Man Linux: Main Page and Category List

NAME

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

SYNOPSIS

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

           CHARACTER       DIAG, TRANS, UPLO

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

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

           REAL            BERR( * ), FERR( * ), RWORK( * )

           COMPLEX         A( * ), B( * ), WORK( * ), X( * )

PURPOSE

       PCTRRFS 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 PCTRTRS or some  other  means
       before entering this routine.  PCTRRFS 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 )**H * sub( X ) = sub( B ) (Conjugate 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) COMPLEX 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) COMPLEX 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 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) REAL 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) 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) COMPLEX 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) REAL 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.

               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.