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**H U22**H )


 A  =  ( L11  0 ) (  D   0  ) ( L11**H L21**H )  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**H 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').

UPLO

          UPLO is CHARACTER*1


          Specifies whether the upper or lower triangular part of the


          Hermitian matrix A is stored:


          = 'U':  Upper triangular


          = 'L':  Lower triangular

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

NB

          NB is 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

          KB is INTEGER


          The number of columns of A that were actually factored.


          KB is either NB-1 or NB, or N if N <= NB.

A

          A is 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

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

IPIV

          IPIV is 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 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 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':


             Only the first KB elements of IPIV 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 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

          W is COMPLEX*16 array, dimension (LDW,NB)

LDW

          LDW is INTEGER


          The leading dimension of the array W.  LDW >= max(1,N).

INFO

          INFO is 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

  December 2016,  Igor Kozachenko,


                  Computer Science Division,


                  University of California, Berkeley