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NAME

       PCGEEQU  -  compute  row and column scalings intended to equilibrate an M-by-N distributed
       matrix sub( A ) = A(IA:IA+N-1,JA:JA:JA+N-1) and reduce its condition number

SYNOPSIS

       SUBROUTINE PCGEEQU( M, N, A, IA, JA, DESCA, R, C, ROWCND, COLCND, AMAX, INFO )

           INTEGER         IA, INFO, JA, M, N

           REAL            AMAX, COLCND, ROWCND

           INTEGER         DESCA( * )

           REAL            C( * ), R( * )

           COMPLEX         A( * )

PURPOSE

       PCGEEQU computes row and column scalings intended to  equilibrate  an  M-by-N  distributed
       matrix  sub(  A  ) = A(IA:IA+N-1,JA:JA:JA+N-1) and reduce its condition number.  R returns
       the row scale factors and C the column scale factors, chosen to try to  make  the  largest
       entry  in  each  row  and column of the distributed matrix B with elements B(i,j) = R(i) *
       A(i,j) * C(j) have absolute value 1.

       R(i) and C(j) are restricted to be between SMLNUM = smallest  safe  number  and  BIGNUM  =
       largest  safe  number.   Use  of  these  scaling  factors  is not guaranteed to reduce the
       condition number of sub( A ) but works well in practice.

       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) COMPLEX pointer into the local memory
               to  an  array of dimension ( LLD_A, LOCc(JA+N-1) ), the local pieces of the M-by-N
               distributed matrix whose equilibration factors are to be computed.

       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 output) REAL array, dimension LOCr(M_A)
               If INFO = 0 or INFO > IA+M-1, R(IA:IA+M-1) contains 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 output) REAL array, dimension LOCc(N_A)
               If INFO = 0,  C(JA:JA+N-1) contains the column scale factors for sub( A  ).  C  is
               aligned with the distributed matrix A, and replicated down every process row. C is
               tied to the distri- buted matrix A.

       ROWCND  (global output) REAL
               If INFO = 0 or INFO > IA+M-1, ROWCND contains the ratio of the  smallest  R(i)  to
               the  largest  R(i)  (IA <= i <= IA+M-1).  If ROWCND >= 0.1 and AMAX is neither too
               large nor too small, it is not worth scaling by R(IA:IA+M-1).

       COLCND  (global output) REAL
               If INFO = 0, COLCND contains the ratio of the smallest C(j) to  the  largest  C(j)
               (JA <= j <= JA+N-1). If COLCND >= 0.1, it is not worth scaling by C(JA:JA+N-1).

       AMAX    (global output) REAL
               Absolute  value  of  largest distributed matrix element.  If AMAX is very close to
               overflow or very close to underflow, the matrix should be scaled.

       INFO    (global output) INTEGER
               = 0:  successful exit
               < 0:  If the i-th argument is an array and the j-entry had an illegal value,  then
               INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then
               INFO = -i.  > 0:  If INFO = i,  and i is
               <= M:  the i-th row of the distributed matrix sub( A ) is exactly zero, >  M:  the
               (i-M)-th column of the distributed matrix sub( A ) is exactly zero.