dlahrd.f

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      SUBROUTINE dlahrd( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. 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 auxiliary routine called by DGEHRD.
*
*  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