dlabrd.f

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*> \brief \b DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
*                          LDY )
* 
*       .. Scalar Arguments ..
*       INTEGER            LDA, LDX, LDY, M, N, NB
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
*      $                   TAUQ( * ), X( LDX, * ), Y( LDY, * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLABRD reduces the first NB rows and columns of a real general
*> m by n matrix A to upper or lower bidiagonal form by an orthogonal
*> transformation Q**T * A * P, and returns the matrices X and Y which
*> are needed to apply the transformation to the unreduced part of A.
*>
*> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
*> bidiagonal form.
*>
*> This is an auxiliary routine called by DGEBRD
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows in the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns in the matrix A.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          The number of leading rows and columns of A to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the m by n general matrix to be reduced.
*>          On exit, the first NB rows and columns of the matrix are
*>          overwritten; the rest of the array is unchanged.
*>          If m >= n, elements on and below the diagonal in the first NB
*>            columns, with the array TAUQ, represent the orthogonal
*>            matrix Q as a product of elementary reflectors; and
*>            elements above the diagonal in the first NB rows, with the
*>            array TAUP, represent the orthogonal matrix P as a product
*>            of elementary reflectors.
*>          If m < n, elements below the diagonal in the first NB
*>            columns, with the array TAUQ, represent the orthogonal
*>            matrix Q as a product of elementary reflectors, and
*>            elements on and above the diagonal in the first NB rows,
*>            with the array TAUP, represent the orthogonal matrix P 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 >= max(1,M).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (NB)
*>          The diagonal elements of the first NB rows and columns of
*>          the reduced matrix.  D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (NB)
*>          The off-diagonal elements of the first NB rows and columns of
*>          the reduced matrix.
*> \endverbatim
*>
*> \param[out] TAUQ
*> \verbatim
*>          TAUQ is DOUBLE PRECISION array dimension (NB)
*>          The scalar factors of the elementary reflectors which
*>          represent the orthogonal matrix Q. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP
*> \verbatim
*>          TAUP is DOUBLE PRECISION array, dimension (NB)
*>          The scalar factors of the elementary reflectors which
*>          represent the orthogonal matrix P. See Further Details.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NB)
*>          The m-by-nb matrix X required to update the unreduced part
*>          of A.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X. LDX >= max(1,M).
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*>          Y is DOUBLE PRECISION array, dimension (LDY,NB)
*>          The n-by-nb matrix Y required to update the unreduced part
*>          of A.
*> \endverbatim
*>
*> \param[in] LDY
*> \verbatim
*>          LDY is INTEGER
*>          The leading dimension of the array Y. LDY >= max(1,N).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrices Q and P are represented as products of elementary
*>  reflectors:
*>
*>     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
*>
*>  Each H(i) and G(i) has the form:
*>
*>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
*>
*>  where tauq and taup are real scalars, and v and u are real vectors.
*>
*>  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
*>  A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
*>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*>  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
*>  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
*>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*>  The elements of the vectors v and u together form the m-by-nb matrix
*>  V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
*>  the transformation to the unreduced part of the matrix, using a block
*>  update of the form:  A := A - V*Y**T - X*U**T.
*>
*>  The contents of A on exit are illustrated by the following examples
*>  with nb = 2:
*>
*>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
*>
*>    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
*>    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
*>    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
*>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
*>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
*>    (  v1  v2  a   a   a  )
*>
*>  where a denotes an element of the original matrix which is unchanged,
*>  vi denotes an element of the vector defining H(i), and ui an element
*>  of the vector defining G(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE dlabrd( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
     $                   ldy )
*
*  -- LAPACK auxiliary routine (version 3.4.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     September 2012
*
*     .. Scalar Arguments ..
      INTEGER            LDA, LDX, LDY, M, N, NB
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( lda, * ), D( * ), E( * ), TAUP( * ),
     $                   tauq( * ), x( ldx, * ), y( ldy, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      parameter                ( zero = 0.0d0, one = 1.0d0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
*     ..
*     .. External Subroutines ..
