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NAME

       PDGERFS - improve the computed solution to a system of linear equations
       and  provides  error  bounds  and  backward  error  estimates  for  the
       solutions

SYNOPSIS

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

           CHARACTER       TRANS

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

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

           DOUBLE          PRECISION A( * ), AF( * ), B( * ), BERR( * ), FERR(
                           * ), WORK( * ), X( * )

PURPOSE

       PDGERFS improves the computed solution to a system of linear  equations
       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

       TRANS   (global input) CHARACTER*1
               Specifies the form of the system of equations.  = ’N’: sub( A )
               * sub( X ) = sub( B )          (No transpose)
               = ’T’: sub( A )**T * sub( X ) = sub( B )          (Transpose)
               = ’C’: sub( A )**T * sub( X ) = sub( B ) (Conjugate transpose =
               Transpose)

       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) DOUBLE PRECISION 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 distributed matrix
               sub( A ).

       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) DOUBLE PRECISION pointer into the local
               memory to an array of  local  dimension  (LLD_AF,LOCc(JA+N-1)).
               This array contains the local pieces of the distributed factors
               of the matrix sub( A ) = P * L * U as computed by PDGETRF.

       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.

       IPIV    (local input) INTEGER array of dimension LOCr(M_AF)+MB_AF.
               This array contains the pivoting  information  as  computed  by
               PDGETRF.  IPIV(i)  ->  The  global  row local row i was swapped
               with. This array is tied to the distributed matrix A.

       B       (local input) DOUBLE PRECISION pointer into the local
               memory to an array of local dimension  (LLD_B,LOCc(JB+NRHS-1)).
               This  array contains the local pieces of the distributed matrix
               of 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 and output) DOUBLE PRECISION pointer into the
               local   memory    to    an    array    of    local    dimension
               (LLD_X,LOCc(JX+NRHS-1)).  On  entry,  this  array  contains the
               local pieces of the distributed matrix solution sub(  X  ).  On
               exit, 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) DOUBLE PRECISION 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.