CLABRD(l) ) CLABRD(l)NAME
CLABRD - 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 trans‐
formation Q' * A * P, and returns the matrices X and Y which are needed
to apply the transformation to the unreduced part of A
SYNOPSIS
SUBROUTINE CLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )
INTEGER LDA, LDX, LDY, M, N, NB
REAL D( * ), E( * )
COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ), Y(
LDY, * )
PURPOSE
CLABRD 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 trans‐
formation Q' * 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 CGEBRD
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 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, repre‐
sent the unitary matrix Q as a product of elementary reflec‐
tors, 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).
D (output) REAL array, dimension (NB)
The diagonal elements of the first NB rows and columns of the
reduced matrix. D(i) = A(i,i).
E (output) REAL array, dimension (NB)
The off-diagonal elements of the first NB rows and columns of
the reduced matrix.
TAUQ (output) COMPLEX array dimension (NB)
The scalar factors of the elementary reflectors which represent
the unitary matrix Q. See Further Details. TAUP (output)
COMPLEX array, dimension (NB) The scalar factors of the elemen‐
tary reflectors which represent the unitary matrix P. See Fur‐
ther Details. X (output) COMPLEX 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 array, dimension (LDY,NB)
The n-by-nb matrix Y required to update the unreduced 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 elementary reflec‐
tors:
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 vec‐
tors.
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).
LAPACK version 3.0 15 June 2000 CLABRD(l)
