SLABRD - reduce the first NB rows and columns of a real
general m by n matrix A to upper or lower bidiagonal form
by an orthogonal transformation Q' * A * P, and returns
the matrices X and Y which are needed to apply the trans
formation to the unreduced part of A
SYNOPSIS
SUBROUTINE SLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X,
LDX, Y, LDY )
INTEGER LDA, LDX, LDY, M, N, NB
REAL A( LDA, * ), D( * ), E( * ), TAUP( * ),
TAUQ( * ), X( LDX, * ), Y( LDY, * )
PURPOSE
SLABRD reduces the first NB rows and columns of a real
general m by n matrix A to upper or lower bidiagonal form
by an orthogonal transformation Q' * A * P, and returns
the matrices X and Y which are needed to apply the trans
formation to the unreduced part of A. If m >= n, A is
reduced to upper bidiagonal form; if m < n, to lower bidi
agonal form.
This is an auxiliary routine called by SGEBRD
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) REAL 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 orthogonal matrix Q as a prod
uct of elementary reflectors; and elements above
the diagonal in the first NB rows, with the array
TAUP, represent the orthogonal matrix P as a prod
uct of elementary reflectors. If m < n, elements
below the diagonal in the first NB columns, with
the array TAUQ, represent the orthogonal matrix Q
as a product of elementary reflectors, and ele
ments on and above the diagonal in the first NB
nal matrix P as a product of elementary reflec
tors. See Further Details. LDA (input) INTE
GER 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) REAL array dimension (NB)
The scalar factors of the elementary reflectors
which represent the orthogonal matrix Q. See Fur
ther Details. TAUP (output) REAL array, dimen
sion (NB) The scalar factors of the elementary
reflectors which represent the orthogonal matrix
P. See Further Details. X (output) REAL
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 >= M.
Y (output) REAL 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 >= N.
FURTHER DETAILS
The matrices Q and P are represented as products of ele
mentary 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 real scalars, and v and u are real
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
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).
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