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.
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 f11dcforf11def.
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,f11befandf11bff. 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.
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 ETNA1 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
1: – Character(*)Input
On entry: specifies the iterative method to be used.
The arrays istb, indb and a together with 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.
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.
19: – IntegerInput
On entry: the maximum number of iterations allowed.
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.
if , ;
if , ;
if , ;
if , .
25: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or 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).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
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
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
On successful termination, the final residual , where , satisfies the termination criterion
The value of the final residual norm is returned in rnorm.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
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).
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 .