zhetrf_aa.f

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*> \brief \b ZHETRF_AA
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZHETRF_AA( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER    UPLO
*       INTEGER      N, LDA, LWORK, INFO
*       ..
*       .. Array Arguments ..
*       INTEGER      IPIV( * )
*       COMPLEX*16   A( LDA, * ), WORK( * )
*       ..
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZHETRF_AA computes the factorization of a complex hermitian matrix A
*> using the Aasen's algorithm.  The form of the factorization is
*>
*>    A = U**H*T*U  or  A = L*T*L**H
*>
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and T is a hermitian tridiagonal matrix.
*>
*> 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 tridiagonal matrix is stored in the diagonals
*>          and the subdiagonals of A just below (or above) the diagonals,
*>          and L is stored below (or above) the subdiagonals, when UPLO
*>          is 'L' (or 'U').
*> \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)
*>          On exit, it contains the details of the interchanges, i.e.,
*>          the row and column k of A were interchanged with the
*>          row and column IPIV(k).
*> \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, if N >= 1, and LWORK >= 2*N, otherwise.
*>          For optimum performance LWORK >= N*(1+NB), where NB is
*>          the optimal blocksize, returned by ILAENV.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hetrf_aa
*
*  =====================================================================
      SUBROUTINE zhetrf_aa( 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..--
*
      IMPLICIT NONE
*
*     .. Scalar Arguments ..
      CHARACTER    UPLO
      INTEGER      N, LDA, LWORK, INFO
*     ..
*     .. Array Arguments ..
      INTEGER      IPIV( * )
      COMPLEX*16   A( LDA, * ), WORK( * )
*     ..
*
*  =====================================================================
*     .. Parameters ..
      COMPLEX*16   ZERO, ONE
      parameter( zero = (0.0d+0, 0.0d+0), one = (1.0d+0, 0.0d+0) )
*
*     .. Local Scalars ..
      LOGICAL      LQUERY, UPPER
      INTEGER      J, LWKMIN, LWKOPT
      INTEGER      NB, MJ, NJ, K1, K2, J1, J2, J3, JB
      COMPLEX*16   ALPHA
*     ..
*     .. External Functions ..
      LOGICAL      LSAME
      INTEGER      ILAENV
      EXTERNAL     lsame, ilaenv
*     ..
*     .. External Subroutines ..
      EXTERNAL     zlahef_aa, zgemm, zgemv, zcopy, zscal, zswap,
     $                   xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC    dble, dconjg, max
*     ..
*     .. Executable Statements ..
*
*     Determine the block size
*
      nb = ilaenv( 1, 'ZHETRF_AA', uplo, n, -1, -1, -1 )
*
*     Test the input parameters.
*
      info = 0
      upper = lsame( uplo, 'U' )
      lquery = ( lwork.EQ.-1 )
      IF( n.LE.1 ) THEN
         lwkmin = 1
         lwkopt = 1
      ELSE
         lwkmin = 2*n
         lwkopt = (nb+1)*n
      END IF
*
      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.lwkmin .AND. .NOT.lquery ) THEN
         info = -7
      END IF
*
      IF( info.EQ.0 ) THEN
         work( 1 ) = lwkopt
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZHETRF_AA', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return
*
      IF( n.EQ.0 ) THEN
          RETURN
      ENDIF
      ipiv( 1 ) = 1
      IF( n.EQ.1 ) THEN
         a( 1, 1 ) = dble( a( 1, 1 ) )
         RETURN
      END IF
*
*     Adjust block size based on the workspace size
*
      IF( lwork.LT.((1+nb)*n) ) THEN
         nb = ( lwork-n ) / n
      END IF
*
      IF( upper ) THEN
*
*        .....................................................
*        Factorize A as U**H*D*U using the upper triangle of A
*        .....................................................
