SLABRD(1) LAPACK auxiliary routine (version 3.2) SLABRD(1)NAME
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 transforma‐
tion 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 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 transforma‐
tion 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 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 product of elemen‐
tary reflectors; and elements above the diagonal in the first
NB rows, with the array TAUP, represent the orthogonal 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 orthogonal matrix Q as a product of elemen‐
tary reflectors, and elements on and above the diagonal in the
first NB rows, with the array TAUP, represent the orthogonal
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) REAL array dimension (NB)
The scalar factors of the elementary reflectors which represent
the orthogonal matrix Q. See Further Details. TAUP (output)
REAL array, dimension (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 unreduced part of
A.
LDY (input) INTEGER
The leading dimension of the array Y. LDY >= 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 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 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 auxiliary routine (versioNovember 2008 SLABRD(1)
