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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 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 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 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 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 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) 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 elementary reflectors
which represent the unitary matrix P. See Further
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 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).