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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: computes a partial factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method

```NAME
LAPACK-3  -  computes  a  partial  factorization of a complex Hermitian
matrix A using the Bunch-Kaufman diagonal pivoting method

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

CHARACTER      UPLO

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

INTEGER        IPIV( * )

COMPLEX*16     A( LDA, * ), W( LDW, * )

PURPOSE
ZLAHEF 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.
ZLAHEF is an auxiliary routine called by ZHETRF. 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*16 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*16 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 routine (version 3.2)     March 2011                       LAPACK-3(3)
```