*
*        copy first row A(1, 1:N) into H(1:n) (stored in WORK(1:N))
*
         CALL zcopy( n, a( 1, 1 ), lda, work( 1 ), 1 )
*
*        J is the main loop index, increasing from 1 to N in steps of
*        JB, where JB is the number of columns factorized by ZLAHEF;
*        JB is either NB, or N-J+1 for the last block
*
         j = 0
 10      CONTINUE
         IF( j.GE.n )
     $      GO TO 20
*
*        each step of the main loop
*         J is the last column of the previous panel
*         J1 is the first column of the current panel
*         K1 identifies if the previous column of the panel has been
*          explicitly stored, e.g., K1=1 for the first panel, and
*          K1=0 for the rest
*
         j1 = j + 1
         jb = min( n-j1+1, nb )
         k1 = max(1, j)-j
*
*        Panel factorization
*
         CALL zlahef_aa( uplo, 2-k1, n-j, jb,
     $                      a( max(1, j), j+1 ), lda,
     $                      ipiv( j+1 ), work, n, work( n*nb+1 ) )
*
*        Adjust IPIV and apply it back (J-th step picks (J+1)-th pivot)
*
         DO j2 = j+2, min(n, j+jb+1)
            ipiv( j2 ) = ipiv( j2 ) + j
            IF( (j2.NE.ipiv(j2)) .AND. ((j1-k1).GT.2) ) THEN
               CALL zswap( j1-k1-2, a( 1, j2 ), 1,
     $                              a( 1, ipiv(j2) ), 1 )
            END IF
         END DO
         j = j + jb
*
*        Trailing submatrix update, where
*         the row A(J1-1, J2-1:N) stores U(J1, J2+1:N) and
*         WORK stores the current block of the auxiriarly matrix H
*
         IF( j.LT.n ) THEN
*
*          if the first panel and JB=1 (NB=1), then nothing to do
*
            IF( j1.GT.1 .OR. jb.GT.1 ) THEN
*
*              Merge rank-1 update with BLAS-3 update
*
               alpha = dconjg( a( j, j+1 ) )
               a( j, j+1 ) = one
               CALL zcopy( n-j, a( j-1, j+1 ), lda,
     $                          work( (j+1-j1+1)+jb*n ), 1 )
               CALL zscal( n-j, alpha, work( (j+1-j1+1)+jb*n ), 1 )
*
*              K1 identifies if the previous column of the panel has been
*               explicitly stored, e.g., K1=0 and K2=1 for the first panel,
*               and K1=1 and K2=0 for the rest
*
               IF( j1.GT.1 ) THEN
*
*                 Not first panel
*
                  k2 = 1
               ELSE
*
*                 First panel
*
                  k2 = 0
*
*                 First update skips the first column
*
                  jb = jb - 1
               END IF
*
               DO j2 = j+1, n, nb
                  nj = min( nb, n-j2+1 )
*
*                 Update (J2, J2) diagonal block with ZGEMV
*
                  j3 = j2
                  DO mj = nj-1, 1, -1
                     CALL zgemm( 'Conjugate transpose', 'Transpose',
     $                            1, mj, jb+1,
     $                           -one, a( j1-k2, j3 ), lda,
     $                                 work( (j3-j1+1)+k1*n ), n,
     $                            one, a( j3, j3 ), lda )
                     j3 = j3 + 1
                  END DO
*
*                 Update off-diagonal block of J2-th block row with ZGEMM
*
                  CALL zgemm( 'Conjugate transpose', 'Transpose',
     $                        nj, n-j3+1, jb+1,
     $                       -one, a( j1-k2, j2 ), lda,
     $                             work( (j3-j1+1)+k1*n ), n,
     $                        one, a( j2, j3 ), lda )
               END DO
*
*              Recover T( J, J+1 )
*
               a( j, j+1 ) = dconjg( alpha )
            END IF
*
*           WORK(J+1, 1) stores H(J+1, 1)
*
            CALL zcopy( n-j, a( j+1, j+1 ), lda, work( 1 ), 1 )
         END IF
         GO TO 10
      ELSE
*
*        .....................................................
*        Factorize A as L*D*L**H using the lower triangle of A
*        .....................................................
