CXML
        	    
LAPACK version 3.0
dgebd2(3)

PURPOSE
  DGEBD2 - reduce a real general m by n matrix A to upper or lower bidiagonal
  form B by an orthogonal transformation

SYNTAX
  SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )

      INTEGER	     INFO, LDA, M, N

      DOUBLE	     PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), TAUQ(
		     * ), WORK( * )

DESCRIPTION
  DGEBD2 reduces a real general m by n matrix A to upper or lower bidiagonal
  form B by an orthogonal transformation: Q' * A * P = B.  If m >= n, B is
  upper bidiagonal; if m < n, B is lower bidiagonal.

ARGUMENTS
  M	  (input) INTEGER
	  The number of rows in the matrix A.  M >= 0.

  N	  (input) INTEGER
	  The number of columns in the matrix A.  N >= 0.

  A	  (input/output) DOUBLE PRECISION array, dimension (LDA,N)
	  On entry, the m by n general matrix to be reduced.  On exit, if m
	  >= n, the diagonal and the first superdiagonal are overwritten with
	  the upper bidiagonal matrix B; the elements below the diagonal,
	  with the array TAUQ, represent the orthogonal matrix Q as a product
	  of elementary reflectors, and the elements above the first
	  superdiagonal, with the array TAUP, represent the orthogonal matrix
	  P as a product of elementary reflectors; if m < n, the diagonal and
	  the first subdiagonal are overwritten with the lower bidiagonal
	  matrix B; the elements below the first subdiagonal, with the array
	  TAUQ, represent the orthogonal matrix Q as a product of elementary
	  reflectors, and the elements above the diagonal, 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 (min(M,N))
	  The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).

  E	  (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
	  The off-diagonal elements of the bidiagonal matrix B: if m >= n,
	  E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for
	  i = 1,2,...,m-1.

  TAUQ	  (output) DOUBLE PRECISION array dimension (min(M,N))
	  The scalar factors of the elementary reflectors which represent the
	  orthogonal matrix Q. See Further Details.  TAUP    (output) DOUBLE
	  PRECISION array, dimension (min(M,N)) The scalar factors of the
	  elementary reflectors which represent the orthogonal matrix P. See
	  Further Details.  WORK    (workspace) DOUBLE PRECISION array,
	  dimension (max(M,N))

  INFO	  (output) INTEGER
	  = 0: successful exit.
	  < 0: if INFO = -i, the i-th argument had an illegal value.

FURTHER DETAILS
  The matrices Q and P are represented as products of elementary reflectors:

  If m >= n,

     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

  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; v(1:i-
  1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); u(1:i) = 0,
  u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); tauq is stored in
  TAUQ(i) and taup in TAUP(i).

  If m < n,

     Q = H(1) H(2) . . . H(m-1)	 and  P = G(1) G(2) . . . G(m)

  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; v(1:i)
  = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); u(1:i-1) =
  0, u(i) = 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).

  The contents of A on exit are illustrated by the following examples:

  m = 6 and n = 5 (m > n):	    m = 5 and n = 6 (m < n):

    (  d   e   u1  u1  u1 )	      (	 d   u1	 u1  u1	 u1  u1 )
    (  v1  d   e   u2  u2 )	      (	 e   d	 u2  u2	 u2  u2 )
    (  v1  v2  d   e   u3 )	      (	 v1  e	 d   u3	 u3  u3 )
    (  v1  v2  v3  d   e  )	      (	 v1  v2	 e   d	 u4  u4 )
    (  v1  v2  v3  v4  d  )	      (	 v1  v2	 v3  e	 d   u5 )
    (  v1  v2  v3  v4  v5 )

  where d and e denote diagonal and off-diagonal elements of B, vi denotes an
  element of the vector defining H(i), and ui an element of the vector
  defining G(i).