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

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 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 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 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 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  ele-
       ment 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)


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