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NAME

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

SYNOPSIS

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

           CHARACTER       UPLO

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

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

           REAL            A( * ), AF( * ), B( * ), BERR( * ), FERR( * ), WORK( * ), X( * )

PURPOSE

       PSPORFS  improves  the  computed  solution  to  a  system  of  linear  equations  when the
       coefficient matrix is symmetric 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 symmetric 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) REAL 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  symmetric  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) REAL 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**T  or
               U**T*U, as computed by PSPOTRF.

       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) REAL 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) REAL 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) REAL 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) REAL 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) REAL 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
               >= 3*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.

       IWORK   (local workspace/local output) INTEGER array,
               dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK.

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

               If  LIWORK = -1, then LIWORK 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.