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NAME

       PDLACONSB  -  look  for  two  consecutive small subdiagonal elements by
       seeing the effect of starting a double shift  QR  iteration   given  by
       H44, H33, & H43H34 and see if this would make a  subdiagonal negligible

SYNOPSIS

       SUBROUTINE PDLACONSB( A, DESCA, I, L, M, H44, H33, H43H34, BUF, LWORK )

           INTEGER           I, L, LWORK, M

           DOUBLE            PRECISION H33, H43H34, H44

           INTEGER           DESCA( * )

           DOUBLE            PRECISION A( * ), BUF( * )

PURPOSE

       PDLACONSB looks for two consecutive small subdiagonal elements by
          seeing the effect of starting a double shift QR iteration
          given by H44, H33, & H43H34 and see if this would make a
          subdiagonal negligible.

       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) DOUBLE PRECISION 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.

       M       (global output) INTEGER
               On  exit,  this  yields  the starting location of the QR double
               shift.  This will satisfy: L <= M  <= I-2.

               H44 H33 H43H34  (global input)  DOUBLE  PRECISION  These  three
               values are for the double shift QR iteration.

       BUF     (local output) DOUBLE PRECISION array of size LWORK.

       LWORK   (global input) INTEGER
               On exit, LWORK is the size of the work buffer.  This must be at
               least 7*Ceil( Ceil( (I-L)/HBL ) / LCM(NPROW,NPCOL) )  Here  LCM
               is  least  common multiple, and NPROWxNPCOL is the logical grid
               size.

               Logic: ======

               Two  consecutive  small   subdiagonal   elements   will   stall
               convergence  of  a  double  shift  if  their  product  is small
               relatively even  if  each  is  not  very  small.   Thus  it  is
               necessary  to scan the "tridiagonal portion of the matrix."  In
               the LAPACK algorithm DLAHQR, a loop of M goes from I-2 down  to
               L                          and                         examines
               H(m,m),H(m+1,m+1),H(m+1,m),H(m,m+1),H(m-1,m-1),H(m,m-1),    and
               H(m+2,m-1).    Since   these   elements   may  be  on  separate
               processors, the first major loop (10) goes over the tridiagonal
               and  has  each  node store whatever values of the 7 it has that
               the node owning H(m,m) does not.  This will occur on  a  border
               and  can  happen in no more than 3 locations per block assuming
               square blocks.  There are 5 buffers that each node stores these
               values:   a  buffer to send diagonally down and right, a buffer
               to send up, a buffer to send left, a buffer to send  diagonally
               up  and left and a buffer to send right.  Each of these buffers
               is actually stored in one buffer BUF where BUF(ISTR1+1)  starts
               the  first buffer, BUF(ISTR2+1) starts the second, etc..  After
               the values are stored, if there are  any  values  that  a  node
               needs,  they  will  be  sent and received.  Then the next major
               loop passes over the data  and  searches  for  two  consecutive
               small subdiagonals.

               Notes:

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

               Implemented by:  G. Henry, November 17, 1996