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NAME

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

SYNOPSIS

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

           CHARACTER       DIRECT, STOREV

           INTEGER         IV, JV, K, N

           INTEGER         DESCV( * )

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

PURPOSE

       PCLARFT forms the triangular factor T of a complex 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) COMPLEX 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) COMPLEX, 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) COMPLEX 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) COMPLEX 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 )