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NAME

       PDLARFT  -  form the triangular factor T of a real block reflector H of
       order n, which is defined as a product of k elementary reflectors

SYNOPSIS

       SUBROUTINE PDLARFT( DIRECT, STOREV, N, K, V, IV,  JV,  DESCV,  TAU,  T,
                           WORK )

           CHARACTER       DIRECT, STOREV

           INTEGER         IV, JV, K, N

           INTEGER         DESCV( * )

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

PURPOSE

       PDLARFT  forms  the  triangular factor T of a real block reflector H of
       order n, which is defined as a product of k elementary reflectors.

       If DIRECT = ’F’, H = H(1) H(2) . . . H(k) and T is upper triangular;

       If DIRECT = ’B’, H = H(k) . . . H(2) H(1) and T is lower triangular.

       If STOREV = ’C’, the vector which defines the elementary reflector H(i)
       is stored in the i-th column of the distributed matrix V, and

          H  =  I - V * T * V’

       If STOREV = ’R’, the vector which defines the elementary reflector H(i)
       is stored in the i-th row of the distributed matrix V, and

          H  =  I - V’ * T * V

       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

ARGUMENTS

       DIRECT  (global input) CHARACTER*1
               Specifies  the  order  in  which  the elementary reflectors are
               multiplied to form the block reflector:
               = ’F’: H = H(1) H(2) . . . H(k) (Forward)
               = ’B’: H = H(k) . . . H(2) H(1) (Backward)

       STOREV  (global input) CHARACTER*1
               Specifies  how  the  vectors  which   define   the   elementary
               reflectors are stored (see also Further Details):
               = ’R’: rowwise

       N       (global input) INTEGER
               The order of the block reflector H. N >= 0.

       K       (global input) INTEGER
               The  order  of  the  triangular  factor  T  (=  the  number  of
               elementary reflectors). 1 <= K <= MB_V (= NB_V).

       V       (input/output) DOUBLE PRECISION pointer into the local memory
               to an array of local dimension  (LOCr(IV+N-1),LOCc(JV+K-1))  if
               STOREV  = ’C’, and (LOCr(IV+K-1),LOCc(JV+N-1)) if STOREV = ’R’.
               The distributed matrix V contains the Householder vectors.  See
               further details.

       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.

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

       T       (local output) DOUBLE PRECISION array, dimension (NB_V,NB_V)
               if  STOREV  = ’Col’, and (MB_V,MB_V) otherwise. It contains the
               k-by-k triangular factor of the block  reflector  asso-  ciated
               with  V.  If  DIRECT  = ’F’, T is upper triangular; if DIRECT =
               ’B’, T is lower triangular.

       WORK    (local workspace) DOUBLE PRECISION array,
               dimension (K*(K-1)/2)

FURTHER DETAILS

       The shape of the matrix V and the storage of the vectors  which  define
       the  H(i) is best illustrated by the following example with n = 5 and k
       = 3. The elements equal to 1 are not stored;  the  corresponding  array
       elements  are  modified  but restored on exit. The rest of the array is
       not used.

       DIRECT = ’F’ and STOREV = ’C’:   DIRECT = ’F’ and STOREV = ’R’:

       V( IV:IV+N-1,    (  1       )    V( IV:IV+K-1,    (  1 v1 v1 v1 v1 )
          JV:JV+K-1 ) = ( v1  1    )       JV:JV+N-1 ) = (     1 v2 v2 v2 )
                        ( v1 v2  1 )                     (        1 v3 v3 )
                        ( v1 v2 v3 )
                        ( v1 v2 v3 )

       DIRECT = ’B’ and STOREV = ’C’:   DIRECT = ’B’ and STOREV = ’R’:

       V( IV:IV+N-1,    ( v1 v2 v3 )    V( IV:IV+K-1,    ( v1 v1  1       )
          JV:JV+K-1 ) = ( v1 v2 v3 )       JV:JV+N-1 ) = ( v2 v2 v2  1    )
                        (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
                        (     1 v3 )
                        (        1 )