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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
       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, 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
               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) 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
               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) 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