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# slasyf

```
NAME
SLASYF - compute 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 fac-
tored.  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 con-
tains 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 tri-
angular 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 ele-
ments 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 inter-
changed 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 inter-
changed 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 fac-
torization has been completed, but the block diago-
nal matrix D is exactly singular.
```