zlahrd.f

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      SUBROUTINE ZLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
*  -- LAPACK auxiliary routine (version 3.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2006
*
*     .. Scalar Arguments ..
      INTEGER            K, LDA, LDT, LDY, N, NB
*     ..
*     .. Array Arguments ..
      COMPLEX*16         A( LDA, * ), T( LDT, NB ), TAU( NB ),
     $                   Y( LDY, NB )
*     ..
*
*  Purpose
*  =======
*
*  ZLAHRD reduces the first NB columns of a complex general n-by-(n-k+1)
*  matrix A so that elements below the k-th subdiagonal are zero. The
*  reduction is performed by a 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 OBSOLETE auxiliary routine. 
*  This routine will be 'deprecated' in a  future release.
*  Please use the new routine ZLAHR2 instead.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix A.
*
*  K       (input) INTEGER
*          The offset for the reduction. Elements below the k-th
*          subdiagonal in the first NB columns are reduced to zero.
*
*  NB      (input) INTEGER
*          The number of columns to be reduced.
*
*  A       (input/output) COMPLEX*16 array, dimension (LDA,N-K+1)
*          On entry, the n-by-(n-k+1) general matrix A.
*          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 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 are
*          unchanged. See Further Details.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  TAU     (output) COMPLEX*16 array, dimension (NB)
*          The scalar factors of the elementary reflectors. See Further
*          Details.
*
*  T       (output) COMPLEX*16 array, dimension (LDT,NB)
*          The upper triangular matrix T.
*
*  LDT     (input) INTEGER
*          The leading dimension of the array T.  LDT >= NB.
*
*  Y       (output) COMPLEX*16 array, dimension (LDY,NB)
*          The n-by-nb matrix Y.
*
*  LDY     (input) INTEGER
*          The leading dimension of the array Y. LDY >= max(1,N).
*
*  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(i+k+1:n,i), and tau in TAU(i).
*
*  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 := (I - V*T*V') * (A - Y*V').
*
*  The contents of A 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, h denotes a
*  modified element of the upper Hessenberg matrix H, and vi denotes an
*  element of the vector defining H(i).
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX*16         ZERO, ONE
      PARAMETER          ( ZERO = ( 0.0D+0, 0.0D+0 ),
     $                   ONE = ( 1.0D+0, 0.0D+0 ) )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      COMPLEX*16         EI
*     ..
*     .. External Subroutines ..
      EXTERNAL           ZAXPY, ZCOPY, ZGEMV, ZLACGV, ZLARFG, ZSCAL,
     $                   ZTRMV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.LE.1 )
     $   RETURN
*
      DO 10 I = 1, NB
         IF( I.GT.1 ) THEN
*
*           Update A(1:n,i)
*
*           Compute i-th column of A - Y * V'
*
            CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
            CALL ZGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,
     $                  A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )
            CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
*
*           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
*
            CALL ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
            CALL ZTRMV( 'Lower', 'Conjugate transpose', 'Unit', I-1,
     $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )
*
*           w := w + V2'*b2
*
            CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1, ONE,
     $                  A( K+I, 1 ), LDA, A( K+I, I ), 1, ONE,
     $                  T( 1, NB ), 1 )
*
*           w := T'*w
*
            CALL ZTRMV( 'Upper', 'Conjugate transpose', 'Non-unit', I-1,
     $                  T, LDT, T( 1, NB ), 1 )
*
*           b2 := b2 - V2*w
*
            CALL ZGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ),
     $                  LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
*
*           b1 := b1 - V1*w
*
            CALL ZTRMV( 'Lower', 'No transpose', 'Unit', I-1,
     $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )
            CALL ZAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
*
            A( K+I-1, I-1 ) = EI
         END IF
*
*        Generate the elementary reflector H(i) to annihilate
*        A(k+i+1:n,i)
*
         EI = A( K+I, I )
         CALL ZLARFG( N-K-I+1, EI, A( MIN( K+I+1, N ), I ), 1,
     $                TAU( I ) )
         A( K+I, I ) = ONE
*
*        Compute  Y(1:n,i)
*
         CALL ZGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA,
     $               A( K+I, I ), 1, ZERO, Y( 1, I ), 1 )
         CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1, ONE,
     $               A( K+I, 1 ), LDA, A( K+I, I ), 1, ZERO, T( 1, I ),
     $               1 )
         CALL ZGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1,
     $               ONE, Y( 1, I ), 1 )
         CALL ZSCAL( N, TAU( I ), Y( 1, I ), 1 )
*
*        Compute T(1:i,i)
*
         CALL ZSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
         CALL ZTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT,
     $               T( 1, I ), 1 )
         T( I, I ) = TAU( I )
*
   10 CONTINUE
      A( K+NB, NB ) = EI
*
      RETURN
*
*     End of ZLAHRD
*
      END