cgebd2(3)

NAME

CGEBD2 - reduce a complex general m by n matrix A to upper

or lower real bidiagonal form B by a unitary transformation

SYNOPSIS

SUBROUTINE  CGEBD2(  M, N, A, LDA, D, E, TAUQ, TAUP, WORK,
INFO )
    INTEGER        INFO, LDA, M, N
    REAL           D( * ), E( * )
    COMPLEX        A( LDA, * ), TAUP(  *  ),  TAUQ(  *  ),
WORK( * )

PURPOSE

CGEBD2 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 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) REAL array, dimension (min(M,N))

The diagonal elements of the bidiagonal matrix B:

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

E (output) REAL 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 array dimension (min(M,N))

The scalar factors of the elementary reflectors

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

(output) COMPLEX array, dimension (min(M,N)) The scalar factors

of the elementary reflectors which represent the unitary matrix

P. See Further Details. WORK (workspace) COMPLEX array, di

mension (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

CGEBD2(3)