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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.