CLAHEF_ROOK computes a partial factorization of a complex Hermitian


 matrix A using the bounded Bunch-Kaufman ("rook") 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.


 CLAHEF_ROOK is an auxiliary routine called by CHETRF_ROOK. 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 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) < 0 and IPIV(k-1) < 0, then rows and


             columns k and -IPIV(k) were interchanged and rows and


             columns k-1 and -IPIV(k-1) were inerchaged,


             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) < 0 and IPIV(k+1) < 0, then rows and


             columns k and -IPIV(k) were interchanged and rows and


             columns k+1 and -IPIV(k+1) were inerchaged,


             D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

W

          W is COMPLEX 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.

November 2013

  November 2013, Igor Kozachenko,


                  Computer Science Division,


                  University of California, Berkeley


  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,


                  School of Mathematics,


                  University of Manchester