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 zlabrd.f(3) LAPACK zlabrd.f(3)

zlabrd.f -

# SYNOPSIS

## Functions/Subroutines

subroutine zlabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)

ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

# Function/Subroutine Documentation

## subroutine zlabrd (integerM, integerN, integerNB, complex*16, dimension( lda, * )A, integerLDA, double precision, dimension( * )D, double precision, dimension( * )E, complex*16, dimension( * )TAUQ, complex*16, dimension( * )TAUP, complex*16, dimension( ldx, * )X, integerLDX, complex*16, dimension( ldy, * )Y, integerLDY)

ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
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**H * A * P, and returns the matrices X and Y which
are needed to apply the transformation 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
```
Parameters:
M
```          M is INTEGER
The number of rows in the matrix A.
```
N
```          N is INTEGER
The number of columns in the matrix A.
```
NB
```          NB is INTEGER
The number of leading rows and columns of A to be reduced.
```
A
```          A is 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 reflectors; 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
```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,M).
```
D
```          D is 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
```          E is DOUBLE PRECISION array, dimension (NB)
The off-diagonal elements of the first NB rows and columns of
the reduced matrix.
```
TAUQ
```          TAUQ is COMPLEX*16 array dimension (NB)
The scalar factors of the elementary reflectors which
represent the unitary matrix Q. See Further Details.
```
TAUP
```          TAUP is COMPLEX*16 array, dimension (NB)
The scalar factors of the elementary reflectors which
represent the unitary matrix P. See Further Details.
```
X
```          X is COMPLEX*16 array, dimension (LDX,NB)
The m-by-nb matrix X required to update the unreduced part
of A.
```
LDX
```          LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,M).
```
Y
```          Y is COMPLEX*16 array, dimension (LDY,NB)
The n-by-nb matrix Y required to update the unreduced part
of A.
```
LDY
```          LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
```
Author:
Univ. of Tennessee
Univ. of California Berkeley
NAG Ltd.
Date:
September 2012
Further Details:
```  The matrices Q and P are represented as products of elementary
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**H  and G(i) = I - taup * u * u**H

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**H 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**H - X*U**H.

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).
```
Definition at line 212 of file zlabrd.f.

# Author

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