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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).