pclahrd.f

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      SUBROUTINE pclahrd( N, K, NB, A, IA, JA, DESCA, TAU, T, Y, IY, JY,
     $                    DESCY, WORK )
*
*  -- ScaLAPACK auxiliary routine (version 1.7) --
*     University of Tennessee, Knoxville, Oak Ridge National Laboratory,
*     and University of California, Berkeley.
*     May 1, 1997
*
*     .. Scalar Arguments ..
      INTEGER             IA, IY, JA, JY, K, N, NB
*     ..
*     .. Array Arguments ..
      INTEGER             DESCA( * ), DESCY( * )
      COMPLEX             A( * ), T( * ), TAU( * ), WORK( * ), Y( * )
*     ..
*
*  Purpose
*  =======
*
*  PCLAHRD reduces the first NB columns of a complex general
*  N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that
*  elements below the k-th subdiagonal are zero. The reduction is
*  performed by an unitary similarity transformation Q' * A * Q. The
*  routine returns the matrices V and T which determine Q as a block
*  reflector I - V*T*V', and also the matrix Y = A * V * T.
*
*  This is an auxiliary routine called by PCGEHRD. In the following
*  comments sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1).
*
*  Arguments
*  =========
*
*  N       (global input) INTEGER
*          The number of rows and columns to be operated on, i.e. the
*          order of the distributed submatrix sub( A ).
*          N >= 0.
*
*  K       (global input) INTEGER
*          The offset for the reduction. Elements below the k-th
*          subdiagonal in the first NB columns are reduced to zero.
*
*  NB      (global input) INTEGER
*          The number of columns to be reduced.
*
*  A       (local input/local output) COMPLEX pointer into
*          the local memory to an array of dimension (LLD_A,
*          LOCc(JA+N-K)). On entry, this array contains the the local
*          pieces of the N-by-(N-K+1) general distributed matrix
*          A(IA:IA+N-1,JA:JA+N-K). On exit, the elements on and above
*          the k-th subdiagonal in the first NB columns are overwritten
*          with the corresponding elements of the reduced distributed
*          matrix; the elements below the k-th subdiagonal, with the
*          array TAU, represent the matrix Q as a product of elementary
*          reflectors. The other columns of A(IA:IA+N-1,JA:JA+N-K) are
*          unchanged. See Further Details.
*
*  IA      (global input) INTEGER
*          The row index in the global array A indicating the first
*          row of sub( A ).
*
*  JA      (global input) INTEGER
*          The column index in the global array A indicating the
*          first column of sub( A ).
*
*  DESCA   (global and local input) INTEGER array of dimension DLEN_.
*          The array descriptor for the distributed matrix A.
*
*  TAU     (local output) COMPLEX array, dimension LOCc(JA+N-2)
*          The scalar factors of the elementary reflectors (see Further
*          Details). TAU is tied to the distributed matrix A.
*
*  T       (local output) COMPLEX array, dimension (NB_A,NB_A)
*          The upper triangular matrix T.
*
*  Y       (local output) COMPLEX pointer into the local memory
*          to an array of dimension (LLD_Y,NB_A). On exit, this array
*          contains the local pieces of the N-by-NB distributed
*          matrix Y. LLD_Y >= LOCr(IA+N-1).
*
*  IY      (global input) INTEGER
*          The row index in the global array Y indicating the first
*          row of sub( Y ).
*
*  JY      (global input) INTEGER
*          The column index in the global array Y indicating the
*          first column of sub( Y ).
*
*  DESCY   (global and local input) INTEGER array of dimension DLEN_.
*          The array descriptor for the distributed matrix Y.
*
*  WORK    (local workspace) COMPLEX array, dimension (NB)
*
*  Further Details
*  ===============
*
*  The matrix Q is represented as a product of nb elementary reflectors
*
*     Q = H(1) H(2) . . . H(nb).
*
*  Each H(i) has the form
*
*     H(i) = I - tau * v * v'
*
*  where tau is a complex scalar, and v is a complex vector with
*  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
*  A(ia+i+k:ia+n-1,ja+i-1), and tau in TAU(ja+i-1).
