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RedHat 9 (Linux i386) - man page for sgebd2 (redhat section l)

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 diago-
	       nal  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  sub-
	       diagonal,  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 Fur-
	       ther 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	Further  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)


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