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PSGEHRD(l)			       )			    PSGEHRD(l)

NAME
       PSGEHRD	-  reduce  a real general distributed matrix sub( A ) to upper
       Hessenberg form H by an orthogonal similarity transforma- tion

SYNOPSIS
       SUBROUTINE PSGEHRD( N, ILO, IHI,	A, IA, JA, DESCA,  TAU,	 WORK,	LWORK,
			   INFO	)

	   INTEGER	   IA, IHI, ILO, INFO, JA, LWORK, N

	   INTEGER	   DESCA( * )

	   REAL		   A( *	), TAU(	* ), WORK( * )

PURPOSE
       PSGEHRD	reduces	 a  real  general distributed matrix sub( A ) to upper
       Hessenberg form H by an orthogonal similarity transforma-  tion:	 Q'  *
       sub( A )	* Q = H, where sub( A )	= A(IA+N-1:IA+N-1,JA+N-1:JA+N-1).

       Notes
       =====

       Each  global data object	is described by	an associated description vec-
       tor.  This vector stores	the information	required to establish the map-
       ping between an object element and its corresponding process and	memory
       location.

       Let A be	a generic term for any 2D block	 cyclicly  distributed	array.
       Such a global array has an associated description vector	DESCA.	In the
       following comments, the character _ should be read as  "of  the	global
       array".

       NOTATION	       STORED IN      EXPLANATION
       ---------------	--------------	--------------------------------------
       DTYPE_A(global) DESCA( DTYPE_ )The descriptor type.  In this case,
				      DTYPE_A =	1.
       CTXT_A (global) DESCA( CTXT_ ) The BLACS	context	handle,	indicating
				      the BLACS	process	grid A is distribu-
				      ted over.	The context itself is glo-
				      bal, but the handle (the integer
				      value) may vary.
       M_A    (global) DESCA( M_ )    The number of rows in the	global
				      array A.
       N_A    (global) DESCA( N_ )    The number of columns in the global
				      array A.
       MB_A   (global) DESCA( MB_ )   The blocking factor used to distribute
				      the rows of the array.
       NB_A   (global) DESCA( NB_ )   The blocking factor used to distribute
				      the columns of the array.
       RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
				      row  of  the  array  A  is  distributed.
       CSRC_A (global) DESCA( CSRC_ ) The process column over which the
				      first column of the array	A is
				      distributed.
       LLD_A  (local)  DESCA( LLD_ )  The leading dimension of the local
				      array.  LLD_A >= MAX(1,LOCr(M_A)).

       Let K be	the number of rows or columns of a distributed matrix, and as-
       sume that its process grid has dimension	p x q.
       LOCr( K ) denotes the number of elements	of K that a process would  re-
       ceive if	K were distributed over	the p processes	of its process column.
       Similarly, LOCc(	K ) denotes the	number of elements of K	that a process
       would receive if	K were distributed over	the q processes	of its process
       row.
       The  values  of	LOCr()	and LOCc() may be determined via a call	to the
       ScaLAPACK tool function,	NUMROC:
	       LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
	       LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).  An	 upper
       bound for these quantities may be computed by:
	       LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
	       LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A

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.

       ILO     (global input) INTEGER
	       IHI     (global input) INTEGER It is assumed that sub( A	 )  is
	       already	upper triangular in rows IA:IA+ILO-2 and IA+IHI:IA+N-1
	       and columns JA:JA+ILO-2 and JA+IHI:JA+N-1. See Further Details.
	       If N > 0,

       A       (local input/local output) REAL pointer into the
	       local memory to an array	of dimension (LLD_A,LOCc(JA+N-1)).  On
	       entry, this array contains the local pieces of the N-by-N  gen-
	       eral  distributed  matrix  sub( A ) to be reduced. On exit, the
	       upper triangle and the first subdiagonal	of sub(	A ) are	 over-
	       written	with the upper Hessenberg matrix H, and	the ele- ments
	       below the first subdiagonal, with the array  TAU,  repre-  sent
	       the  orthogonal matrix Q	as a product of	elementary reflectors.
	       See Further Details.  IA	     (global input)  INTEGER  The  row
	       index  in the global array A indicating 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) REAL array, dimension LOCc(JA+N-2)
	       The  scalar  factors  of	the elementary reflectors (see Further
	       Details). Elements JA:JA+ILO-2 and JA+IHI:JA+N-2	of TAU are set
	       to zero.	TAU is tied to the distributed matrix A.

