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 zlahr2.f(3) LAPACK zlahr2.f(3)

zlahr2.f -

# SYNOPSIS

## Functions/Subroutines

subroutine zlahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)

ZLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

# Function/Subroutine Documentation

## subroutine zlahr2 (integerN, integerK, integerNB, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( nb )TAU, complex*16, dimension( ldt, nb )T, integerLDT, complex*16, dimension( ldy, nb )Y, integerLDY)

ZLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
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**H * A * Q. The routine returns the matrices V and T which determine
Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T.

This is an auxiliary routine called by ZGEHRD.
```
Parameters:
N
```          N is INTEGER
The order of the matrix A.
```
K
```          K is INTEGER
The offset for the reduction. Elements below the k-th
subdiagonal in the first NB columns are reduced to zero.
K < N.
```
NB
```          NB is INTEGER
The number of columns to be reduced.
```
A
```          A is 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
```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,N).
```
TAU
```          TAU is COMPLEX*16 array, dimension (NB)
The scalar factors of the elementary reflectors. See Further
Details.
```
T
```          T is COMPLEX*16 array, dimension (LDT,NB)
The upper triangular matrix T.
```
LDT
```          LDT is INTEGER
The leading dimension of the array T.  LDT >= NB.
```
Y
```          Y is COMPLEX*16 array, dimension (LDY,NB)
The n-by-nb matrix Y.
```
LDY
```          LDY is INTEGER
The leading dimension of the array Y. LDY >= N.
```
Author:
Univ. of Tennessee
Univ. of California Berkeley
NAG Ltd.
Date:
September 2012
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**H

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**H) * (A - Y*V**H).

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 subroutine is a slight modification of LAPACK-3.0's DLAHRD
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-3.0's DLAHRD routine. (This
subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
```
References:
Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the
performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.
Definition at line 182 of file zlahr2.f.

# Author

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