# ?lasyf

Computes a partial factorization of a real/complex symmetric matrix, using the diagonal pivoting method.

## Syntax

call slasyf( uplo, n, nb, kb, a, lda, ipiv, w, ldw, info )

call dlasyf( uplo, n, nb, kb, a, lda, ipiv, w, ldw, info )

call clasyf( uplo, n, nb, kb, a, lda, ipiv, w, ldw, info )

call zlasyf( uplo, n, nb, kb, a, lda, ipiv, w, ldw, info )

## Description

The routine ?lasyf computes a partial factorization of a real/complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The partial factorization has the form:

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`.

This is an auxiliary routine called by ?sytrf. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if `uplo = 'U'`) or A22 (if `uplo = 'L'`).

## Input Parameters

uplo

  CHARACTER*1.

  Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:

  = 'U': Upper triangular

  = 'L': Lower triangular

n

  INTEGER. The order of the matrix A. `n ≥ 0`.

nb

  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.

a

  REAL for slasyf

  DOUBLE PRECISION for dlasyf

  COMPLEX for clasyf

  COMPLEX*16 for zlasyf.

  Array, DIMENSION (lda, n). 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.

lda

  INTEGER. The leading dimension of the array a. `lda ≥ max(1,n)`.

w

  REAL for slasyf

  DOUBLE PRECISION for dlasyf

  COMPLEX for clasyf

  COMPLEX*16 for zlasyf.

  Workspace array, DIMENSION (ldw, nb).

ldw

  INTEGER. The leading dimension of the array w. `ldw ≥ max(1,n)`.

## Output Parameters

kb

  INTEGER. The number of columns of A that were actually factored kb is either nb-1 or nb, or n if `n ≤ nb`.

a

  On exit, a contains details of the partial factorization.

ipiv

  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.

info

  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.

Parent topic: Auxiliary Routines