dlabrd.f

          SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
         $                   LDY )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            LDA, LDX, LDY, M, N, NB
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
         $                   TAUQ( * ), X( LDX, * ), Y( LDY, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  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' * 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
    *
    *  Arguments
    *  =========
    *
    *  M       (input) INTEGER
    *          The number of rows in the matrix A.
    *
    *  N       (input) INTEGER
    *          The number of columns in the matrix A.
    *
    *  NB      (input) INTEGER
    *          The number of leading rows and columns of A to be reduced.
    *
    *  A       (input/output) 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.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,M).
    *
    *  D       (output) DOUBLE PRECISION array, dimension (NB)
    *          The diagonal elements of the first NB rows and columns of
    *          the reduced matrix.  D(i) = A(i,i).
    *
    *  E       (output) DOUBLE PRECISION array, dimension (NB)
    *          The off-diagonal elements of the first NB rows and columns of
    *          the reduced matrix.
    *
    *  TAUQ    (output) DOUBLE PRECISION array dimension (NB)
    *          The scalar factors of the elementary reflectors which
    *          represent the orthogonal matrix Q. See Further Details.
    *
    *  TAUP    (output) DOUBLE PRECISION array, dimension (NB)
    *          The scalar factors of the elementary reflectors which
    *          represent the orthogonal matrix P. See Further Details.
    *
    *  X       (output) DOUBLE PRECISION array, dimension (LDX,NB)
    *          The m-by-nb matrix X required to update the unreduced part
    *          of A.
    *
    *  LDX     (input) INTEGER
    *          The leading dimension of the array X. LDX >= M.
    *
    *  Y       (output) DOUBLE PRECISION array, dimension (LDY,NB)
    *          The n-by-nb matrix Y required to update the unreduced part
    *          of A.
    *
    *  LDY     (input) INTEGER
    *          The leading dimension of the array Y. LDY >= N.
    *
    *  Further Details
    *  ===============
    *
    *  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'  and G(i) = I - taup * u * u'
    *
    *  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' 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' - X*U'.
    *
    *  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).
    *
    *  =====================================================================
    *
    *     .. 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