zlabrd(3)

NAME

ZLABRD - reduce the first NB rows and columns of a complex

general m by n matrix A to upper or lower real bidiagonal form by

a unitary 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 ZLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP,  X,
LDX, Y, LDY )
    INTEGER        LDA, LDX, LDY, M, N, NB
    DOUBLE         PRECISION D( * ), E( * )
    COMPLEX*16      A(  LDA, * ), TAUP( * ), TAUQ( * ), X(
LDX, * ), Y( LDY, * )

PURPOSE

ZLABRD reduces the first NB rows and columns of a complex

general m by n matrix A to upper or lower real bidiagonal form by

a unitary 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 ZGEBRD

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) COMPLEX*16 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 unitary matrix Q as a product of elementary

reflectors; and elements above the diagonal in the first NB rows,

with the array TAUP, represent the unitary 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 uni

tary 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 unitary 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) DOUBLE PRECISION array, dimension (NB)

The diagonal elements of the first NB rows and

columns of the reduced matrix. D(i) = A(i,i).

E (output) DOUBLE PRECISION array, dimension (NB)

The off-diagonal elements of the first NB rows and

columns of the reduced matrix.

TAUQ (output) COMPLEX*16 array dimension (NB)

The scalar factors of the elementary reflectors

which represent the unitary matrix Q. See Further Details. TAUP

(output) COMPLEX*16 array, dimension (NB) The scalar factors of

the elementary reflectors which represent the unitary matrix P.

See Further Details. X (output) COMPLEX*16 array, dimen

sion (LDX,NB) The m-by-nb matrix X required to update the unre

duced part of A.

LDX (input) INTEGER

The leading dimension of the array X. LDX >=

max(1,M).

Y (output) COMPLEX*16 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 >=

max(1,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 complex scalars, and v and u are

complex 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

ZLABRD(3)