SGEBD2(l)								 )								 SGEBD2(l)

NAME

       SGEBD2 - reduce a real general m by n matrix A to upper or lower bidiagonal form B by an orthogonal transformation

SYNOPSIS

       SUBROUTINE SGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )

	   INTEGER	  INFO, LDA, M, N

	   REAL 	  A( LDA, * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), WORK( * )

PURPOSE

       SGEBD2 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) REAL 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) REAL array, dimension (min(M,N))
	       The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).

       E       (output) REAL 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) REAL array dimension (min(M,N))
	       The  scalar  factors  of  the elementary reflectors which represent the orthogonal matrix Q. See Further Details.  TAUP	  (output)
	       REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See  Fur-
	       ther Details.  WORK    (workspace) REAL 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).

LAPACK version 3.0						   15 June 2000 							 SGEBD2(l)