NAME

CLASYF - compute a partial factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method

SYNOPSIS

SUBROUTINE CLASYF(
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, ∗ )

PURPOSE

CLASYF computes a partial factorization of a complex symmetric 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 transpose of U.
 
CLASYF is an auxiliary routine called by CSYTRF. 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 symmetric 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 symmetric 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.

  —  LAPACK version 2.0  —  08 October 1994