slabrd(3)

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 unre

duced 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 unre

duced 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 over

written; 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 elemen

tary 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) IN

TEGER 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 unre

duced 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 ele

mentary 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(3)