dlatrd.f

Go to the documentation of this file.

      SUBROUTINE dlatrd( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            LDA, LDW, N, NB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
*     ..
*
*  Purpose
*  =======
*
*  DLATRD reduces NB rows and columns of a real symmetric matrix A to
*  symmetric tridiagonal form by an orthogonal similarity
*  transformation Q' * A * Q, and returns the matrices V and W which are
*  needed to apply the transformation to the unreduced part of A.
*
*  If UPLO = 'U', DLATRD reduces the last NB rows and columns of a
*  matrix, of which the upper triangle is supplied;
*  if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
*  matrix, of which the lower triangle is supplied.
*
*  This is an auxiliary routine called by DSYTRD.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER
*          Specifies whether the upper or lower triangular part of the
*          symmetric matrix A is stored:
*          = 'U': Upper triangular
*          = 'L': Lower triangular
*
*  N       (input) INTEGER
*          The order of the matrix A.
*
*  NB      (input) INTEGER
*          The number of rows and columns to be reduced.
*
*  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
*          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*          n-by-n upper triangular part of A contains the upper
*          triangular part of the matrix A, and the strictly lower
*          triangular part of A is not referenced.  If UPLO = 'L', the
*          leading n-by-n lower triangular part of A contains the lower
*          triangular part of the matrix A, and the strictly upper
*          triangular part of A is not referenced.
*          On exit:
*          if UPLO = 'U', the last NB columns have been reduced to
*            tridiagonal form, with the diagonal elements overwriting
*            the diagonal elements of A; the elements above the diagonal
*            with the array TAU, represent the orthogonal matrix Q as a
*            product of elementary reflectors;
*          if UPLO = 'L', the first NB columns have been reduced to
*            tridiagonal form, with the diagonal elements overwriting
*            the diagonal elements of A; the elements below the diagonal
*            with the array TAU, represent the  orthogonal matrix Q as a
*            product of elementary reflectors.
*          See Further Details.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= (1,N).
*
*  E       (output) DOUBLE PRECISION array, dimension (N-1)
*          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
*          elements of the last NB columns of the reduced matrix;
*          if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
*          the first NB columns of the reduced matrix.
*
*  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
*          The scalar factors of the elementary reflectors, stored in
*          TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
*          See Further Details.
*
*  W       (output) DOUBLE PRECISION array, dimension (LDW,NB)
*          The n-by-nb matrix W required to update the unreduced part
*          of A.
*
*  LDW     (input) INTEGER
*          The leading dimension of the array W. LDW >= max(1,N).
*
*  Further Details
*  ===============
*
*  If UPLO = 'U', the matrix Q is represented as a product of elementary
*  reflectors
*
*     Q = H(n) H(n-1) . . . H(n-nb+1).
*
*  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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
*  and tau in TAU(i-1).
*
*  If UPLO = 'L', the matrix Q is represented as a product of 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) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
*  and tau in TAU(i).
*
*  The elements of the vectors v together form the n-by-nb matrix V
*  which is needed, with W, to apply the transformation to the unreduced
*  part of the matrix, using a symmetric rank-2k update of the form:
*  A := A - V*W' - W*V'.
*
*  The contents of A on exit are illustrated by the following examples
*  with n = 5 and nb = 2:
*
*  if UPLO = 'U':                       if UPLO = 'L':
*
*    (  a   a   a   v4  v5 )              (  d                  )
*    (      a   a   v4  v5 )              (  1   d              )
*    (          a   1   v5 )              (  v1  1   a          )
*    (              d   1  )              (  v1  v2  a   a      )
*    (                  d  )              (  v1  v2  a   a   a  )
*
*  where d denotes a diagonal element of the reduced matrix, a denotes
*  an element of the original matrix that is unchanged, and vi denotes
*  an element of the vector defining H(i).
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, HALF
      parameter( zero = 0.0d+0, one = 1.0d+0, half = 0.5d+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IW
      DOUBLE PRECISION   ALPHA
*     ..
*     .. External Subroutines ..
