dlahrd.f

          SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            K, LDA, LDT, LDY, N, NB
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), T( LDT, NB ), TAU( NB ),
         $                   Y( LDY, NB )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DLAHRD reduces the first NB columns of a real general n-by-(n-k+1)
    *  matrix A so that elements below the k-th subdiagonal are zero. The
    *  reduction is performed by an orthogonal 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 DLAHR2 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (NB)
    *          The scalar factors of the elementary reflectors. See Further
    *          Details.
    *
    *  T       (output) DOUBLE PRECISION array, dimension (LDT,NB)
    *          The upper triangular matrix T.
    *
    *  LDT     (input) INTEGER
    *          The leading dimension of the array T.  LDT >= NB.
    *
    *  Y       (output) DOUBLE PRECISION array, dimension (LDY,NB)
    *          The n-by-nb matrix Y.
    *
    *  LDY     (input) INTEGER
    *          The leading dimension of the array Y. LDY >= 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 real scalar, and v is a real 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 ..
          DOUBLE PRECISION   ZERO, ONE
          PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I
          DOUBLE PRECISION   EI
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DAXPY, DCOPY, DGEMV, DLARFG, DSCAL, DTRMV
    *     ..
    *     .. 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 DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,
         $                  A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )
    *
    *           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 DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
                CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ),
         $                  LDA, T( 1, NB ), 1 )
    *
    *           w := w + V2'*b2
    *
                CALL DGEMV( '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 DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT,
         $                  T( 1, NB ), 1 )
    *
    *           b2 := b2 - V2*w
    *
                CALL DGEMV( '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 DTRMV( 'Lower', 'No transpose', 'Unit', I-1,
         $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )
                CALL DAXPY( 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)
    *
             CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
         $                TAU( I ) )
             EI = A( K+I, I )
             A( K+I, I ) = ONE
    *
    *        Compute  Y(1:n,i)
    *
             CALL DGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA,
         $               A( K+I, I ), 1, ZERO, Y( 1, I ), 1 )
             CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), LDA,
         $               A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
             CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1,
         $               ONE, Y( 1, I ), 1 )
             CALL DSCAL( N, TAU( I ), Y( 1, I ), 1 )
    *
    *        Compute T(1:i,i)
    *
             CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
             CALL DTRMV( '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 DLAHRD
    *
          END