NAG Library Routine Document
F11DCF
1 Purpose
F11DCF 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 biconjugate gradient (BiCGSTAB), or transposefree quasiminimal residual (TFQMR) method, with incomplete $LU$ preconditioning.
2 Specification
SUBROUTINE F11DCF ( 
METHOD, N, NNZ, A, LA, IROW, ICOL, IPIVP, IPIVQ, ISTR, IDIAG, B, M, TOL, MAXITN, X, RNORM, ITN, WORK, LWORK, IFAIL) 
INTEGER 
N, NNZ, LA, IROW(LA), ICOL(LA), IPIVP(N), IPIVQ(N), ISTR(N+1), IDIAG(N), M, MAXITN, ITN, LWORK, IFAIL 
REAL (KIND=nag_wp) 
A(LA), B(N), TOL, X(N), RNORM, WORK(LWORK) 
CHARACTER(*) 
METHOD 

3 Description
F11DCF solves a real sparse nonsymmetric linear system of equations:
using a preconditioned RGMRES (see
Saad and Schultz (1986)), CGS (see
Sonneveld (1989)), BiCGSTAB(
$\ell $) (see
Van der Vorst (1989) and
Sleijpen and Fokkema (1993)), or TFQMR (see
Freund and Nachtigal (1991) and
Freund (1993)) method.
F11DCF uses the incomplete
$LU$ factorization determined by
F11DAF as the preconditioning matrix. A call to F11DCF must always be preceded by a call to
F11DAF. Alternative preconditioners for the same storage scheme are available by calling
F11DEF.
The matrix
$A$, and the preconditioning matrix
$M$, 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
F11DAF. The array
A holds the nonzero entries in these matrices, while
IROW and
ICOL hold the corresponding row and column indices.
F11DCF 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 transposefree quasiminimal residual algorithm for nonHermitian linear systems SIAM J. Sci. Comput. 14 470–482
Freund R W and Nachtigal N (1991) QMR: a QuasiMinimal Residual Method for NonHermitian 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$\left(\ell \right)$ for linear equations involving matrices with complex spectrum ETNA 1 11–32
Sonneveld P (1989) CGS, a fast Lanczostype solver for nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 10 36–52
Van der Vorst H (1989) BiCGSTAB, a fast and smoothly converging variant of BiCG for the solution of nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 13 631–644
5 Parameters
 1: METHOD – CHARACTER(*)Input
On entry: specifies the iterative method to be used.
 ${\mathbf{METHOD}}=\text{'RGMRES'}$
 Restarted generalized minimum residual method.
 ${\mathbf{METHOD}}=\text{'CGS'}$
 Conjugate gradient squared method.
 ${\mathbf{METHOD}}=\text{'BICGSTAB'}$
 Biconjugate gradient stabilized ($\ell $) method.
 ${\mathbf{METHOD}}=\text{'TFQMR'}$
 Transposefree quasiminimal residual method.
Constraint:
${\mathbf{METHOD}}=\text{'RGMRES'}$, $\text{'CGS'}$, $\text{'BICGSTAB'}$ or $\text{'TFQMR'}$.
 2: N – INTEGERInput
On entry:
$n$, the order of the matrix
$A$. This
must be the same value as was supplied in the preceding call to
F11DAF.
Constraint:
${\mathbf{N}}\ge 1$.
 3: NNZ – INTEGERInput
On entry: the number of nonzero elements in the matrix
$A$. This
must be the same value as was supplied in the preceding call to
F11DAF.
Constraint:
$1\le {\mathbf{NNZ}}\le {{\mathbf{N}}}^{2}$.
 4: A(LA) – REAL (KIND=nag_wp) arrayInput
On entry: the values returned in the array
A by a previous call to
F11DAF.
 5: LA – INTEGERInput
On entry: the dimension of the arrays
A,
IROW and
ICOL as declared in the (sub)program from which F11DCF is called. This
must be the same value as was supplied in the preceding call to
F11DAF.
Constraint:
${\mathbf{LA}}\ge 2\times {\mathbf{NNZ}}$.
 6: IROW(LA) – INTEGER arrayInput
 7: ICOL(LA) – INTEGER arrayInput
 8: IPIVP(N) – INTEGER arrayInput
 9: IPIVQ(N) – INTEGER arrayInput
 10: ISTR(${\mathbf{N}}+1$) – INTEGER arrayInput
 11: IDIAG(N) – INTEGER arrayInput
On entry: the values returned in arrays
IROW,
ICOL,
IPIVP,
IPIVQ,
ISTR and
IDIAG by a previous call to
F11DAF.
IPIVP and
IPIVQ are restored on exit.
 12: B(N) – REAL (KIND=nag_wp) arrayInput
On entry: the righthand side vector $b$.
 13: M – INTEGERInput
On entry: if
${\mathbf{METHOD}}=\text{'RGMRES'}$,
M is the dimension of the restart subspace.
If
${\mathbf{METHOD}}=\text{'BICGSTAB'}$,
M is the order
$\ell $ of the polynomial BiCGSTAB method; otherwise,
M is not referenced.
Constraints:
 if ${\mathbf{METHOD}}=\text{'RGMRES'}$, $0<{\mathbf{M}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{N}},50\right)$;
 if ${\mathbf{METHOD}}=\text{'BICGSTAB'}$, $0<{\mathbf{M}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{N}},10\right)$.
