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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.