.TH ZLABRD 1 "November 2006" " LAPACK auxiliary routine (version 3.1) " " LAPACK auxiliary routine (version 3.1) "
.SH NAME
ZLABRD - 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\(aq * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A
.SH SYNOPSIS
.TP 19
SUBROUTINE ZLABRD(
M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
LDY )
.TP 19
.ti +4
INTEGER
LDA, LDX, LDY, M, N, NB
.TP 19
.ti +4
DOUBLE
PRECISION D( * ), E( * )
.TP 19
.ti +4
COMPLEX*16
A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ),
Y( LDY, * )
.SH 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\(aq * 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.
.br
This is an auxiliary routine called by ZGEBRD
.br
.SH ARGUMENTS
.TP 8
M (input) INTEGER
The number of rows in the matrix A.
.TP 8
N (input) INTEGER
The number of columns in the matrix A.
.TP 8
NB (input) INTEGER
The number of leading rows and columns of A to be reduced.
.TP 8
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 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 (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
.TP 8
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).
.TP 8
E (output) DOUBLE PRECISION array, dimension (NB)
The off-diagonal elements of the first NB rows and columns of
the reduced matrix.
.TP 8
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, dimension (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.
.TP 8
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,M).
.TP 8
Y (output) COMPLEX*16 array, dimension (LDY,NB)
The n-by-nb matrix Y required to update the unreduced part
of A.
.TP 8
LDY (input) INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
.SH FURTHER DETAILS
The matrices Q and P are represented as products of elementary
reflectors:
.br
Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
Each H(i) and G(i) has the form:
.br
H(i) = I - tauq * v * v\(aq and G(i) = I - taup * u * u\(aq
where tauq and taup are complex scalars, and v and u are complex
vectors.
.br
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\(aq 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\(aq - X*U\(aq.
.br
The contents of A on exit are illustrated by the following examples
with nb = 2:
.br
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 )
.br
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).
.br