*
*  The elements of the vectors v together form the (n-k+1)-by-nb matrix
*  V which is needed, with T and Y, to apply the transformation to the
*  unreduced part of the matrix, using an update of the form:
*  A(ia:ia+n-1,ja:ja+n-k) := (I-V*T*V')*(A(ia:ia+n-1,ja:ja+n-k)-Y*V').
*
*  The contents of A(ia:ia+n-1,ja:ja+n-k) on exit are illustrated by the
*  following example with n = 7, k = 3 and nb = 2:
*
*     ( a   h   a   a   a )
*     ( a   h   a   a   a )
*     ( a   h   a   a   a )
*     ( h   h   a   a   a )
*     ( v1  h   a   a   a )
*     ( v1  v2  a   a   a )
*     ( v1  v2  a   a   a )
*
*  where a denotes an element of the original matrix
*  A(ia:ia+n-1,ja:ja+n-k), h denotes a modified element of the upper
*  Hessenberg matrix H, and vi denotes an element of the vector
*  defining H(i).
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
     $                   lld_, mb_, m_, nb_, n_, rsrc_
      parameter( block_cyclic_2d = 1, dlen_ = 9, dtype_ = 1,
     $                     ctxt_ = 2, m_ = 3, n_ = 4, mb_ = 5, nb_ = 6,
     $                     rsrc_ = 7, csrc_ = 8, lld_ = 9 )
      COMPLEX            ONE, ZERO
      parameter( one = ( 1.0e+0, 0.0e+0 ),
     $                   zero = ( 0.0e+0, 0.0e+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            IPROC
      INTEGER            I, IACOL, IAROW, ICTXT, IOFF, II, J, JJ, JL,
     $                   jt, jw, l, myrow, mycol, npcol, nprow, nq
      COMPLEX            EI
*     ..
*     .. Local Arrays ..
      INTEGER            DESCW( DLEN_ )
*     ..
*     .. External Functions ..
      INTEGER            NUMROC
      EXTERNAL           numroc
*     ..
*     .. External Subroutines ..
      EXTERNAL           blacs_gridinfo, caxpy, ccopy, cscal,
     $                   ctrmv, descset, infog2l, pcelset,
     $                   pcgemv, pclacgv, pclarfg, pcscal
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          min, mod
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( n.LE.1 )
     $   RETURN
*
      ictxt = desca( ctxt_ )
      CALL blacs_gridinfo( ictxt, nprow, npcol, myrow, mycol )
*
      ioff = mod( ja-1, desca( nb_ ) )
      CALL infog2l( ia+k, ja, desca, nprow, npcol, myrow, mycol, ii,
     $              jj, iarow, iacol )
*
      iproc = ( myrow.EQ.iarow .AND. mycol.EQ.iacol )
      nq = numroc( n+ja-1, desca( nb_ ), mycol, iacol, npcol )
      IF( mycol.EQ.iacol )
     $   nq = nq - ioff
*
      ei = zero
 
      jw = ioff + 1
      CALL descset( descw, 1, desca( mb_ ), 1, desca( mb_ ), iarow,
     $              iacol, ictxt, 1 )
*
      DO 10 l = 1, nb
         i = ia + k + l - 2
         j = ja + l - 1
*
         IF( l.GT.1 ) THEN
*
*           Update A(ia:ia+n-1,j)
*
*           Compute i-th column of A - Y * V'
*
            CALL pclacgv( l-1, a, i, ja, desca, desca( m_ ) )
            CALL pcgemv( 'No transpose', n, l-1, -one, y, iy, jy, descy,
     $                   a, i, ja, desca, desca( m_ ), one, a, ia, j,
     $                   desca, 1 )
            CALL pclacgv( l-1, a, i, ja, desca, desca( m_ ) )
*
*           Apply I - V * T' * V' to this column (call it b) from the
