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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.