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



NAME
       PZPOEQU  -  compute  row  and column scalings intended to equilibrate a
       distributed  Hermitian  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 PZPOEQU( N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX, INFO )

           INTEGER         IA, INFO, JA, N

           DOUBLE          PRECISION AMAX, SCOND

           INTEGER         DESCA( * )

           DOUBLE          PRECISION SC( * ), SR( * )

           COMPLEX*16      A( * )

PURPOSE
       PZPOEQU computes row and column scalings intended to equilibrate a dis-
       tributed    Hermitian   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 scal-
       ings.

       The scaling factor are stored along process rows in SR and  along  pro-
       cess  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  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
       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) COMPLEX*16 pointer into the local memory to an
               array  of  local  dimension ( LLD_A, LOCc(JA+N-1) ), the N-by-N
               Hermitian 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) DOUBLE PRECISION 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  dis-
               tributed matrix A.

       SC      (local output) DOUBLE PRECISION 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) DOUBLE PRECISION
               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) DOUBLE PRECISION
               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.



ScaLAPACK routine               31 October 2017                     PZPOEQU(3)