clahef(3S)

NAME
     CLAHEF - compute a partial factorization of a complex Hermitian matrix A
     using the Bunch-Kaufman diagonal pivoting method

SYNOPSIS
     SUBROUTINE CLAHEF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )

         CHARACTER      UPLO

         INTEGER        INFO, KB, LDA, LDW, N, NB

         INTEGER        IPIV( * )

         COMPLEX        A( LDA, * ), W( LDW, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     CLAHEF computes a partial factorization of a complex Hermitian matrix A
     using the Bunch-Kaufman diagonal pivoting method. The partial
     factorization has the form:

     A  =  ( I  U12 ) ( A11  0  ) (  I    0   )  if UPLO = 'U', or:
           ( 0  U22 ) (  0   D  ) ( U12' U22' )

     A  =  ( L11  0 ) (  D   0  ) ( L11' L21' )  if UPLO = 'L'
           ( L21  I ) (  0  A22 ) (  0    I   )

     where the order of D is at most NB. The actual order is returned in the
     argument KB, and is either NB or NB-1, or N if N <= NB.  Note that U'
     denotes the conjugate transpose of U.

     CLAHEF is an auxiliary routine called by CHETRF. It uses blocked code
     (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22
     (if UPLO = 'L').


ARGUMENTS
     UPLO    (input) CHARACTER*1
             Specifies whether the upper or lower triangular part of the
             Hermitian matrix A is stored:
             = 'U':  Upper triangular
             = 'L':  Lower triangular

     N       (input) INTEGER
             The order of the matrix A.  N >= 0.

     NB      (input) INTEGER
             The maximum number of columns of the matrix A that should be
             factored.  NB should be at least 2 to allow for 2-by-2 pivot
             blocks.

     KB      (output) INTEGER
             The number of columns of A that were actually factored.  KB is
             either NB-1 or NB, or N if N <= NB.

     A       (input/output) COMPLEX array, dimension (LDA,N)
             On entry, the Hermitian matrix A.  If UPLO = 'U', the leading n-
             by-n upper triangular part of A contains the upper triangular
             part of the matrix A, and the strictly lower triangular part of A
             is not referenced.  If UPLO = 'L', the leading n-by-n lower
             triangular part of A contains the lower triangular part of the
             matrix A, and the strictly upper triangular part of A is not
             referenced.  On exit, A contains details of the partial
             factorization.

     LDA     (input) INTEGER
             The leading dimension of the array A.  LDA >= max(1,N).

     IPIV    (output) INTEGER array, dimension (N)
             Details of the interchanges and the block structure of D.  If
             UPLO = 'U', only the last KB elements of IPIV are set; if UPLO =
             'L', only the first KB elements are set.

             If IPIV(k) > 0, then rows and columns k and IPIV(k) were
             interchanged and D(k,k) is a 1-by-1 diagonal block.  If UPLO =
             'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
             -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2
             diagonal block.  If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then
             rows and columns k+1 and -IPIV(k) were interchanged and
             D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

     W       (workspace) COMPLEX array, dimension (LDW,NB)

     LDW     (input) INTEGER
             The leading dimension of the array W.  LDW >= max(1,N).

     INFO    (output) INTEGER
             = 0: successful exit
             > 0: if INFO = k, D(k,k) is exactly zero.  The factorization has
             been completed, but the block diagonal matrix D is exactly
             singular.

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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