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# zlabrd

```
NAME
ZLABRD - reduce the first NB rows and columns of a complex
general m by n matrix A to upper or lower real bidiagonal
form by a unitary transformation Q' * A * P, and returns the
matrices X and Y which are needed to apply the transforma-
tion to the unreduced part of A

SYNOPSIS
SUBROUTINE ZLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X,
LDX, Y, LDY )

INTEGER        LDA, LDX, LDY, M, N, NB

DOUBLE         PRECISION D( * ), E( * )

COMPLEX*16     A( LDA, * ), TAUP( * ), TAUQ( * ), X(
LDX, * ), Y( LDY, * )

PURPOSE
ZLABRD reduces the first NB rows and columns of a complex
general m by n matrix A to upper or lower real bidiagonal
form by a unitary transformation Q' * A * P, and returns the
matrices X and Y which are needed to apply the transforma-
tion to the unreduced part of A.

If m >= n, A is reduced to upper bidiagonal form; if m < n,
to lower bidiagonal form.

This is an auxiliary routine called by ZGEBRD

ARGUMENTS
M       (input) INTEGER
The number of rows in the matrix A.

N       (input) INTEGER
The number of columns in the matrix A.

NB      (input) INTEGER
The number of leading rows and columns of A to be
reduced.

A       (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the m by n general matrix to be reduced.
On exit, the first NB rows and columns of the matrix
are overwritten; the rest of the array is unchanged.
If m >= n, elements on and below the diagonal in the
first NB columns, with the array TAUQ, represent the
unitary matrix Q as a product of elementary reflec-
tors; and elements above the diagonal in the first
NB rows, with the array TAUP, represent the unitary
matrix P as a product of elementary reflectors.  If

m < n, elements below the diagonal in the first NB
columns, with the array TAUQ, represent the unitary
matrix Q as a product of elementary reflectors, and
elements on and above the diagonal in the first NB
rows, with the array TAUP, represent the unitary
matrix P as a product of elementary reflectors.  See
Further Details.  LDA     (input) INTEGER The lead-
ing dimension of the array A.  LDA >= max(1,M).

D       (output) DOUBLE PRECISION array, dimension (NB)
The diagonal elements of the first NB rows and
columns of the reduced matrix.  D(i) = A(i,i).

E       (output) DOUBLE PRECISION array, dimension (NB)
The off-diagonal elements of the first NB rows and
columns of the reduced matrix.

TAUQ    (output) COMPLEX*16 array dimension (NB)
The scalar factors of the elementary reflectors
which represent the unitary matrix Q. See Further
Details.  TAUP    (output) COMPLEX*16 array, dimen-
sion (NB) The scalar factors of the elementary
reflectors which represent the unitary matrix P. See
Further Details.  X       (output) COMPLEX*16 array,
dimension (LDX,NB) The m-by-nb matrix X required to
update the unreduced part of A.

LDX     (input) INTEGER
The leading dimension of the array X. LDX >=
max(1,M).

Y       (output) COMPLEX*16 array, dimension (LDY,NB)
The n-by-nb matrix Y required to update the unre-
duced part of A.

LDY     (output) INTEGER
The leading dimension of the array Y. LDY >=
max(1,N).

FURTHER DETAILS
The matrices Q and P are represented as products of elemen-
tary reflectors:

Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

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.

If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on
exit in A(i:m,i); u(1:i) = 0, u(i+1) = 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).

If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on
exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is
stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and
taup in TAUP(i).

The elements of the vectors v and u together form the m-by-
nb matrix V and the nb-by-n matrix U' which are needed, with
X and Y, to apply the transformation to the unreduced part
of the matrix, using a block update of the form:  A := A -
V*Y' - X*U'.

The contents of A on exit are illustrated by the following
examples with nb = 2:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

(  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1
u1 )
(  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2
u2 )
(  v1  v2  a   a   a  )           (  v1  1   a   a   a   a
)
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a
)
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a
)
(  v1  v2  a   a   a  )

where a denotes an element of the original matrix which is
unchanged, vi denotes an element of the vector defining
H(i), and ui an element of the vector defining G(i).
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