CLAHRD(l) ) CLAHRD(l)NAME
CLAHRD - reduce the first NB columns of a complex general n-by-(n-k+1)
matrix A so that elements below the k-th subdiagonal are zero
SYNOPSIS
SUBROUTINE CLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
INTEGER K, LDA, LDT, LDY, N, NB
COMPLEX A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )
PURPOSE
CLAHRD reduces the first NB columns of a complex general n-by-(n-k+1)
matrix A so that elements below the k-th subdiagonal are zero. The
reduction is performed by a unitary 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 CGEHRD.
ARGUMENTS
N (input) INTEGER
The order of the matrix A.
K (input) INTEGER
The offset for the reduction. Elements below the k-th subdiago‐
nal in the first NB columns are reduced to zero.
NB (input) INTEGER
The number of columns to be reduced.
A (input/output) COMPLEX array, dimension (LDA,N-K+1)
On entry, the n-by-(n-k+1) general matrix A. On exit, the ele‐
ments on and above the k-th subdiagonal in the first NB columns
are overwritten with the corresponding elements of the reduced
matrix; the elements below the k-th subdiagonal, with the array
TAU, represent the matrix Q as a product of elementary reflec‐
tors. 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) COMPLEX array, dimension (NB)
The scalar factors of the elementary reflectors. See Further
Details.
T (output) COMPLEX array, dimension (LDT,NB)
The upper triangular matrix T.
LDT (input) INTEGER
The leading dimension of the array T. LDT >= NB.
Y (output) COMPLEX array, dimension (LDY,NB)
The n-by-nb matrix Y.
LDY (input) INTEGER
The leading dimension of the array Y. LDY >= max(1,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 complex scalar, and v is a complex vector with
v(1:i+k-1) = 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 (n-k+1)-by-nb matrix V
which is needed, with T and Y, to apply the transformation to the unre‐
duced 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 modi‐
fied element of the upper Hessenberg matrix H, and vi denotes an ele‐
ment of the vector defining H(i).
LAPACK version 3.0 15 June 2000 CLAHRD(l)
