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# zlahrd

```
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 subdi-
agonal 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 subdiag-
onal are zero. The reduction is performed by a unitary simi-
larity 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 elemen-
tary 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 (NB,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
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).
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