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# zgebrd

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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).
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