 SLASYF(3) computes a partial factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method

## SYNOPSIS

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

CHARACTER UPLO

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

INTEGER IPIV( * )

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

## PURPOSE

SLASYF computes a partial factorization of a real 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. SLASYF is an auxiliary routine called by SSYTRF. 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) REAL 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) REAL 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.