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NAME
ZGEBRD - reduce a general complex M-by-N matrix A to upper
or lower bidiagonal form B by a unitary transformation
SYNOPSIS
SUBROUTINE ZGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK,
LWORK, INFO )
INTEGER INFO, LDA, LWORK, M, N
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK(
LWORK )
PURPOSE
ZGEBRD reduces a general complex M-by-N matrix A to upper or
lower bidiagonal form B by a unitary transformation: Q**H *
A * P = B.
If m >= n, B is upper bidiagonal; if m < n, B is lower bidi-
agonal.
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 bidiag-
onal 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 ele-
mentary 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 uni-
tary 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, dimen-
sion (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 (LWORK) On exit, if INFO = 0,
WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The length of the array WORK. LWORK >= max(1,M,N).
For optimum performance LWORK >= (M+N)*NB, where NB
is the optimal blocksize.
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 elemen-
tary 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, and 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).