*           left, using the last column of T as workspace
*
*           Let  V = ( V1 )   and   b = ( b1 )   (first I-1 rows)
*                    ( V2 )             ( b2 )
*
*           where V1 is unit lower triangular
*
*           w := V1' * b1
*
            IF( iproc ) THEN
               CALL ccopy( l-1, a( (jj+l-2)*desca( lld_ )+ii ), 1,
     $                     work( jw ), 1 )
               CALL ctrmv( 'Lower', 'Conjugate transpose', 'unit', L-1,
     $                     A( (JJ-1)*DESCA( LLD_ )+II ), DESCA( LLD_ ),
     $                     WORK( JW ), 1 )
            END IF
*
*           w := w + V2'*b2
*
            CALL PCGEMV( 'conjugate transpose', N-K-L+1, L-1, ONE, A,
     $                   I+1, JA, DESCA, A, I+1, J, DESCA, 1, ONE, WORK,
     $                   1, JW, DESCW, DESCW( M_ ) )
*
*           w := T'*w
*
            IF( IPROC )
     $         CALL CTRMV( 'upper', 'conjugate transpose', 'non-unit',
     $                     L-1, T, DESCA( NB_ ), WORK( JW ), 1 )
*
*           b2 := b2 - V2*w
*
            CALL PCGEMV( 'no transpose', N-K-L+1, L-1, -ONE, A, I+1, JA,
     $                   DESCA, WORK, 1, JW, DESCW, DESCW( M_ ), ONE,
     $                   A, I+1, J, DESCA, 1 )
*
*           b1 := b1 - V1*w
*
            IF( IPROC ) THEN
               CALL CTRMV( 'lower', 'no transpose', 'unit', L-1,
     $                     A( (JJ-1)*DESCA( LLD_ )+II ), DESCA( LLD_ ),
     $                     WORK( JW ), 1 )
               CALL CAXPY( L-1, -ONE, WORK( JW ), 1,
     $                     A( ( JJ+L-2 )*DESCA( LLD_ )+II ), 1 )
            END IF
            CALL PCELSET( A, I, J-1, DESCA, EI )
         END IF
*
*        Generate the elementary reflector H(i) to annihilate
*        A(ia+k+i:ia+n-1,j)
*
         CALL PCLARFG( N-K-L+1, EI, I+1, J, A, MIN( I+2, N+IA-1 ), J,
     $                 DESCA, 1, TAU )
         CALL PCELSET( A, I+1, J, DESCA, ONE )
*
*        Compute  Y(iy:y+n-1,jy+l-1)
*
         CALL PCGEMV( 'no transpose', N, N-K-L+1, ONE, A, IA, J+1,
     $                DESCA, A, I+1, J, DESCA, 1, ZERO, Y, IY, JY+L-1,
     $                DESCY, 1 )
         CALL PCGEMV( 'conjugate transpose', N-K-L+1, L-1, ONE, A, I+1,
     $                JA, DESCA, A, I+1, J, DESCA, 1, ZERO, WORK, 1, JW,
     $                DESCW, DESCW( M_ ) )
         CALL PCGEMV( 'no transpose', N, L-1, -ONE, Y, IY, JY, DESCY,
     $                WORK, 1, JW, DESCW, DESCW( M_ ), ONE, Y, IY,
     $                JY+L-1, DESCY, 1 )
         JL = MIN( JJ+L-1, JA+NQ-1 )
         CALL PCSCAL( N, TAU( JL ), Y, IY, JY+L-1, DESCY, 1 )
*
*        Compute T(1:i,i)
*
         IF( IPROC ) THEN
            JT = ( L-1 ) * DESCA( NB_ )
            CALL CSCAL( L-1, -TAU( JL ), WORK( JW ), 1 )
            CALL CCOPY( L-1, WORK( JW ), 1, T( JT+1 ), 1 )
            CALL CTRMV( 'upper', 'no transpose', 'non-unit', L-1, T,
     $                  DESCA( NB_ ), T( JT+1 ), 1 )
            T( JT+L ) = TAU( JL )
         END IF
   10 CONTINUE
*
      CALL PCELSET( A, K+NB+IA-1, J, DESCA, EI )
*
      RETURN
*
*     End of PCLAHRD
*
      END