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NAME

       PZLAQSY  -  equilibrate  a  symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1)
       using the scaling factors in the vectors SR and SC

SYNOPSIS

       SUBROUTINE PZLAQSY( UPLO, N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX, EQUED )

           CHARACTER       EQUED, UPLO

           INTEGER         IA, JA, N

           DOUBLE          PRECISION AMAX, SCOND

           INTEGER         DESCA( * )

           DOUBLE          PRECISION SC( * ), SR( * )

           COMPLEX*16      A( * )

PURPOSE

       PZLAQSY equilibrates a symmetric distributed matrix  sub(  A  )  =  A(IA:IA+N-1,JA:JA+N-1)
       using the scaling factors in the vectors SR and SC.

       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

       UPLO    (global input) CHARACTER
               Specifies whether the upper or lower triangular part of the symmetric  distributed
               matrix sub( A ) is to be referenced:
               = 'U':  Upper triangular
               = 'L':  Lower triangular

       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       (input/output) COMPLEX*16 pointer into the local
               memory to an array of local dimension (LLD_A,LOCc(JA+N-1)).  On entry,  the  local
               pieces of the distributed symmetric matrix sub( A ). If UPLO = 'U', the leading N-
               by-N upper triangular part of sub( A ) contains the upper triangular part  of  the
               matrix,  and the strictly lower triangular part of sub( A ) is not referenced.  If
               UPLO = 'L', the leading N-by-N lower triangular part of  sub(  A  )  contains  the
               lower  triangular  part of the matrix, and the strictly upper trian- gular part of
               sub( A ) is not referenced.  On exit, if EQUED = 'Y', the equilibrated matrix:
               diag(SR(IA:IA+N-1)) * sub( A ) * diag(SC(JA:JA+N-1)).

       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 input) DOUBLE PRECISION array, dimension LOCr(M_A)
               The scale factors for A(IA:IA+M-1,JA:JA+N-1). SR is aligned with  the  distributed
               matrix  A,  and  replicated  across  every  process  column.  SR  is  tied  to the
               distributed matrix A.

       SC      (local input) DOUBLE PRECISION array, dimension LOCc(N_A)
               The scale factors for sub( A ). SC is aligned with the dis- tributed matrix A, and
               replicated down every process row.  SC is tied to the distributed matrix A.

       SCOND   (global input) DOUBLE PRECISION
               Ratio   of   the   smallest  SR(i)  (respectively  SC(j))  to  the  largest  SR(i)
               (respectively SC(j)), with IA <= i <= IA+N-1 and JA <= j <= JA+N-1.

       AMAX    (global input) DOUBLE PRECISION
               Absolute value of the largest distributed submatrix entry.

       EQUED   (output) CHARACTER*1
               Specifies whether or not equilibration was done.  = 'N':  No equilibration.
               = 'Y':  Equilibration was done, i.e., sub( A ) has been re-
               placed by:
               diag(SR(IA:IA+N-1)) * sub( A ) * diag(SC(JA:JA+N-1)).

PARAMETERS

       THRESH is a threshold value used to decide if scaling should be done based on the ratio of
       the scaling factors.  If SCOND < THRESH, scaling is done.

       LARGE and SMALL are threshold values used to decide if scaling should be done based on the
       absolute size of the largest matrix element.  If AMAX > LARGE or AMAX < SMALL, scaling  is
       done.