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NAME

       PDLAQGE   -   equilibrate   a   general   M-by-N   distributed   matrix   sub(   A   )   =
       A(IA:IA+M-1,JA:JA+N-1) using the row and scaling factors in the vectors R and C

SYNOPSIS

       SUBROUTINE PDLAQGE( M, N, A, IA, JA, DESCA, R, C, ROWCND, COLCND, AMAX, EQUED )

           CHARACTER       EQUED

           INTEGER         IA, JA, M, N

           DOUBLE          PRECISION AMAX, COLCND, ROWCND

           INTEGER         DESCA( * )

           DOUBLE          PRECISION A( * ), C( * ), R( * )

PURPOSE

       PDLAQGE equilibrates a general M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1)
       using the row and scaling factors in the vectors R and C.

       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

       M       (global input) INTEGER
               The number of rows to be operated on i.e the number of  rows  of  the  distributed
               submatrix sub( A ). M >= 0.

       N       (global input) INTEGER
               The  number  of  columns  to  be  operated  on  i.e  the  number of columns of the
               distributed submatrix sub( A ). N >= 0.

       A       (local input/local output) DOUBLE PRECISION pointer into the
               local memory to an array of dimension (LLD_A,LOCc(JA+N-1)) containing on entry the
               M-by-N  matrix  sub( A ). On exit, the equilibrated distributed matrix.  See EQUED
               for the form of the equilibrated distributed submatrix.

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

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

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

       R       (local input) DOUBLE PRECISION array, dimension LOCr(M_A)
               The row scale factors for sub( A ). R is aligned with the  distributed  matrix  A,
               and replicated across every process column. R is tied to the distributed matrix A.

       C       (local input) DOUBLE PRECISION array, dimension LOCc(N_A)
               The  column scale factors of sub( A ). C is aligned with the distributed matrix A,
               and replicated down every process row. C is tied to the distributed matrix A.

       ROWCND  (global input) DOUBLE PRECISION
               The global ratio of the smallest R(i) to the largest R(i), IA <= i <= IA+M-1.

       COLCND  (global input) DOUBLE PRECISION
               The global ratio of the smallest C(i) to the largest C(i), JA <= j <= JA+N-1.

       AMAX    (global input) DOUBLE PRECISION
               Absolute value of largest distributed submatrix entry.

       EQUED   (global output) CHARACTER
               Specifies the form of equilibration that was done.  = 'N':  No equilibration
               = 'R':  Row equilibration, i.e., sub( A ) has been pre-
               multiplied by diag(R(IA:IA+M-1)),
               = 'C':  Column equilibration, i.e., sub( A ) has been post-
               multiplied by diag(C(JA:JA+N-1)),
               = 'B':  Both row and column equilibration, i.e., sub( A )  has  been  replaced  by
               diag(R(IA:IA+M-1)) * sub( A ) * diag(C(JA:JA+N-1)).

PARAMETERS

       THRESH  is  a threshold value used to decide if row or column scaling should be done based
       on the ratio of the row or column scaling factors.  If ROWCND <  THRESH,  row  scaling  is
       done, and if COLCND < THRESH, column scaling is done.

       LARGE and SMALL are threshold values used to decide if row scaling should be done based on
       the absolute size of the largest matrix element.  If AMAX > LARGE or  AMAX  <  SMALL,  row
       scaling is done.