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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