slahr2.f

          SUBROUTINE SLAHR2( 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 ..
          REAL              A( LDA, * ), T( LDT, NB ), TAU( NB ),
         $                   Y( LDY, NB )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SLAHR2 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 SGEHRD.
    *
    *  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.
    *          K < N.
    *
    *  NB      (input) INTEGER
    *          The number of columns to be reduced.
    *
    *  A       (input/output) REAL 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) REAL array, dimension (NB)
    *          The scalar factors of the elementary reflectors. See Further
    *          Details.
    *
    *  T       (output) REAL array, dimension (LDT,NB)
    *          The upper triangular matrix T.
    *
    *  LDT     (input) INTEGER
    *          The leading dimension of the array T.  LDT >= NB.
    *
    *  Y       (output) REAL 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   a   a   a   a )
    *     ( a   a   a   a   a )
    *     ( a   a   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).
    *
    *  This file is a slight modification of LAPACK-3.0's SLAHRD
    *  incorporating improvements proposed by Quintana-Orti and Van de
    *  Gejin. Note that the entries of A(1:K,2:NB) differ from those
    *  returned by the original LAPACK routine. This function is
    *  not backward compatible with LAPACK3.0.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          REAL              ZERO, ONE
          PARAMETER          ( ZERO = 0.0E+0,
         $                     ONE = 1.0E+0 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I
          REAL              EI
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SAXPY, SCOPY, SGEMM, SGEMV, SLACPY,
         $                   SLARFG, SSCAL, STRMM, STRMV
    *     ..
    *     .. 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(K+1:N,I)
    *
    *           Update I-th column of A - Y * V'
    *
                CALL SGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
         $                  A( K+I-1, 1 ), LDA, ONE, A( K+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 SCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
                CALL STRMV( 'Lower', 'Transpose', 'UNIT',
         $                  I-1, A( K+1, 1 ),
         $                  LDA, T( 1, NB ), 1 )
    *
    *           w := w + V2'*b2
    *
                CALL SGEMV( '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 STRMV( 'Upper', 'Transpose', 'NON-UNIT',
         $                  I-1, T, LDT,
         $                  T( 1, NB ), 1 )
    *
    *           b2 := b2 - V2*w
    *
                CALL SGEMV( '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 STRMV( 'Lower', 'NO TRANSPOSE',
         $                  'UNIT', I-1,
         $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )
                CALL SAXPY( 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 SLARFG( 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(K+1:N,I)
    *
             CALL SGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
         $               ONE, A( K+1, I+1 ),
         $               LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
             CALL SGEMV( 'Transpose', N-K-I+1, I-1,
         $               ONE, A( K+I, 1 ), LDA,
         $               A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
             CALL SGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
         $               Y( K+1, 1 ), LDY,
         $               T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
             CALL SSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
    *
    *        Compute T(1:I,I)
    *
             CALL SSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
             CALL STRMV( '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
    *
    *     Compute Y(1:K,1:NB)
    *
          CALL SLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
          CALL STRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
         $            'UNIT', K, NB,
         $            ONE, A( K+1, 1 ), LDA, Y, LDY )
          IF( N.GT.K+NB )
         $   CALL SGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
         $               NB, N-K-NB, ONE,
         $               A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
         $               LDY )
          CALL STRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
         $            'NON-UNIT', K, NB,
         $            ONE, T, LDT, Y, LDY )
    *
          RETURN
    *
    *     End of SLAHR2
    *
          END