
SLAHRD(l) ) SLAHRD(l)
NAME
SLAHRD  reduce the first NB columns of a real general nby(nk+1) matrix A so that ele
ments below the kth subdiagonal are zero
SYNOPSIS
SUBROUTINE SLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
INTEGER K, LDA, LDT, LDY, N, NB
REAL A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )
PURPOSE
SLAHRD reduces the first NB columns of a real general nby(nk+1) matrix A so that ele
ments below the kth subdiagonal are zero. The reduction is performed by an orthogonal
similarity transformation Q' * A * Q. The routine returns the matrices V and T which
determine Q as a block reflector I  V*T*V', and also the matrix Y = A * V * T.
This is an auxiliary routine called by SGEHRD.
ARGUMENTS
N (input) INTEGER
The order of the matrix A.
K (input) INTEGER
The offset for the reduction. Elements below the kth subdiagonal in the first NB
columns are reduced to zero.
NB (input) INTEGER
The number of columns to be reduced.
A (input/output) REAL array, dimension (LDA,NK+1)
On entry, the nby(nk+1) general matrix A. On exit, the elements on and above
the kth subdiagonal in the first NB columns are overwritten with the correspond
ing elements of the reduced matrix; the elements below the kth subdiagonal, with
the array TAU, represent the matrix Q as a product of elementary reflectors. The
other columns of A are unchanged. See Further Details. LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
TAU (output) REAL array, dimension (NB)
The scalar factors of the elementary reflectors. See Further Details.
T (output) REAL array, dimension (LDT,NB)
The upper triangular matrix T.
LDT (input) INTEGER
The leading dimension of the array T. LDT >= NB.
Y (output) REAL array, dimension (LDY,NB)
The nbynb matrix Y.
LDY (input) INTEGER
The leading dimension of the array Y. LDY >= N.
FURTHER DETAILS
The matrix Q is represented as a product of nb elementary reflectors
Q = H(1) H(2) . . . H(nb).
Each H(i) has the form
H(i) = I  tau * v * v'
where tau is a real scalar, and v is a real vector with
v(1:i+k1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in
TAU(i).
The elements of the vectors v together form the (nk+1)bynb matrix V which is needed,
with T and Y, to apply the transformation to the unreduced part of the matrix, using an
update of the form: A := (I  V*T*V') * (A  Y*V').
The contents of A on exit are illustrated by the following example with n = 7, k = 3 and
nb = 2:
( a h a a a )
( a h a a a )
( a h a a a )
( h h a a a )
( v1 h a a a )
( v1 v2 a a a )
( v1 v2 a a a )
where a denotes an element of the original matrix A, h denotes a modified element of the
upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i).
LAPACK version 3.0 15 June 2000 SLAHRD(l) 
