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version 1.4, 2010/08/06 15:32:43 version 1.13, 2012/12/14 14:22:50
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   *> \brief \b ZLAHEF computes a partial factorization of a complex Hermitian indefinite matrix, using the diagonal pivoting method.
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZLAHEF + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlahef.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlahef.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlahef.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZLAHEF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          UPLO
   *       INTEGER            INFO, KB, LDA, LDW, N, NB
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IPIV( * )
   *       COMPLEX*16         A( LDA, * ), W( LDW, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> 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').
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] UPLO
   *> \verbatim
   *>          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
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix A.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in] NB
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[out] KB
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] IPIV
   *> \verbatim
   *>          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 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.
   *> \endverbatim
   *>
   *> \param[out] W
   *> \verbatim
   *>          W is COMPLEX*16 array, dimension (LDW,NB)
   *> \endverbatim
   *>
   *> \param[in] LDW
   *> \verbatim
   *>          LDW is INTEGER
   *>          The leading dimension of the array W.  LDW >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          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.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date September 2012
   *
   *> \ingroup complex16HEcomputational
   *
   *  =====================================================================
       SUBROUTINE ZLAHEF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )        SUBROUTINE ZLAHEF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- LAPACK computational routine (version 3.4.2) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     November 2006  *     September 2012
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          UPLO        CHARACTER          UPLO
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       COMPLEX*16         A( LDA, * ), W( LDW, * )        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.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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 *  *
 *        Update the upper triangle of A11 (= A(1:k,1:k)) as  *        Update the upper triangle of A11 (= A(1:k,1:k)) as
 *  *
 *        A11 := A11 - U12*D*U12' = A11 - U12*W'  *        A11 := A11 - U12*D*U12**H = A11 - U12*W**H
 *  *
 *        computing blocks of NB columns at a time (note that conjg(W) is  *        computing blocks of NB columns at a time (note that conjg(W) is
 *        actually stored)  *        actually stored)
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 *  *
 *        Update the lower triangle of A22 (= A(k:n,k:n)) as  *        Update the lower triangle of A22 (= A(k:n,k:n)) as
 *  *
 *        A22 := A22 - L21*D*L21' = A22 - L21*W'  *        A22 := A22 - L21*D*L21**H = A22 - L21*W**H
 *  *
 *        computing blocks of NB columns at a time (note that conjg(W) is  *        computing blocks of NB columns at a time (note that conjg(W) is
 *        actually stored)  *        actually stored)
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  Added in v.1.13

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