PDLARF(l)					      LAPACK auxiliary routine (version 1.5)						 PDLARF(l)

NAME

       PDLARF  - applie a real elementary reflector Q (or Q**T) to a real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either
       the left or the right

SYNOPSIS

       SUBROUTINE PDLARF( SIDE, M, N, V, IV, JV, DESCV, INCV, TAU, C, IC, JC, DESCC, WORK )

	   CHARACTER	  SIDE

	   INTEGER	  IC, INCV, IV, JC, JV, M, N

	   INTEGER	  DESCC( * ), DESCV( * )

	   DOUBLE	  PRECISION C( * ), TAU( * ), V( * ), WORK( * )

PURPOSE

       PDLARF applies a real elementary reflector Q (or Q**T) to a real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1),  from  either
       the left or the right. Q is represented in the form

	     Q = I - tau * v * v'

       where tau is a real scalar and v is a real vector.

       If tau = 0, then Q is taken to be the unit matrix.

       Notes
       =====

       Each  global  data  object  is described by an associated description vector.  This vector stores the information required to establish the
       mapping 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 assume that its process grid has dimension p x q.
       LOCr(  K  ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process col-
       umn.
       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

       Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix.

       Restrictions
       ============

       If  SIDE  =  'Left'  and INCV = 1, then the row process having the first entry V(IV,JV) must also have the first row of sub( C ). Moreover,
       MOD(IV-1,MB_V) must be equal to MOD(IC-1,MB_C), if INCV=M_V, only the last equality must be satisfied.

       If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also have the first column of sub(  C  )  and
       MOD(JV-1,NB_V) must be equal to MOD(JC-1,NB_C), if INCV = 1 only the last equality must be satisfied.

ARGUMENTS

       SIDE    (global input) CHARACTER
	       = 'L': form  Q * sub( C ),
	       = 'R': form  sub( C ) * Q, Q = Q**T.

       M       (global input) INTEGER
	       The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). M >= 0.

       N       (global input) INTEGER
	       The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( C ). N >= 0.

       V       (local input) DOUBLE PRECISION pointer into the local memory
	       to  an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transfor-
	       mation Q, V(IV:IV+M-1,JV) if SIDE = 'L' and INCV = 1,
	       V(IV,JV:JV+M-1) if SIDE = 'L' and INCV = M_V,
	       V(IV:IV+N-1,JV) if SIDE = 'R' and INCV = 1,
	       V(IV,JV:JV+N-1) if SIDE = 'R' and INCV = M_V,

	       The vector v in the representation of Q. V is not used if TAU = 0.

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

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

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

       INCV    (global input) INTEGER
	       The global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V.  INCV must not
	       be zero.

       TAU     (local input) DOUBLE PRECISION, array, dimension  LOCc(JV) if
	       INCV  =	1, and LOCr(IV) otherwise. This array contains the Householder scalars related to the Householder vectors.  TAU is tied to
	       the distributed matrix V.

       C       (local input/local output) DOUBLE PRECISION pointer into the
	       local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C )  is  over-
	       written by the Q * sub( C ) if SIDE = 'L', or sub( C ) * Q if SIDE = 'R'.

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

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

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

       WORK    (local workspace) DOUBLE PRECISION array, dimension (LWORK)
	       If INCV = 1, if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0 else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = 'R', LWORK >=
	       NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV = M_V, if SIDE  =  'L',
	       LWORK  >=  MpC0	+  MAX( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) ) else if SIDE = 'R', if IVROW =
	       ICROW, LWORK >= MpC0 else LWORK >= NqC0 + MAX( 1, MpC0 ) end if end if end if

	       where LCM is the least common multiple of NPROW and NPCOL and LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM / NPCOL,

	       IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ),  ICCOL  =  INDXG2P(  JC,
	       NB_C,  MYCOL,  CSRC_C,  NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL,
	       NPCOL ),

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

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

	       The  distributed  submatrices  V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following
	       expressions should be true:

	       MB_V = NB_V,

	       If INCV = 1, If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_A .AND.
	       MB_V.EQ.NB_C  .AND.  IROFFV.EQ.ICOFFC  )  else  if  INCV  =  M_V,  If  SIDE  =  'Left',	(  MB_V.EQ.NB_V  .AND.	MB_V.EQ.MB_C .AND.
	       ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if

LAPACK version 1.5						    12 May 1997 							 PDLARF(l)