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NAME

       PDLARFG  - generate a real elementary reflector H of order n, such that   H * sub( X ) = H
       * ( x(iax,jax) ) = ( alpha ), H' * H = I

SYNOPSIS

       SUBROUTINE PDLARFG( N, ALPHA, IAX, JAX, X, IX, JX, DESCX, INCX, TAU )

           INTEGER         IAX, INCX, IX, JAX, JX, N

           DOUBLE          PRECISION ALPHA

           INTEGER         DESCX( * )

           DOUBLE          PRECISION TAU( * ), X( * )

PURPOSE

       PDLARFG generates a real elementary reflector H of order n, such that
                             (      x     )   (   0   )

       where alpha is a scalar, and  sub(  X  )  is  an  (N-1)-element  real  distributed  vector
       X(IX:IX+N-2,JX)  if INCX = 1 and X(IX,JX:JX+N-2) if INCX = DESCX(M_).  H is represented in
       the form

             H = I - tau * ( 1 ) * ( 1 v' ) ,
                           ( v )

       where tau is a real scalar and v is a real (N-1)-element
       vector.

       If the elements of sub( X ) are all zero, then tau = 0 and H  is  taken  to  be  the  unit
       matrix.

       Otherwise  1 <= tau <= 2.

       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.

ARGUMENTS

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

       ALPHA   (local output) DOUBLE PRECISION
               On exit, alpha is computed in the process scope having the vector sub( X ).

       IAX     (global input) INTEGER
               The global row index in X of X(IAX,JAX).

       JAX     (global input) INTEGER
               The global column index in X of X(IAX,JAX).

       X       (local input/local output) DOUBLE PRECISION, pointer into the
               local  memory  to  an  array of dimension (LLD_X,*). This array contains the local
               pieces of the distributed vector sub( X ).  Before entry,  the  incremented  array
               sub( X ) must contain the vector x. On exit, it is overwritten with the vector v.

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

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

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

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

       TAU     (local output) DOUBLE PRECISION, array, dimension  LOCc(JX)
               if INCX = 1, and LOCr(IX) otherwise. This array contains the  Householder  scalars
               related to the Householder vectors.  TAU is tied to the distributed matrix X.