dlatrd.f

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*> \brief \b DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            LDA, LDW, N, NB
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLATRD reduces NB rows and columns of a real symmetric matrix A to
*> symmetric tridiagonal form by an orthogonal similarity
*> transformation Q**T * 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.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored:
*>          = 'U': Upper triangular
*>          = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          The number of rows and columns to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= (1,N).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is 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.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is 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.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (LDW,NB)
*>          The n-by-nb matrix W required to update the unreduced part
*>          of A.
*> \endverbatim
*>
*> \param[in] LDW
*> \verbatim
*>          LDW is INTEGER
*>          The leading dimension of the array W. LDW >= max(1,N).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup latrd
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  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**T
*>
*>  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**T
*>
*>  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**T - W*V**T.
*>
*>  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).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE dlatrd( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            LDA, LDW, N, NB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
*     ..
*
*  =====================================================================
*
*     .. 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