zhetrf.f

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*> \brief \b ZHETRF
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZHETRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       COMPLEX*16         A( LDA, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZHETRF computes the factorization of a complex Hermitian matrix A
*> using the Bunch-Kaufman diagonal pivoting method.  The form of the
*> factorization is
*>
*>    A = U*D*U**H  or  A = L*D*L**H
*>
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and D is Hermitian and block diagonal with
*> 1-by-1 and 2-by-2 diagonal blocks.
*>
*> This is the blocked version of the algorithm, calling Level 3 BLAS.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \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, the block diagonal matrix D and the multipliers used
*>          to obtain the factor U or L (see below for further details).
*> \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 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] WORK
*> \verbatim
*>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The length of WORK. LWORK >= 1. For best performance
*>          LWORK >= N*NB, where NB is the block size returned by ILAENV.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
*>                has been completed, but the block diagonal matrix D is
*>                exactly singular, and division by zero will occur if it
*>                is used to solve a system of equations.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetrf
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  If UPLO = 'U', then A = U*D*U**H, where
*>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
*>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
*>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
*>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*>             (   I    v    0   )   k-s
*>     U(k) =  (   0    I    0   )   s
*>             (   0    0    I   )   n-k
*>                k-s   s   n-k
*>
*>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
*>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
*>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
*>
*>  If UPLO = 'L', then A = L*D*L**H, where
*>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
*>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
*>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
*>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*>             (   I    0     0   )  k-1
*>     L(k) =  (   0    I     0   )  s
*>             (   0    v     I   )  n-k-s+1
*>                k-1   s  n-k-s+1
*>
*>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
*>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
*>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE zhetrf( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      COMPLEX*16         A( LDA, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY, UPPER
      INTEGER            IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           lsame, ilaenv
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, zhetf2, zlahef
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      upper = lsame( uplo, 'U' )
      lquery = ( lwork.EQ.-1 )
      IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -4
      ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
         info = -7
      END IF
*
      IF( info.EQ.0 ) THEN
*
*        Determine the block size
*
         nb = ilaenv( 1, 'ZHETRF', uplo, n, -1, -1, -1 )
         lwkopt = max( 1, n*nb )
         work( 1 ) = lwkopt
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZHETRF', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
      nbmin = 2
      ldwork = n
      IF( nb.GT.1 .AND. nb.LT.n ) THEN
         iws = ldwork*nb
         IF( lwork.LT.iws ) THEN
            nb = max( lwork / ldwork, 1 )
            nbmin = max( 2, ilaenv( 2, 'ZHETRF', uplo, n, -1, -1,
     $                   -1 ) )
         END IF
      ELSE
         iws = 1
      END IF
      IF( nb.LT.nbmin )
     $   nb = n
*
      IF( upper ) THEN
*
*        Factorize A as U*D*U**H using the upper triangle of A
*
*        K is the main loop index, decreasing from N to 1 in steps of
*        KB, where KB is the number of columns factorized by ZLAHEF;
*        KB is either NB or NB-1, or K for the last block
*
         k = n
   10    CONTINUE
*
*        If K < 1, exit from loop
*
         IF( k.LT.1 )
     $      GO TO 40
*
         IF( k.GT.nb ) THEN
*
*           Factorize columns k-kb+1:k of A and use blocked code to
*           update columns 1:k-kb
*
            CALL zlahef( uplo, k, nb, kb, a, lda, ipiv, work, n,
     $                   iinfo )
         ELSE
*
*           Use unblocked code to factorize columns 1:k of A
*
            CALL zhetf2( uplo, k, a, lda, ipiv, iinfo )
            kb = k
         END IF
*
*        Set INFO on the first occurrence of a zero pivot
*
         IF( info.EQ.0 .AND. iinfo.GT.0 )
     $      info = iinfo
*
*        Decrease K and return to the start of the main loop
*
         k = k - kb
         GO TO 10
*
      ELSE
*
*        Factorize A as L*D*L**H using the lower triangle of A
*
*        K is the main loop index, increasing from 1 to N in steps of
*        KB, where KB is the number of columns factorized by ZLAHEF;
*        KB is either NB or NB-1, or N-K+1 for the last block
*
         k = 1
   20    CONTINUE
*
*        If K > N, exit from loop
*
         IF( k.GT.n )
     $      GO TO 40
*
         IF( k.LE.n-nb ) THEN
*
*           Factorize columns k:k+kb-1 of A and use blocked code to
*           update columns k+kb:n
*
            CALL zlahef( uplo, n-k+1, nb, kb, a( k, k ), lda,
     $                   ipiv( k ),
     $                   work, n, iinfo )
         ELSE
*
*           Use unblocked code to factorize columns k:n of A
*
            CALL zhetf2( uplo, n-k+1, a( k, k ), lda, ipiv( k ),
     $                   iinfo )
            kb = n - k + 1
         END IF
*
*        Set INFO on the first occurrence of a zero pivot
*
         IF( info.EQ.0 .AND. iinfo.GT.0 )
     $      info = iinfo + k - 1
*
*        Adjust IPIV
*
         DO 30 j = k, k + kb - 1
            IF( ipiv( j ).GT.0 ) THEN
               ipiv( j ) = ipiv( j ) + k - 1
            ELSE
               ipiv( j ) = ipiv( j ) - k + 1
            END IF
   30    CONTINUE
*
*        Increase K and return to the start of the main loop
*
         k = k + kb
         GO TO 20
*
      END IF
*
   40 CONTINUE
*
      work( 1 ) = lwkopt
      RETURN
*
*     End of ZHETRF
*
      SUBROUTINE zhetrf( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO ) …
      END