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NAME

       PSLASMSUB  -  look  for a small subdiagonal element from the bottom  of the matrix that it
       can safely set to zero

SYNOPSIS

       SUBROUTINE PSLASMSUB( A, DESCA, I, L, K, SMLNUM, BUF, LWORK )

           INTEGER           I, K, L, LWORK

           REAL              SMLNUM

           INTEGER           DESCA( * )

           REAL              A( * ), BUF( * )

PURPOSE

       PSLASMSUB looks for a small subdiagonal element from the bottom
          of the matrix that it can safely set to zero.

       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

       A       (global input) REAL array, dimension
               (DESCA(LLD_),*)  On  entry,  the Hessenberg matrix whose tridiagonal part is being
               scanned.  Unchanged on exit.

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

       I       (global input) INTEGER
               The global location of the bottom of the unreduced submatrix of A.   Unchanged  on
               exit.

       L       (global input) INTEGER
               The  global  location  of  the  top of the unreduced submatrix of A.  Unchanged on
               exit.

       K       (global output) INTEGER
               On exit, this yields the bottom portion of the  unreduced  submatrix.   This  will
               satisfy: L <= M  <= I-1.

       SMLNUM  (global input) REAL
               On entry, a "small number" for the given matrix.  Unchanged on exit.

       BUF     (local output) REAL array of size LWORK.

       LWORK   (global input) INTEGER
               On  exit,  LWORK  is  the  size of the work buffer.  This must be at least 2*Ceil(
               Ceil( (I-L)/HBL ) / LCM(NPROW,NPCOL) ) Here LCM  is  least  common  multiple,  and
               NPROWxNPCOL is the logical grid size.

               Notes:

               This routine does a global maximum and must be called by all processes.

               This code is basically a parallelization of the following snip of LAPACK code from
               SLAHQR:

               Look for a single small subdiagonal element.

               DO 20 K = I, L + 1, -1 TST1 = ABS( H( K-1, K-1 )  )  +  ABS(  H(  K,  K  )  )  IF(
               TST1.EQ.ZERO  )  $          TST1  = SLANHS( '1', I-L+1, H( L, L ), LDH, WORK ) IF(
               ABS( H( K, K-1 ) ).LE.MAX( ULP*TST1, SMLNUM ) ) $         GO TO 30 20     CONTINUE
               30    CONTINUE

               Implemented by:  G. Henry, November 17, 1996