SLABRD(l)								 )								 SLABRD(l)

NAME

       SLABRD  -  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 trans-
       formation 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 SLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )

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

	   REAL 	  A( LDA, * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), X( LDX, * ), Y( LDY, * )

PURPOSE

       SLABRD 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 transfor-
       mation  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 SGEBRD

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) REAL 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) REAL array, dimension (NB)
	       The diagonal elements of the first NB rows and columns of the reduced matrix.  D(i) = A(i,i).

       E       (output) REAL array, dimension (NB)
	       The off-diagonal elements of the first NB rows and columns of the reduced matrix.

       TAUQ    (output) REAL array dimension (NB)
	       The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See  Further  Details.	 TAUP	  (output)
	       REAL  array,  dimension	(NB)  The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further
	       Details.  X	 (output) REAL 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) REAL 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).

LAPACK version 3.0						   15 June 2000 							 SLABRD(l)