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NAME

       PDTRRFS  -  provide error bounds and backward error estimates for the solution to a system
       of linear equations with a triangular coefficient matrix

SYNOPSIS

       SUBROUTINE PDTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, X, IX,
                           JX, DESCX, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO )

           CHARACTER       DIAG, TRANS, UPLO

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

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

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

PURPOSE

       PDTRRFS provides error bounds and backward error estimates for the solution to a system of
       linear equations with a triangular coefficient matrix.

       The solution matrix X must be computed by PDTRTRS or some other means before entering this
       routine.   PDTRRFS  does  not  do iterative refinement because doing so cannot improve the
       backward error.

       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
               = 'U':  sub( A ) is upper triangular;
               = 'L':  sub( A ) is lower triangular.

       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)

       DIAG    (global input) CHARACTER*1
               = 'N':  sub( A ) is non-unit triangular;
               = 'U':  sub( A ) is unit 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) 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 original triangular distributed 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  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.  If DIAG = 'U', the diagonal elements  of
               sub( A ) are also not referenced and are assumed to be 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.

       B       (local input) DOUBLE PRECISION 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) 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 the local pieces of the solution vectors sub( X ).

       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 bounds for each solution vector of
               sub( X ).  If XTRUE is the true solution, FERR bounds the magnitude of the largest
               entry  in (sub( X ) - XTRUE) divided by the magnitude of the largest entry 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.

               Notes =====

               This routine temporarily returns when N <= 1.

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

               Moreover,  this  routine  requires the distributed submatrices sub( A ), 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( IB, DESCB( MB_ ) ) = f( JB,
               DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0.