zlahrd(3)

NAME

ZLAHRD - 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 ZLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
    INTEGER        K, LDA, LDT, LDY, N, NB
    COMPLEX*16      A(  LDA, * ), T( LDT, NB ), TAU( NB ),
Y( LDY, NB )

PURPOSE

ZLAHRD 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 ZGEHRD.

ARGUMENTS

N (input) INTEGER

The order of the matrix A.

K (input) INTEGER

The offset for the reduction. Elements below the

k-th subdiagonal in the first NB columns are reduced to zero.

NB (input) INTEGER

The number of columns to be reduced.

A (input/output) COMPLEX*16 array, dimension (LDA,N

K+1)

On entry, the n-by-(n-k+1) general matrix A. On

exit, the elements 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 re

flectors. The other columns of A are unchanged. See Further De

tails. LDA (input) INTEGER The leading dimension of the ar

ray A. LDA >= max(1,N).

TAU (output) COMPLEX*16 array, dimension (NB)

The scalar factors of the elementary reflectors.

See Further Details.

T (output) COMPLEX*16 array, dimension (LDT,NB)

The upper triangular matrix T.

LDT (input) INTEGER

The leading dimension of the array T. LDT >= NB.

Y (output) COMPLEX*16 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 unreduced part of the matrix, using an up

date 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 de

notes 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

ZLAHRD(3)