[ 367.76, 368.12
  674.06, 674.72 ]
@end example

@node BLAS References and Further Reading
@section References and Further Reading
@noindent
Information on the @sc{blas} standards, including both the legacy and
draft interface standards, is available online from the @sc{blas}
Homepage and @sc{blas} Technical Forum web-site.

@itemize @asis
@item
@cite{BLAS Homepage} 
@url{http://www.netlib.org/blas/}
@item
@cite{BLAS Technical Forum} 
@url{http://www.netlib.org/cgi-bin/checkout/blast/blast.pl}
@end itemize
@noindent
The following papers contain the specifications for Level 1, Level 2 and
Level 3 @sc{blas}.

@itemize @asis
@item
C. Lawson, R. Hanson, D. Kincaid, F. Krogh, "Basic Linear Algebra
Subprograms for Fortran Usage", @cite{ACM Transactions on Mathematical
Software}, Vol. 5 (1979), Pages 308-325.

@item
J.J. Dongarra, J. DuCroz, S. Hammarling, R. Hanson, "An Extended Set of
Fortran Basic Linear Algebra Subprograms", @cite{ACM Transactions on
Mathematical Software}, Vol. 14, No. 1 (1988), Pages 1-32.

@item
J.J. Dongarra, I. Duff, J. DuCroz, S. Hammarling, "A Set of
Level 3 Basic Linear Algebra Subprograms", @cite{ACM Transactions on
Mathematical Software}, Vol. 16 (1990), Pages 1-28.
@end itemize
@noindent
Postscript versions of the latter two papers are available from
@url{http://www.netlib.org/blas/}. A @sc{cblas} wrapper for Fortran @sc{blas}
libraries is available from the same location.