       WORK    (local workspace/local output) REAL array,
	       dimension  (LWORK)  On  exit, WORK( 1 ) returns the minimal and
	       optimal LWORK.

       LWORK   (local or global	input) INTEGER
	       The dimension of	the array WORK.	 LWORK is local	input and must
	       be at least LWORK >= NB*NB + NB*MAX( IHIP+1, IHLP+INLQ )

	       where NB	= MB_A = NB_A, IROFFA =	MOD( IA-1, NB ), ICOFFA	= MOD(
	       JA-1, NB	), IOFF	= MOD( IA+ILO-2, NB ), IAROW  =	 INDXG2P(  IA,
	       NB,  MYROW, RSRC_A, NPROW ), IHIP = NUMROC( IHI+IROFFA, NB, MY-
	       ROW, IAROW, NPROW ), ILROW  =  INDXG2P(	IA+ILO-1,  NB,	MYROW,
	       RSRC_A,	NPROW ), IHLP =	NUMROC(	IHI-ILO+IOFF+1,	NB, MYROW, IL-
	       ROW, NPROW ), ILCOL = INDXG2P(  JA+ILO-1,  NB,  MYCOL,  CSRC_A,
	       NPCOL  ),  INLQ = NUMROC( N-ILO+IOFF+1, NB, MYCOL, ILCOL, NPCOL
	       ),

	       INDXG2P and NUMROC are ScaLAPACK	tool functions;	MYROW,	MYCOL,
	       NPROW  and  NPCOL  can  be determined by	calling	the subroutine
	       BLACS_GRIDINFO.

	       If LWORK	= -1, then LWORK is global input and a workspace query
	       is assumed; the routine only calculates the minimum and optimal
	       size for	all work arrays. Each of these values is  returned  in
	       the  first  entry of the	corresponding work array, and no error
	       message is issued by PXERBLA.

       INFO    (global output) INTEGER
	       = 0:  successful	exit
	       < 0:  If	the i-th argument is an	array and the j-entry  had  an
	       illegal	value, then INFO = -(i*100+j), if the i-th argument is
	       a scalar	and had	an illegal value, then INFO = -i.

FURTHER	DETAILS
       The matrix Q is represented as a	product	of  (ihi-ilo)  elementary  re-
       flectors

	  Q = H(ilo) H(ilo+1) .	. . H(ihi-1).

       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) =	0, v(I+1) = 1 and v(IHI+1:N) = 0; v(I+2:IHI) is	stored on exit
       in A(IA+ILO+I:IA+IHI-1,JA+ILO+I-2), and tau in TAU(JA+ILO+I-2).

       The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by  the  follow-
       ing example, with N = 7,	ILO = 2	and IHI	= 6:

       on entry				on exit

       ( a   a	 a   a	 a   a	 a )	(  a   a   h   h   h   h   a ) (     a
       a   a   a   a   a )    (	     a	 h   h	 h   h	 a ) (	   a	a    a
       a    a	 a )	(      h   h   h   h   h   h ) (     a	 a   a	 a   a
       a )    (	     v2	 h   h	 h   h	 h ) (	   a   a   a	a    a	  a  )
       (       v2   v3	 h    h	   h	h ) (	  a   a	  a   a	  a   a	)    (
       v2   v3	 v4   h	   h	h  )  (				  a   )	     (
       a )

       where a denotes an element of the original matrix sub( A	), H denotes a
       modified	element	of the upper Hessenberg	matrix H, and  vi  denotes  an
       element of the vector defining H(JA+ILO+I-2).

       Alignment requirements
       ======================

       The  distributed	 submatrix sub(	A ) must verify	some alignment proper-
       ties, namely the	following expression should be true:
       ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA )

ScaLAPACK version 1.7		13 August 2001			    PSGEHRD(l)

NAME | SYNOPSIS | PURPOSE | ARGUMENTS | FURTHER DETAILS

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