 14: TOL – REAL (KIND=nag_wp)Input
On entry: the required tolerance. Let
${x}_{k}$ denote the approximate solution at iteration
$k$, and
${r}_{k}$ the corresponding residual. The algorithm is considered to have converged at iteration
$k$ if
If
${\mathbf{TOL}}\le 0.0$,
$\tau =\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\sqrt{\epsilon},\sqrt{n}\epsilon \right)$ is used, where
$\epsilon $ is the
machine precision. Otherwise
$\tau =\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{TOL}},10\epsilon ,\sqrt{n}\epsilon \right)$ is used.
Constraint:
${\mathbf{TOL}}<1.0$.
 15: MAXITN – INTEGERInput
On entry: the maximum number of iterations allowed.
Constraint:
${\mathbf{MAXITN}}\ge 1$.
 16: X(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: an initial approximation to the solution vector $x$.
On exit: an improved approximation to the solution vector $x$.
 17: RNORM – REAL (KIND=nag_wp)Output
On exit: the final value of the residual norm
${\Vert {r}_{k}\Vert}_{\infty}$, where
$k$ is the output value of
ITN.
 18: ITN – INTEGEROutput
On exit: the number of iterations carried out.
 19: WORK(LWORK) – REAL (KIND=nag_wp) arrayWorkspace
 20: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F11DCF is called.
Constraints:
 if ${\mathbf{METHOD}}=\text{'RGMRES'}$, ${\mathbf{LWORK}}\ge 4\times {\mathbf{N}}+{\mathbf{M}}\times \left({\mathbf{M}}+{\mathbf{N}}+5\right)+101$;
 if ${\mathbf{METHOD}}=\text{'CGS'}$, ${\mathbf{LWORK}}\ge 8\times {\mathbf{N}}+100$;
 if ${\mathbf{METHOD}}=\text{'BICGSTAB'}$, ${\mathbf{LWORK}}\ge 2\times {\mathbf{N}}\times \left({\mathbf{M}}+3\right)+{\mathbf{M}}\times \left({\mathbf{M}}+2\right)+100$;
 if ${\mathbf{METHOD}}=\text{'TFQMR'}$, ${\mathbf{LWORK}}\ge 11\times {\mathbf{N}}+100$.
 21: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$
On entry,  ${\mathbf{METHOD}}\ne \text{'RGMRES'},\text{'CGS'},\text{'BICGSTAB'}$, or 'TFQMR', 
or  ${\mathbf{N}}<1$, 
or  ${\mathbf{NNZ}}<1$, 
or  ${\mathbf{NNZ}}>{{\mathbf{N}}}^{2}$, 
or  ${\mathbf{LA}}<2\times {\mathbf{NNZ}}$, 
or  ${\mathbf{M}}<1$ and ${\mathbf{METHOD}}=\text{'RGMRES'}$ or ${\mathbf{METHOD}}=\text{'BICGSTAB'}$, 
or  ${\mathbf{M}}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{N}},50\right)$, with ${\mathbf{METHOD}}=\text{'RGMRES'}$, 
or  ${\mathbf{M}}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{N}},10\right)$, with ${\mathbf{METHOD}}=\text{'BICGSTAB'}$, 
or  ${\mathbf{TOL}}\ge 1.0$, 
or  ${\mathbf{MAXITN}}<1$, 
or  LWORK too small. 
 ${\mathbf{IFAIL}}=2$
On entry, the CS representation of
$A$ is invalid. Further details are given in the error message. Check that the call to F11DCF has been preceded by a valid call to
F11DAF, and that the arrays
A,
IROW, and
ICOL have not been corrupted between the two calls.
 ${\mathbf{IFAIL}}=3$
On entry, the CS representation of the preconditioning matrix
$M$ is invalid. Further details are given in the error message. Check that the call to F11DCF has been preceded by a valid call to
F11DAF and that the arrays
A,
IROW,
ICOL,
IPIVP,
IPIVQ,
ISTR and
IDIAG have not been corrupted between the two calls.
 ${\mathbf{IFAIL}}=4$
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.
 ${\mathbf{IFAIL}}=5$
Required accuracy not obtained in
MAXITN iterations.
 ${\mathbf{IFAIL}}=6$
Algorithmic breakdown. A solution is returned, although it is possible that it is completely inaccurate.
 ${\mathbf{IFAIL}}=7$ (F11BDF, F11BEF or F11BFF)
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.
7 Accuracy
On successful termination, the final residual
${r}_{k}=bA{x}_{k}$, where
$k={\mathbf{ITN}}$, satisfies the termination criterion
The value of the final residual norm is returned in
RNORM.
The time taken by F11DCF for each iteration is roughly proportional to the value of
NNZC returned from the preceding call to
F11DAF.
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 $\stackrel{}{A}={{\mathbf{M}}}^{1}A$.
Some illustrations of the application of F11DCF to linear systems arising from the discretization of twodimensional elliptic partial differential equations, and to randomvalued randomly structured linear systems, can be found in
Salvini and Shaw (1996).
9 Example
This example solves a sparse linear system of equations using the CGS method, with incomplete $LU$ preconditioning.
9.1 Program Text
Program Text (f11dcfe.f90)
9.2 Program Data
Program Data (f11dcfe.d)
9.3 Program Results
Program Results (f11dcfe.r)