      EXTERNAL           daxpy, dgemv, dlarfg, dscal, dsymv
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT
      EXTERNAL           lsame, ddot
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          min
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( n.LE.0 )
     $   RETURN
*
      IF( lsame( uplo, 'U' ) ) THEN
*
*        Reduce last NB columns of upper triangle
*
         DO 10 i = n, n - nb + 1, -1
            iw = i - n + nb
            IF( i.LT.n ) THEN
*
*              Update A(1:i,i)
*
               CALL dgemv( 'No transpose', i, n-i, -one, a( 1, i+1 ),
     $                     lda, w( i, iw+1 ), ldw, one, a( 1, i ), 1 )
               CALL dgemv( 'No transpose', i, n-i, -one, w( 1, iw+1 ),
     $                     ldw, a( i, i+1 ), lda, one, a( 1, i ), 1 )
            END IF
            IF( i.GT.1 ) THEN
*
*              Generate elementary reflector H(i) to annihilate
*              A(1:i-2,i)
*
               CALL dlarfg( i-1, a( i-1, i ), a( 1, i ), 1, tau( i-1 ) )
               e( i-1 ) = a( i-1, i )
               a( i-1, i ) = one
*
*              Compute W(1:i-1,i)
*
               CALL dsymv( 'Upper', i-1, one, a, lda, a( 1, i ), 1,
     $                     zero, w( 1, iw ), 1 )
               IF( i.LT.n ) THEN
                  CALL dgemv( 'Transpose', i-1, n-i, one, w( 1, iw+1 ),
     $                        ldw, a( 1, i ), 1, zero, w( i+1, iw ), 1 )
                  CALL dgemv( 'No transpose', i-1, n-i, -one,
     $                        a( 1, i+1 ), lda, w( i+1, iw ), 1, one,
     $                        w( 1, iw ), 1 )
                  CALL dgemv( 'Transpose', i-1, n-i, one, a( 1, i+1 ),
     $                        lda, a( 1, i ), 1, zero, w( i+1, iw ), 1 )
                  CALL dgemv( 'No transpose', i-1, n-i, -one,
     $                        w( 1, iw+1 ), ldw, w( i+1, iw ), 1, one,
     $                        w( 1, iw ), 1 )
               END IF
               CALL dscal( i-1, tau( i-1 ), w( 1, iw ), 1 )
               alpha = -half*tau( i-1 )*ddot( i-1, w( 1, iw ), 1,
     $                 a( 1, i ), 1 )
               CALL daxpy( i-1, alpha, a( 1, i ), 1, w( 1, iw ), 1 )
            END IF
*
   10    CONTINUE
      ELSE
*
*        Reduce first NB columns of lower triangle
*
         DO 20 i = 1, nb
*
*           Update A(i:n,i)
*
            CALL dgemv( 'No transpose', n-i+1, i-1, -one, a( i, 1 ),
     $                  lda, w( i, 1 ), ldw, one, a( i, i ), 1 )
            CALL dgemv( 'No transpose', n-i+1, i-1, -one, w( i, 1 ),
     $                  ldw, a( i, 1 ), lda, one, a( i, i ), 1 )
            IF( i.LT.n ) THEN
*
*              Generate elementary reflector H(i) to annihilate
*              A(i+2:n,i)
*
               CALL dlarfg( n-i, a( i+1, i ), a( min( i+2, n ), i ), 1,
     $                      tau( i ) )
               e( i ) = a( i+1, i )
               a( i+1, i ) = one
*
*              Compute W(i+1:n,i)
*
               CALL dsymv( 'Lower', n-i, one, a( i+1, i+1 ), lda,
     $                     a( i+1, i ), 1, zero, w( i+1, i ), 1 )
               CALL dgemv( 'Transpose', n-i, i-1, one, w( i+1, 1 ), ldw,
     $                     a( i+1, i ), 1, zero, w( 1, i ), 1 )
               CALL dgemv( 'No transpose', n-i, i-1, -one, a( i+1, 1 ),
     $                     lda, w( 1, i ), 1, one, w( i+1, i ), 1 )
               CALL dgemv( 'Transpose', n-i, i-1, one, a( i+1, 1 ), lda,
     $                     a( i+1, i ), 1, zero, w( 1, i ), 1 )
               CALL dgemv( 'No transpose', n-i, i-1, -one, w( i+1, 1 ),
     $                     ldw, w( 1, i ), 1, one, w( i+1, i ), 1 )
               CALL dscal( n-i, tau( i ), w( i+1, i ), 1 )
               alpha = -half*tau( i )*ddot( n-i, w( i+1, i ), 1,
     $                 a( i+1, i ), 1 )
               CALL daxpy( n-i, alpha, a( i+1, i ), 1, w( i+1, i ), 1 )
            END IF
*
   20    CONTINUE
      END IF
*
      RETURN
*
*     End of DLATRD
*
      END