NAG Library Routine Document
F11DGF
1 Purpose
F11DGF solves a real sparse nonsymmetric system of linear equations, represented in coordinate storage format, using a restarted generalized minimal residual (RGMRES), conjugate gradient squared (CGS), stabilized bi-conjugate gradient (Bi-CGSTAB), or transpose-free quasi-minimal residual (TFQMR) method, with block Jacobi or additive Schwarz preconditioning.
2 Specification
SUBROUTINE F11DGF ( |
METHOD, N, NNZ, A, LA, IROW, ICOL, NB, ISTB, INDB, LINDB, IPIVP, IPIVQ, ISTR, IDIAG, B, M, TOL, MAXITN, X, RNORM, ITN, WORK, LWORK, IFAIL) |
INTEGER |
N, NNZ, LA, IROW(LA), ICOL(LA), NB, ISTB(NB+1), INDB(LINDB), LINDB, IPIVP(LINDB), IPIVQ(LINDB), ISTR(LINDB+1), IDIAG(LINDB), M, MAXITN, ITN, LWORK, IFAIL |
REAL (KIND=nag_wp) |
A(LA), B(N), TOL, X(N), RNORM, WORK(LWORK) |
CHARACTER(*) |
METHOD |
|
3 Description
F11DGF solves a real sparse nonsymmetric linear system of equations:
using a preconditioned RGMRES (see
Saad and Schultz (1986)), CGS (see
Sonneveld (1989)), Bi-CGSTAB(
) (see
Van der Vorst (1989) and
Sleijpen and Fokkema (1993)), or TFQMR (see
Freund and Nachtigal (1991) and
Freund (1993)) method.
F11DGF uses the incomplete (possibly overlapping) block
factorization determined by
F11DFF as the preconditioning matrix. A call to F11DGF must always be preceded by a call to
F11DFF. Alternative preconditioners for the same storage scheme are available by calling
F11DCF or
F11DEF.
The matrix
, and the preconditioning matrix
, are represented in coordinate storage (CS) format (see
Section 2.1.1 in the F11 Chapter Introduction) in the arrays
A,
IROW and
ICOL, as returned from
F11DFF. The array
A holds the nonzero entries in these matrices, while
IROW and
ICOL hold the corresponding row and column indices.
F11DGF is a Black Box routine which calls
F11BDF,
F11BEF and
F11BFF. If you wish to use an alternative storage scheme, preconditioner, or termination criterion, or require additional diagnostic information, you should call these underlying routines directly.
4 References
Freund R W (1993) A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems SIAM J. Sci. Comput. 14 470–482
Freund R W and Nachtigal N (1991) QMR: a Quasi-Minimal Residual Method for Non-Hermitian Linear Systems Numer. Math. 60 315–339
Saad Y and Schultz M (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 7 856–869
Salvini S A and Shaw G J (1996) An evaluation of new NAG Library solvers for large sparse unsymmetric linear systems NAG Technical Report TR2/96
Sleijpen G L G and Fokkema D R (1993) BiCGSTAB for linear equations involving matrices with complex spectrum ETNA 1 11–32
Sonneveld P (1989) CGS, a fast Lanczos-type solver for nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 10 36–52
Van der Vorst H (1989) Bi-CGSTAB, a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 13 631–644
5 Parameters
- 1: – CHARACTER(*)Input
-
On entry: specifies the iterative method to be used.
- Restarted generalized minimum residual method.
- Conjugate gradient squared method.
- Bi-conjugate gradient stabilized () method.
- Transpose-free quasi-minimal residual method.
Constraint:
, , or .
- 2: – INTEGERInput
- 3: – INTEGERInput
- 4: – REAL (KIND=nag_wp) arrayInput
- 5: – INTEGERInput
- 6: – INTEGER arrayInput
- 7: – INTEGER arrayInput
- 8: – INTEGERInput
- 9: – INTEGER arrayInput
- 10: – INTEGER arrayInput
- 11: – INTEGERInput
- 12: – INTEGER arrayInput
- 13: – INTEGER arrayInput
- 14: – INTEGER arrayInput
- 15: – INTEGER arrayInput
-
On entry: the values returned in arrays
IROW,
ICOL,
IPIVP,
IPIVQ,
ISTR and
IDIAG by a previous call to
F11DFF.
The arrays
ISTB,
INDB and
A together with the the scalars
N,
NNZ,
LA,
NB and
LINDB must be the same values that were supplied in the preceding call to
F11DFF.
- 16: – REAL (KIND=nag_wp) arrayInput
-
On entry: the right-hand side vector .
