zgebd2(3)

NAME

ZGEBD2 - reduce a complex general m by n matrix A to upper

or lower real bidiagonal form B by a unitary transformation

SYNOPSIS

SUBROUTINE  ZGEBD2(  M, N, A, LDA, D, E, TAUQ, TAUP, WORK,
INFO )
    INTEGER        INFO, LDA, M, N
    DOUBLE         PRECISION D( * ), E( * )
    COMPLEX*16     A( LDA, * ), TAUP(  *  ),  TAUQ(  *  ),
WORK( * )

PURPOSE

ZGEBD2 reduces a complex general m by n matrix A to upper

or lower real bidiagonal form B by a unitary transformation: Q' *

A * P = B. If m >= n, B is upper bidiagonal; if m < n, B is low

er 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) COMPLEX*16 array, dimension (LDA,N)

On entry, the m by n general matrix to be reduced.

On exit, if m >= n, the diagonal and the first superdiagonal are

overwritten with the upper bidiagonal matrix B; the elements be

low the diagonal, with the array TAUQ, represent the unitary ma

trix Q as a product of elementary reflectors, and the elements

above the first superdiagonal, with the array TAUP, represent the

unitary 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 subdiago

nal, with the array TAUQ, represent the unitary matrix Q as a

product of elementary reflectors, and the elements above the di

agonal, 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

(min(M,N))

The diagonal elements of the bidiagonal matrix B:

D(i) = A(i,i).

E (output) DOUBLE PRECISION 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) COMPLEX*16 array dimension (min(M,N))

The scalar factors of the elementary reflectors

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

(output) COMPLEX*16 array, dimension (min(M,N)) The scalar fac

tors of the elementary reflectors which represent the unitary ma

trix P. See Further Details. WORK (workspace) COMPLEX*16 ar

ray, dimension (max(M,N))

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value.

FURTHER DETAILS

The matrices Q and P are represented as products of ele

mentary 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 complex scalars, and v and u are

complex 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 complex scalars, v and u are com

plex 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

ZGEBD2(3)