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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.