ditsol_ptfqmr 

Preconditioned transpose-free quasi-minimal method (Serial and Parallel Versions)

FORMAT

  DITSOL_PTFQMR (matvec, pcondl, pcondr, mstop, a, ia, x, b, n,
                ql, iql , qr, iqr, iparam, rparam, iwork, rwork, ierror)

Arguments

        DITSOL_PTFQMR has the standard parameter list for an iterative
  solver.

Description

  The quasi-minimal residual (QMR) method [Freund and Nachtigal 1991] is one
  of the algorithms proposed as a remedy for the irregular convergence
  behavior of the bi-conjugate gradient and the conjugate graient squared
  algorithms. Since these algorithms are not characterized by a minimization
  property, the residual norm often oscillates wildly. The QMR algorithm, by
  generating iterates that are defined by a quasi-minimization of the
  residual norm, results in smooth convergence curves.

  CXML includes TFQMR, the transpose-free variant of the QMR method,
  implemented without look-ahead [Freund 1993].  The implementation of the
  transpose-free quasi-minimal residual method requires the routine MATVEC to
  provide operations for job= 0.  The routines MATVEC, PCONDL (if used),
  PCONDR (if used) and MSTOP  (if used) should be declared external in your
  calling (sub)program.

  An upper bound for the two norm of the residual of the system being solved,
  is obtained during the implementation of the TFQMR method at no extra cost.
  This is the residual of the system to which the method is applied, which in
  the left and split preconditioned case is the preconditioned residual,

           inverse(QL) * r.

       To obtain the true residual, a non-negligible amount of extra
       computation would be required. Hence, for this method, only stopping
       criteria 3 and 4 are allowed. In the unpreconditioned case, the
       stopping criteria default to 1 and 2, respectively. Thus only istop =
       3 and istop = 4 are permitted for both the preconditioned and
       unpreconditioned case.  Additionally, a user-defined MSTOP is allowed,
       but the vectors r and z, corresponding the the real and preconditioned
       residuals, respectively, and passed as input parameters to the routine
       MSTOP, are undefined.

  CXML provides the following four forms of the method:

  •  Unpreconditioned transpose-free quasi-minimal residual method:

     This is the transpose-free quasi-minimal residual method applied to

           A * x =  b

     where A is a general matrix. As no preconditioning is used, both PCONDL
     and PCONDR are dummy input parameters.

     For the unpreconditioned transpose-free quasi-minimal residual method,
     the length of the real work space array, defined by the   variable nrwk
     (IPARAM(4)), should be at least 7*n,  where n is the order of the matrix
     A.

     The vectors r and z, passed as input arguments to the routine MSTOP, are
     not defined.

  •  Transpose-free quasi-minimal residual method with left preconditioning:

     This is the transpose-free quasi-minimal residual method applied to

         (inverse(QL) * A )* x  = (inverse(QL) * b)

     The routine PCONDL, with job= 0 should evaluate

         v = inverse(QL) * u

     The routine PCONDR is not used and is therefore a dummy input parameter.

     For the transpose-free quasi-minimal residual method, with left
     preconditioning, the length of the real work space array, defined  by
     the variable nrwk (IPARAM(4)), should be at least 8*n, where n is the
     order of the matrix A. This does not include the memory requirements of
     the preconditioner.

     The vectors r and z, passed as input arguments to the routine MSTOP, are
     not defined.

  •  Transpose-free quasi-minimal residual method with right preconditioning:

     This is the transpose-free quasi-minimal residual method applied to

         ( A * inverse(QR))  * y  =  b

     where

         y = QR * x

     The routine PCONDR, with job= 0 should evaluate

         v = inverse(QR) * u

     The routine PCONDL is not used and is therefore a dummy input parameter.

     For the transpose-free quasi-minimal residual method, with right
     preconditioning, the length of the real work space array, defined  by
     the variable nrwk (IPARAM(4)), should be at least 8*n,  where n is the
     order of the  matrix A. This does not include  the memory requirements
     of the preconditioner.

     The vectors r and z, passed as input arguments to the routine  MSTOP,
     are not defined.

  •  Transpose-free quasi-minimal residual method with split preconditioning:

     This is the transpose-free quasi-minimal residual method applied to

     (inverse(QL)  * A * inverse(QR)) * y  = (inverse(QL) * b)

     where

         y = QR * x

     The routine PCONDL, with job= 0 should evaluate

         v = inverse(QL) * u

     and the routine PCONDR, with job= 0 should evaluate

         v = inverse(QR) * u

     For the transpose-free quasi-minimal residual method, with split
     preconditioning, the length of the real work space array, defined  by
     the variable nrwk (IPARAM(4)), should be at least 9*n,  where n is the
     order of the matrix A. This does not include  the memory requirements of
     the preconditioner.

     The vectors r and z, passed as input arguments to the routine  MSTOP,
     are not defined.

  This routine is available in both serial and parallel versions. The routine
  names and parameter list are identical for both versions. For information
  about linking to the serial or to the parallel library, refer to the CXML
  Reference Manual.