- 17: – INTEGERInput
-
On entry: if
,
M is the dimension of the restart subspace.
If
,
M is the order
of the polynomial Bi-CGSTAB method. Otherwise,
M is not referenced.
Constraints:
- if , ;
- if , .
- 18: – REAL (KIND=nag_wp)Input
-
On entry: the required tolerance. Let
denote the approximate solution at iteration
, and
the corresponding residual. The algorithm is considered to have converged at iteration
if
If
,
is used, where
is the
machine precision. Otherwise
is used.
Constraint:
.
- 19: – INTEGERInput
-
On entry: the maximum number of iterations allowed.
Constraint:
.
- 20: – REAL (KIND=nag_wp) arrayInput/Output
-
On entry: an initial approximation to the solution vector .
On exit: an improved approximation to the solution vector .
- 21: – REAL (KIND=nag_wp)Output
-
On exit: the final value of the residual norm
, where
is the output value of
ITN.
- 22: – INTEGEROutput
-
On exit: the number of iterations carried out.
- 23: – REAL (KIND=nag_wp) arrayWorkspace
- 24: – INTEGERInput
-
On entry: the dimension of the array
WORK as declared in the (sub)program from which F11DGF is called.
Constraints:
- if , ;
- if , ;
- if , ;
- if , .
- 25: – INTEGERInput/Output
-
On entry:
IFAIL must be set to
,
. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
.
When the value is used it is essential to test the value of IFAIL on exit.
On exit:
unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
or
, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
-
On entry, for , and .
Constraint: , for .
On entry, and .
Constraint: , for
On entry, .
Constraint: .
On entry, and .
Constraint: .
On entry, , and .
Constraint: .
On entry, .
Constraint: .
On entry, and .
Constraint: if , .
If , .
On entry, .
Constraint: .
On entry, .
Constraint: , or .
On entry, .
Constraint: .
On entry, and .
Constraint: .
On entry, .
Constraint: .
On entry, and .
Constraint: .
On entry, .
Constraint: .
-
On entry, element
of
A was out of order.
Check that
A,
IROW,
ICOL,
IPIVP,
IPIVQ,
ISTR and
IDIAG have not been corrupted between calls to
F11DFF and F11DGF.
On entry,
and
.
Constraint:
, for
.
Check that
A,
IROW,
ICOL,
IPIVP,
IPIVQ,
ISTR and
IDIAG have not been corrupted between calls to
F11DFF and F11DGF.
On entry,
and
.
Constraint:
, for
.
Check that
A,
IROW,
ICOL,
IPIVP,
IPIVQ,
ISTR and
IDIAG have not been corrupted between calls to
F11DFF and F11DGF.
On entry, location
of
was a duplicate.
Check that
A,
IROW,
ICOL,
IPIVP,
IPIVQ,
ISTR and
IDIAG have not been corrupted between calls to
F11DFF and F11DGF.
-
The CS representation of the preconditioner is invalid.
Check that
A,
IROW,
ICOL,
IPIVP,
IPIVQ,
ISTR and
IDIAG have not been corrupted between calls to
F11DFF and F11DGF.
-
The required accuracy could not be obtained. However a reasonable accuracy may have been achieved. You should check the output value of
RNORM for acceptability. This error code usually implies that your problem has been fully and satisfactorily solved to within or close to the accuracy available on your system. Further iterations are unlikely to improve on this situation.
-
The solution has not converged after iterations.
-
Algorithmic breakdown. A solution is returned, although it is possible that it is completely inaccurate.
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
7 Accuracy
On successful termination, the final residual
, where
, satisfies the termination criterion
The value of the final residual norm is returned in
RNORM.
8 Parallelism and Performance
F11DGF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F11DGF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The time taken by F11DGF for each iteration is roughly proportional to the value of
NNZC returned from the preceding call to
F11DFF.
The number of iterations required to achieve a prescribed accuracy cannot be easily determined a priori, as it can depend dramatically on the conditioning and spectrum of the preconditioned coefficient matrix .
Some illustrations of the application of F11DGF to linear systems arising from the discretization of two-dimensional elliptic partial differential equations, and to random-valued randomly structured linear systems, can be found in
Salvini and Shaw (1996).
10 Example
This example program reads in a sparse matrix
and a vector
. It calls
F11DFF, with the array
and the array
, to compute an overlapping incomplete
factorization. This is then used as an additive Schwarz preconditioner on a call to F11DGF which uses the Bi-CGSTAB method to solve
.
10.1 Program Text
Program Text (f11dgfe.f90)
10.2 Program Data
Program Data (f11dgfe.d)
10.3 Program Results
Program Results (f11dgfe.r)