# FreeBSD Manual Pages

```PDLAHRD(l)			       )			    PDLAHRD(l)

NAME
PDLAHRD	-  reduce  the first NB	columns	of a real 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 PDLAHRD( 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( *	)

DOUBLE	   PRECISION A(	* ), T(	* ), TAU( * ), WORK( * ), Y( *
)

PURPOSE
PDLAHRD	reduces	 the  first  NB	columns	of a real 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 orthogo- nal
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 PDGEHRD. 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension	(NB_A,NB_A)
The upper triangular matrix T.

Y       (local output) DOUBLE PRECISION 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) DOUBLE	PRECISION 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 real scalar, and v is a real 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			    PDLAHRD(l)
```

NAME | SYNOPSIS | PURPOSE | ARGUMENTS | FURTHER DETAILS

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