ZGEBD2(1) LAPACK routine (version 3.2) ZGEBD2(1)NAME
ZGEBD2 - reduces 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 lower 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 overwrit‐
ten with the upper bidiagonal matrix B; the elements below the
diagonal, with the array TAUQ, represent the unitary matrix 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
subdiagonal, with the array TAUQ, represent the unitary matrix
Q as a product of elementary reflectors, and the elements above
the diagonal, 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 factors of
the elementary reflectors which represent the unitary matrix P.
See Further Details. WORK (workspace) COMPLEX*16 array,
dimension (max(M,N))
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
FURTHER DETAILS
The matrices Q and P are represented as products of elementary reflec‐
tors:
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 complex 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 routine (version 3.2) November 2008 ZGEBD2(1)
