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NAME

       PZPORFS - improve the computed solution to a system of linear equations
       when the coefficient matrix is Hermitian positive definite and provides
       error bounds and backward error estimates for the solutions

SYNOPSIS

       SUBROUTINE PZPORFS( UPLO,  N,  NRHS,  A,  IA,  JA, DESCA, AF, IAF, JAF,
                           DESCAF, B, IB, JB, DESCB, X, IX, JX,  DESCX,  FERR,
                           BERR, WORK, LWORK, RWORK, LRWORK, INFO )

           CHARACTER       UPLO

           INTEGER         IA,  IAF,  IB,  INFO,  IX, JA, JAF, JB, JX, LRWORK,
                           LWORK, N, NRHS

           INTEGER         DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * )

           COMPLEX*16      A( * ), AF( * ), B( * ), BERR(  *  ),  FERR(  *  ),
                           WORK( * ), X( * )

           DOUBLE          PRECISION RWORK( * )

PURPOSE

       PZPORFS  improves the computed solution to a system of linear equations
       when the coefficient matrix is Hermitian positive definite and provides
       error bounds and backward error estimates for the solutions.

       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

       In the following comments, sub( A ), sub( X  )  and  sub(  B  )  denote
       respectively   A(IA:IA+N-1,JA:JA+N-1),   X(IX:IX+N-1,JX:JX+NRHS-1)  and
       B(IB:IB+N-1,JB:JB+NRHS-1).

ARGUMENTS

       UPLO    (global input) CHARACTER*1
               Specifies whether the upper or lower  triangular  part  of  the
               Hermitian matrix sub( A ) is stored.  = ’U’:  Upper triangular
               = ’L’:  Lower triangular

       N       (global input) INTEGER
               The order of the matrix sub( A ).  N >= 0.

       NRHS    (global input) INTEGER
               The  number of right hand sides, i.e., the number of columns of
               the matrices sub( B ) and sub( X ).  NRHS >= 0.

       A       (local input) COMPLEX*16 pointer into the local
               memory to an array of local  dimension  (LLD_A,LOCc(JA+N-1)  ).
               This  array  contains  the local pieces of the N-by-N Hermitian
               distributed matrix sub( A ) to be factored.  If UPLO = ’U’, the
               leading  N-by-N  upper triangular part of sub( A ) contains the
               upper triangular part of the matrix,  and  its  strictly  lower
               triangular  part is not referenced.  If UPLO = ’L’, the leading
               N-by-N lower triangular part of sub( A  )  contains  the  lower
               triangular  part  of the distribu- ted matrix, and its strictly
               upper triangular part is not 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.

       AF      (local input) COMPLEX*16 pointer into the local memory
               to an  array  of  local  dimension  (LLD_AF,LOCc(JA+N-1)).   On
               entry, this array contains the factors L or U from the Cholesky
               factorization sub( A ) =  L*L**H  or  U**H*U,  as  computed  by
               PZPOTRF.

       IAF     (global input) INTEGER
               The  row  index in the global array AF indicating the first row
               of sub( AF ).

       JAF     (global input) INTEGER
               The column index in the global array AF  indicating  the  first
               column of sub( AF ).

       DESCAF  (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix AF.

       B       (local input) COMPLEX*16 pointer into the local memory
               to  an  array of local dimension (LLD_B, LOCc(JB+NRHS-1) ).  On
               entry, this array contains the the local pieces  of  the  right
               hand sides sub( B ).

       IB      (global input) INTEGER
               The row index in the global array B indicating the first row of
               sub( B ).

       JB      (global input) INTEGER
               The column index in the global array  B  indicating  the  first
               column of sub( B ).

       DESCB   (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix B.

       X       (local input) COMPLEX*16 pointer into the local memory
               to  an  array of local dimension (LLD_X, LOCc(JX+NRHS-1) ).  On
               entry, this array contains the the local pieces of the solution
               vectors  sub(  X  ). On exit, it contains the improved solution
               vectors.

       IX      (global input) INTEGER
               The row index in the global array X indicating the first row of
               sub( X ).

       JX      (global input) INTEGER
               The  column  index  in  the global array X indicating the first
               column of sub( X ).

       DESCX   (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix X.

       FERR    (local output) DOUBLE PRECISION array of local dimension
               LOCc(JB+NRHS-1).  The estimated forward error  bound  for  each
               solution  vector  of  sub(  X ).  If XTRUE is the true solution
               corresponding to sub( X ), FERR is an estimated upper bound for
               the  magnitude  of  the  largest  element in (sub( X ) - XTRUE)
               divided by the magnitude of the largest element in  sub(  X  ).
               The  estimate  is as reliable as the estimate for RCOND, and is
               almost always a slight overestimate of the  true  error.   This
               array is tied to the distributed matrix X.

       BERR    (local output) DOUBLE PRECISION array of local dimension
               LOCc(JB+NRHS-1).  The  componentwise relative backward error of
               each solution vector (i.e., the smallest re- lative  change  in
               any  entry of sub( A ) or sub( B ) that makes sub( X ) an exact
               solution).  This array is tied to the distributed matrix X.

       WORK    (local workspace/local output) COMPLEX*16 array,
               dimension (LWORK) On exit,  WORK(1)  returns  the  minimal  and
               optimal LWORK.

       LWORK   (local or global input) INTEGER
               The dimension of the array WORK.  LWORK is local input and must
               be at least LWORK >= 2*LOCr( N + MOD( IA-1, MB_A ) )

               If LWORK = -1, then LWORK is global input and a workspace query
               is assumed; the routine only calculates the minimum and optimal
               size for all work arrays. Each of these values is  returned  in
               the  first  entry of the corresponding work array, and no error
               message is issued by PXERBLA.

       RWORK   (local workspace/local output) DOUBLE PRECISION array,
               dimension (LRWORK) On exit, RWORK(1) returns  the  minimal  and
               optimal LRWORK.

       LRWORK  (local or global input) INTEGER
               The  dimension  of  the array RWORK.  LRWORK is local input and
               must be at least LRWORK >= LOCr( N + MOD( IB-1, MB_B ) ).

               If LRWORK = -1, then LRWORK is global  input  and  a  workspace
               query  is  assumed; the routine only calculates the minimum and
               optimal size for all work  arrays.  Each  of  these  values  is
               returned  in  the  first entry of the corresponding work array,
               and no error message is issued by PXERBLA.

       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.

PARAMETERS

       ITMAX is the maximum number of steps of iterative refinement.

       Notes =====

       This routine temporarily returns when N <= 1.

       The distributed submatrices op( A ) and op( AF ) (respectively sub( X )
       and  sub(  B  )  )  should  be  distributed  the  same  way on the same
       processes. These conditions ensure that sub( A ) and sub( AF  )  (resp.
       sub( X ) and sub( B ) ) are "perfectly" aligned.

       Moreover,  this  routine requires the distributed submatrices sub( A ),
       sub( AF ), sub( X ), and sub( B ) to be aligned on  a  block  boundary,
       i.e.,  if f(x,y) = MOD( x-1, y ): f( IA, DESCA( MB_ ) ) = f( JA, DESCA(
       NB_ ) ) = 0, f( IAF, DESCAF( MB_ ) ) = f( JAF, DESCAF( NB_ ) ) = 0,  f(
       IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ )
       ) = f( JX, DESCX( NB_ ) ) = 0.