DLABRD (3lapack)

reduce 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

SYNOPSIS

  SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )

      INTEGER        LDA, LDX, LDY, M, N, NB

      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     (output) 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).