ZLAHR2(3) 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

## SYNOPSIS

SUBROUTINE ZLAHR2(
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

ZLAHR2 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 an 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. K < N.
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 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) 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 >= 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 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   a   a   a   a )

( a   a   a   a   a )

( a   a   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).
This file is a slight modification of LAPACK-3.0's ZLAHRD incorporating improvements proposed by Quintana-Orti and Van de Gejin. Note that the entries of A(1:K,2:NB) differ from those returned by the original LAPACK routine. This function is not backward compatible with LAPACK3.0.