      EXTERNAL           dgemv, dlarfg, dscal
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          min
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( m.LE.0 .OR. n.LE.0 )
     $   RETURN
*
      IF( m.GE.n ) THEN
*
*        Reduce to upper bidiagonal form
*
         DO 10 i = 1, nb
*
*           Update A(i:m,i)
*
            CALL dgemv( 'No transpose', m-i+1, i-1, -one, a( i, 1 ),
     $                  lda, y( i, 1 ), ldy, one, a( i, i ), 1 )
            CALL dgemv( 'No transpose', m-i+1, i-1, -one, x( i, 1 ),
     $                  ldx, a( 1, i ), 1, one, a( i, i ), 1 )
*
*           Generate reflection Q(i) to annihilate A(i+1:m,i)
*
            CALL dlarfg( m-i+1, a( i, i ), a( min( i+1, m ), i ), 1,
     $                   tauq( i ) )
            d( i ) = a( i, i )
            IF( i.LT.n ) THEN
               a( i, i ) = one
*
*              Compute Y(i+1:n,i)
*
               CALL dgemv( 'Transpose', m-i+1, n-i, one, a( i, i+1 ),
     $                     lda, a( i, i ), 1, zero, y( i+1, i ), 1 )
               CALL dgemv( 'Transpose', m-i+1, i-1, one, a( i, 1 ), lda,
     $                     a( i, i ), 1, zero, y( 1, i ), 1 )
               CALL dgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
     $                     ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
               CALL dgemv( 'Transpose', m-i+1, i-1, one, x( i, 1 ), ldx,
     $                     a( i, i ), 1, zero, y( 1, i ), 1 )
               CALL dgemv( 'Transpose', i-1, n-i, -one, a( 1, i+1 ),
     $                     lda, y( 1, i ), 1, one, y( i+1, i ), 1 )
               CALL dscal( n-i, tauq( i ), y( i+1, i ), 1 )
*
*              Update A(i,i+1:n)
*
               CALL dgemv( 'No transpose', n-i, i, -one, y( i+1, 1 ),
     $                     ldy, a( i, 1 ), lda, one, a( i, i+1 ), lda )
               CALL dgemv( 'Transpose', i-1, n-i, -one, a( 1, i+1 ),
     $                     lda, x( i, 1 ), ldx, one, a( i, i+1 ), lda )
*
*              Generate reflection P(i) to annihilate A(i,i+2:n)
*
               CALL dlarfg( n-i, a( i, i+1 ), a( i, min( i+2, n ) ),
     $                      lda, taup( i ) )
               e( i ) = a( i, i+1 )
               a( i, i+1 ) = one
*
*              Compute X(i+1:m,i)
*
               CALL dgemv( 'No transpose', m-i, n-i, one, a( i+1, i+1 ),
     $                     lda, a( i, i+1 ), lda, zero, x( i+1, i ), 1 )
               CALL dgemv( 'Transpose', n-i, i, one, y( i+1, 1 ), ldy,
     $                     a( i, i+1 ), lda, zero, x( 1, i ), 1 )
               CALL dgemv( 'No transpose', m-i, i, -one, a( i+1, 1 ),
     $                     lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
               CALL dgemv( 'No transpose', i-1, n-i, one, a( 1, i+1 ),
     $                     lda, a( i, i+1 ), lda, zero, x( 1, i ), 1 )
               CALL dgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
     $                     ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
               CALL dscal( m-i, taup( i ), x( i+1, i ), 1 )
            END IF
   10    CONTINUE
      ELSE
*
*        Reduce to lower bidiagonal form
*
         DO 20 i = 1, nb
*
*           Update A(i,i:n)
*
            CALL dgemv( 'No transpose', n-i+1, i-1, -one, y( i, 1 ),
     $                  ldy, a( i, 1 ), lda, one, a( i, i ), lda )
            CALL dgemv( 'Transpose', i-1, n-i+1, -one, a( 1, i ), lda,
     $                  x( i, 1 ), ldx, one, a( i, i ), lda )
*
*           Generate reflection P(i) to annihilate A(i,i+1:n)
*
            CALL dlarfg( n-i+1, a( i, i ), a( i, min( i+1, n ) ), lda,
     $                   taup( i ) )
            d( i ) = a( i, i )
            IF( i.LT.m ) THEN
               a( i, i ) = one
*
*              Compute X(i+1:m,i)
*
               CALL dgemv( 'No transpose', m-i, n-i+1, one, a( i+1, i ),
     $                     lda, a( i, i ), lda, zero, x( i+1, i ), 1 )
               CALL dgemv( 'Transpose', n-i+1, i-1, one, y( i, 1 ), ldy,
     $                     a( i, i ), lda, zero, x( 1, i ), 1 )
               CALL dgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
     $                     lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
               CALL dgemv( 'No transpose', i-1, n-i+1, one, a( 1, i ),
     $                     lda, a( i, i ), lda, zero, x( 1, i ), 1 )
               CALL dgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
     $                     ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
               CALL dscal( m-i, taup( i ), x( i+1, i ), 1 )
*
*              Update A(i+1:m,i)
*
               CALL dgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
     $                     lda, y( i, 1 ), ldy, one, a( i+1, i ), 1 )
               CALL dgemv( 'No transpose', m-i, i, -one, x( i+1, 1 ),
     $                     ldx, a( 1, i ), 1, one, a( i+1, i ), 1 )
*
*              Generate reflection Q(i) to annihilate A(i+2:m,i)
*
               CALL dlarfg( m-i, a( i+1, i ), a( min( i+2, m ), i ), 1,
     $                      tauq( i ) )
               e( i ) = a( i+1, i )
               a( i+1, i ) = one
*
*              Compute Y(i+1:n,i)
*
               CALL dgemv( 'Transpose', m-i, n-i, one, a( i+1, i+1 ),
     $                     lda, a( i+1, i ), 1, zero, y( i+1, i ), 1 )
               CALL dgemv( 'Transpose', m-i, i-1, one, a( i+1, 1 ), lda,
     $                     a( i+1, i ), 1, zero, y( 1, i ), 1 )
               CALL dgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
     $                     ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
               CALL dgemv( 'Transpose', m-i, i, one, x( i+1, 1 ), ldx,
     $                     a( i+1, i ), 1, zero, y( 1, i ), 1 )
               CALL dgemv( 'Transpose', i, n-i, -one, a( 1, i+1 ), lda,
     $                     y( 1, i ), 1, one, y( i+1, i ), 1 )
               CALL dscal( n-i, tauq( i ), y( i+1, i ), 1 )
            END IF
   20    CONTINUE
      END IF
      RETURN
*
*     End of DLABRD
*
      END