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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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