*
*        copy first column A(1:N, 1) into H(1:N, 1)
*         (stored in WORK(1:N))
*
         CALL zcopy( n, a( 1, 1 ), 1, work( 1 ), 1 )
*
*        J is the main loop index, increasing from 1 to N in steps of
*        JB, where JB is the number of columns factorized by ZLAHEF;
*        JB is either NB, or N-J+1 for the last block
*
         j = 0
 11      CONTINUE
         IF( j.GE.n )
     $      GO TO 20
*
*        each step of the main loop
*         J is the last column of the previous panel
*         J1 is the first column of the current panel
*         K1 identifies if the previous column of the panel has been
*          explicitly stored, e.g., K1=1 for the first panel, and
*          K1=0 for the rest
*
         j1 = j+1
         jb = min( n-j1+1, nb )
         k1 = max(1, j)-j
*
*        Panel factorization
*
         CALL zlahef_aa( uplo, 2-k1, n-j, jb,
     $                      a( j+1, max(1, j) ), lda,
     $                      ipiv( j+1 ), work, n, work( n*nb+1 ) )
*
*        Adjust IPIV and apply it back (J-th step picks (J+1)-th pivot)
*
         DO j2 = j+2, min(n, j+jb+1)
            ipiv( j2 ) = ipiv( j2 ) + j
            IF( (j2.NE.ipiv(j2)) .AND. ((j1-k1).GT.2) ) THEN
               CALL zswap( j1-k1-2, a( j2, 1 ), lda,
     $                              a( ipiv(j2), 1 ), lda )
            END IF
         END DO
         j = j + jb
*
*        Trailing submatrix update, where
*          A(J2+1, J1-1) stores L(J2+1, J1) and
*          WORK(J2+1, 1) stores H(J2+1, 1)
*
         IF( j.LT.n ) THEN
*
*          if the first panel and JB=1 (NB=1), then nothing to do
*
            IF( j1.GT.1 .OR. jb.GT.1 ) THEN
*
*              Merge rank-1 update with BLAS-3 update
*
               alpha = dconjg( a( j+1, j ) )
               a( j+1, j ) = one
               CALL zcopy( n-j, a( j+1, j-1 ), 1,
     $                          work( (j+1-j1+1)+jb*n ), 1 )
               CALL zscal( n-j, alpha, work( (j+1-j1+1)+jb*n ), 1 )
*
*              K1 identifies if the previous column of the panel has been
*               explicitly stored, e.g., K1=0 and K2=1 for the first panel,
*               and K1=1 and K2=0 for the rest
*
               IF( j1.GT.1 ) THEN
*
*                 Not first panel
*
                  k2 = 1
               ELSE
*
*                 First panel
*
                  k2 = 0
*
*                 First update skips the first column
*
                  jb = jb - 1
               END IF
*
               DO j2 = j+1, n, nb
                  nj = min( nb, n-j2+1 )
*
*                 Update (J2, J2) diagonal block with ZGEMV
*
                  j3 = j2
                  DO mj = nj-1, 1, -1
                     CALL zgemm( 'No transpose',
     $                           'Conjugate transpose',
     $                           mj, 1, jb+1,
     $                          -one, work( (j3-j1+1)+k1*n ), n,
     $                                a( j3, j1-k2 ), lda,
     $                           one, a( j3, j3 ), lda )
                     j3 = j3 + 1
                  END DO
*
*                 Update off-diagonal block of J2-th block column with ZGEMM
*
                  CALL zgemm( 'No transpose', 'Conjugate transpose',
     $                        n-j3+1, nj, jb+1,
     $                       -one, work( (j3-j1+1)+k1*n ), n,
     $                             a( j2, j1-k2 ), lda,
     $                        one, a( j3, j2 ), lda )
               END DO
*
*              Recover T( J+1, J )
*
               a( j+1, j ) = dconjg( alpha )
            END IF
*
*           WORK(J+1, 1) stores H(J+1, 1)
*
            CALL zcopy( n-j, a( j+1, j+1 ), 1, work( 1 ), 1 )
         END IF
         GO TO 11
      END IF
*
   20 CONTINUE
      work( 1 ) = lwkopt
      RETURN
*
*     End of ZHETRF_AA
*
      END