
SLABRD(l) ) SLABRD(l)
NAME
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 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 transformation 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 elementary 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 elementary 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 offdiagonal 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 mby
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 nbynb matrix Y required to update the unreduced 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 elementary 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:i1) = 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:i1) =
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 mbynb matrix V and the nbbyn
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 ele
ment of the vector defining H(i), and ui an element of the vector defining G(i).
LAPACK version 3.0 15 June 2000 SLABRD(l) 
