clahr2 man page on Scientific

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```CLAHR2(1)	    LAPACK auxiliary routine (version 3.2)	     CLAHR2(1)

NAME
CLAHR2 - 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 CLAHR2( 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
CLAHR2 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 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.	 K < N.

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 >= 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	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	 modi‐
fied  element  of the upper Hessenberg matrix H, and vi denotes an ele‐
ment of the vector defining H(i).
This file is a slight modification of LAPACK-3.0's CLAHRD 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.

LAPACK auxiliary routine (versioNovember 2008			     CLAHR2(1)
```
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