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# cgebrd

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NAME
CGEBRD - reduce a general complex M-by-N matrix A to upper
or lower bidiagonal form B by a unitary transformation

SYNOPSIS
SUBROUTINE CGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK,
LWORK, INFO )

INTEGER        INFO, LDA, LWORK, M, N

REAL           D( * ), E( * )

COMPLEX        A( LDA, * ), TAUP( * ), TAUQ( * ), WORK(
LWORK )

PURPOSE
CGEBRD 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 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) 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,
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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