NAG Library Routine Document
F11DSF
1 Purpose
F11DSF solves a complex sparse non-Hermitian 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, without preconditioning, with Jacobi, or with SSOR preconditioning.
2 Specification
SUBROUTINE F11DSF ( |
METHOD, PRECON, N, NNZ, A, IROW, ICOL, OMEGA, B, M, TOL, MAXITN, X, RNORM, ITN, WORK, LWORK, IWORK, IFAIL) |
INTEGER |
N, NNZ, IROW(NNZ), ICOL(NNZ), M, MAXITN, ITN, LWORK, IWORK(2*N+1), IFAIL |
REAL (KIND=nag_wp) |
OMEGA, TOL, RNORM |
COMPLEX (KIND=nag_wp) |
A(NNZ), B(N), X(N), WORK(LWORK) |
CHARACTER(*) |
METHOD |
CHARACTER(1) |
PRECON |
|
3 Description
F11DSF solves a complex sparse non-Hermitian system of linear equations:
using an 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.
F11DSF allows the following choices for the preconditioner:
- – no preconditioning;
- – Jacobi preconditioning (see Young (1971));
- – symmetric successive-over-relaxation (SSOR) preconditioning (see Young (1971)).
For incomplete
(ILU) preconditioning see
F11DQF.
The matrix
is represented in coordinate storage (CS) format (see
Section 2.1.1 in the F11 Chapter Introduction) in the arrays
A,
IROW and
ICOL. The array
A holds the nonzero entries in the matrix, while
IROW and
ICOL hold the corresponding row and column indices.
F11DSF is a Black Box routine which calls
F11BRF,
F11BSF and
F11BTF. 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
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
Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York
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: – CHARACTER(1)Input
-
On entry: specifies the type of preconditioning to be used.
- No preconditioning.
- Jacobi.
- Symmetric successive-over-relaxation (SSOR).
Constraint:
, or .
- 3: – INTEGERInput
-
On entry: , the order of the matrix .
Constraint:
.
- 4: – INTEGERInput
-
On entry: the number of nonzero elements in the matrix .
Constraint:
.
- 5: – COMPLEX (KIND=nag_wp) arrayInput
-
On entry: the nonzero elements of the matrix
, ordered by increasing row index, and by increasing column index within each row. Multiple entries for the same row and column indices are not permitted. The routine
F11ZNF may be used to order the elements in this way.
- 6: – INTEGER arrayInput
- 7: – INTEGER arrayInput
-
On entry: the row and column indices of the nonzero elements supplied in
A.
Constraints:
IROW and
ICOL must satisfy the following constraints (which may be imposed by a call to
F11ZNF):
- and , for ;
- either or both and , for .
- 8: – REAL (KIND=nag_wp)Input
-
On entry: if
,
OMEGA is the relaxation parameter
to be used in the SSOR method. Otherwise
OMEGA need not be initialized and is not referenced.
Constraint:
.
- 9: – COMPLEX (KIND=nag_wp) arrayInput
-
On entry: the right-hand side vector .
- 10: – 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 , .
- 11: – 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:
.
- 12: – INTEGERInput
-
On entry: the maximum number of iterations allowed.
Constraint:
.
- 13: – COMPLEX (KIND=nag_wp) arrayInput/Output
-
On entry: an initial approximation to the solution vector .
On exit: an improved approximation to the solution vector .
- 14: – REAL (KIND=nag_wp)Output
-
On exit: the final value of the residual norm
, where
is the output value of
ITN.
- 15: – INTEGEROutput
-
On exit: the number of iterations carried out.
- 16: – COMPLEX (KIND=nag_wp) arrayWorkspace
- 17: – INTEGERInput
-
On entry: the dimension of the array
WORK as declared in the (sub)program from which F11DSF is called.
Constraints:
- if , ;
- if , ;
- if , ;
- if , .
Where for or and otherwise.
- 18: – INTEGER arrayWorkspace
-
- 19: – 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, | , , or , |
or | , or , |
or | , |
or | , |
or | , |
or | and OMEGA lies outside the interval , |
or | , |
or | , when , |
or | , when , |
or | , |
or | , |
or | LWORK is too small. |
-
On entry, the arrays
IROW and
ICOL fail to satisfy the following constraints:
- and , for ;
- , or and , for .
Therefore a nonzero element has been supplied which does not lie within the matrix
, is out of order, or has duplicate row and column indices. Call
F11ZNF to reorder and sum or remove duplicates.
-
On entry, the matrix has a zero diagonal element. Jacobi and SSOR preconditioners are therefore not appropriate for this problem.
-
The required accuracy could not be obtained. However, a reasonable accuracy may have been obtained, and further iterations could not improve the result. 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.
-
Required accuracy not obtained in
MAXITN iterations.
-
Algorithmic breakdown. A solution is returned, although it is possible that it is completely inaccurate.
- (F11BRF, F11BSF or F11BTF)
-
A serious error has occurred in an internal call to one of the specified routines. Check all subroutine calls and array sizes. Seek expert help.
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
F11DSF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F11DSF 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 F11DSF for each iteration is roughly proportional to
NNZ.
The number of iterations required to achieve a prescribed accuracy cannot easily be determined a priori, as it can depend dramatically on the conditioning and spectrum of the preconditioned coefficient matrix , for some preconditioning matrix .
10 Example
This example solves a complex sparse non-Hermitian system of equations using the CGS method, with no preconditioning.
10.1 Program Text
Program Text (f11dsfe.f90)
10.2 Program Data
Program Data (f11dsfe.d)
10.3 Program Results
Program Results (f11dsfe.r)