# FreeBSD Manual Pages

```PZLAHRD(l)			       )			    PZLAHRD(l)

NAME
PZLAHRD - reduce	the first NB columns of	a complex general N-by-(N-K+1)
distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below	the k-
th subdiagonal are zero

SYNOPSIS
SUBROUTINE PZLAHRD( N,  K, NB, A, IA, JA, DESCA,	TAU, T,	Y, IY, JY, DE-
SCY,	WORK )

INTEGER	   IA, IY, JA, JY, K, N, NB

INTEGER	   DESCA( * ), DESCY( *	)

COMPLEX*16	   A( *	), T( *	), TAU(	* ), WORK( * ),	Y( * )

PURPOSE
PZLAHRD reduces the first NB columns of a complex general  N-by-(N-K+1)
distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below	the k-
th subdiagonal are zero.	The reduction is performed by an 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	PZGEHRD. In the	following com-
ments sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1).

ARGUMENTS
N       (global input) INTEGER
The number of rows and columns to be operated on, i.e. the  or-
der of the distributed submatrix	sub( A ).  N >=	0.

K       (global input) INTEGER
The offset for the reduction. Elements below the	k-th subdiago-
nal in the first	NB columns are reduced to zero.

NB      (global input) INTEGER
The number of columns to	be reduced.

A       (local input/local output) COMPLEX*16 pointer into
the local memory	to an array of	dimension  (LLD_A,  LOCc(JA+N-
K)).  On	entry, this array contains the the local pieces	of the
N-by-(N-K+1) general distributed	matrix A(IA:IA+N-1,JA:JA+N-K).
On  exit, the elements on and above the k-th subdiagonal	in the
first NB	columns	are overwritten	with  the  corresponding  ele-
ments of	the reduced distributed	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(IA:IA+N-1,JA:JA+N-K) are unchanged. See Further Details.   IA
(global	input) INTEGER The row index in	the global array A in-
dicating	the first row of sub( A	).

JA      (global input) INTEGER
The column index	in the global array  A	indicating  the	 first
column of sub( A	).

DESCA   (global and local input)	INTEGER	array of dimension DLEN_.
The array descriptor for	the distributed	matrix A.

TAU     (local output) COMPLEX*16 array,	dimension LOCc(JA+N-2)
The  scalar  factors  of	the elementary reflectors (see Further
Details). TAU is	tied to	the distributed	matrix A.

T       (local output) COMPLEX*16 array,	dimension (NB_A,NB_A)
The upper triangular matrix T.

Y       (local output) COMPLEX*16 pointer into the local	memory
to an array of dimension	(LLD_Y,NB_A). On exit, this array con-
tains  the  local  pieces  of the N-by-NB distributed matrix Y.
LLD_Y >=	LOCr(IA+N-1).

IY      (global input) INTEGER
The row index in	the global array Y indicating the first	row of
sub( Y ).

JY      (global input) INTEGER
The  column  index  in  the global array	Y indicating the first
column of sub( Y	).

DESCY   (global and local input)	INTEGER	array of dimension DLEN_.
The array descriptor for	the distributed	matrix Y.

WORK    (local workspace) COMPLEX*16 array, dimension (NB)

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(ia+i+k:ia+n-1,ja+i-1),	and tau	in TAU(ja+i-1).

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(ia:ia+n-1,ja:ja+n-k) := (I-V*T*V')*(A(ia:ia+n-1,ja:ja+n-k)-Y*V').

The  contents  of A(ia:ia+n-1,ja:ja+n-k)	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(ia:ia+n-1,ja:ja+n-k), h denotes a modified element of the upper  Hes-
senberg	matrix	H,  and	 vi  denotes an	element	of the vector defining
H(i).

ScaLAPACK version 1.7		13 August 2001			    PZLAHRD(l)
```

NAME | SYNOPSIS | PURPOSE | ARGUMENTS | FURTHER DETAILS

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