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NAME

       PSPOEQU  -  compute  row  and column scalings intended to equilibrate a
       distributed  symmetric  positive   definite   matrix   sub(   A   )   =
       A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number (with respect to
       the two-norm)

SYNOPSIS

       SUBROUTINE PSPOEQU( N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX, INFO )

           INTEGER         IA, INFO, JA, N

           REAL            AMAX, SCOND

           INTEGER         DESCA( * )

           REAL            A( * ), SC( * ), SR( * )

PURPOSE

       PSPOEQU computes row and column  scalings  intended  to  equilibrate  a
       distributed   symmetric   positive   definite   matrix   sub(   A  )  =
       A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number (with respect to
       the   two-norm).    SR  and  SC  contain  the  scale  factors,  S(i)  =
       1/sqrt(A(i,i)), chosen so that the scaled distri- buted matrix  B  with
       elements  B(i,j)  =  S(i)*A(i,j)*S(j)  has ones on the  diagonal.  This
       choice of SR and SC puts the condition number of B within a factor N of
       the  smallest  possible  condition  number  over  all possible diagonal
       scalings.

       The scaling factor are stored  along  process  rows  in  SR  and  along
       process  columns  in  SC.  The  duplication  of  information simplifies
       greatly the application of the factors.

       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

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

       A       (local input) REAL pointer into the local memory to an
               array of local dimension ( LLD_A, LOCc(JA+N-1)  ),  the  N-by-N
               symmetric  positive  definite distributed matrix sub( A ) whose
               scaling factors are to be computed.  Only the diagonal elements
               of sub( A ) are referenced.

       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.

       SR      (local output) REAL array, dimension LOCr(M_A)
               If  INFO  = 0, SR(IA:IA+N-1) contains the row scale factors for
               sub( A ). SR is aligned with  the  distributed  matrix  A,  and
               replicated  across  every  process  column.  SR  is tied to the
               distributed matrix A.

       SC      (local output) REAL array, dimension LOCc(N_A)
               If INFO = 0, SC(JA:JA+N-1) contains the column scale factors
               for A(IA:IA+M-1,JA:JA+N-1). SC is aligned  with  the  distribu-
               ted matrix A, and replicated down every process row. SC is tied
               to the distributed matrix A.

       SCOND   (global output) REAL
               If INFO = 0, SCOND contains the ratio of the smallest SR(i) (or
               SC(j))  to the largest SR(i) (or SC(j)), with IA <= i <= IA+N-1
               and JA <= j <= JA+N-1. If SCOND >= 0.1 and AMAX is neither  too
               large nor too small, it is not worth scaling by SR (or SC).

       AMAX    (global output) REAL
               Absolute  value  of  largest  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 = K, the K-th diagonal entry of sub( A ) is nonpositive.