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NAME

       PCLARF  -  applie  a complex elementary reflector Q to a complex 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 PCLARF( 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( * )

           COMPLEX        C( * ), TAU( * ), V( * ), WORK( * )

PURPOSE

       PCLARF applies a complex elementary reflector Q to a  complex  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 complex scalar and v is a complex 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 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

       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.

       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) COMPLEX pointer into the local memory
               to an array of dimension (LLD_V,*) containing the local pieces of the  distributed
               vectors V representing the Householder transformation 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) COMPLEX, 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) COMPLEX 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 overwritten 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) COMPLEX 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