PCGEEQU(3)    ScaLAPACK routine of NEC Numeric Library Collection   PCGEEQU(3)



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 num-
       ber 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  vec-
       tor.  This vector stores the information required to establish the map-
       ping 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 col-
       umn.
       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.



ScaLAPACK routine               31 October 2017                     PCGEEQU(3)