F11 Chapter Contents
F11 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF11DSF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 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:
 $Ax=b,$
using an RGMRES (see Saad and Schultz (1986)), CGS (see Sonneveld (1989)), Bi-CGSTAB($\ell$) (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 $LU$ (ILU) preconditioning see F11DQF.
The matrix $A$ 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$\left(\ell \right)$ 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:     METHOD – CHARACTER(*)Input
On entry: specifies the iterative method to be used.
${\mathbf{METHOD}}=\text{'RGMRES'}$
Restarted generalized minimum residual method.
${\mathbf{METHOD}}=\text{'CGS'}$
${\mathbf{METHOD}}=\text{'BICGSTAB'}$
Bi-conjugate gradient stabilized ($\ell$) method.
${\mathbf{METHOD}}=\text{'TFQMR'}$
Transpose-free quasi-minimal residual method.
Constraint: ${\mathbf{METHOD}}=\text{'RGMRES'}$, $\text{'CGS'}$, $\text{'BICGSTAB'}$ or $\text{'TFQMR'}$.
2:     PRECON – CHARACTER(1)Input
On entry: specifies the type of preconditioning to be used.
${\mathbf{PRECON}}=\text{'N'}$
No preconditioning.
${\mathbf{PRECON}}=\text{'J'}$
Jacobi.
${\mathbf{PRECON}}=\text{'S'}$
Symmetric successive-over-relaxation (SSOR).
Constraint: ${\mathbf{PRECON}}=\text{'N'}$, $\text{'J'}$ or $\text{'S'}$.
3:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 1$.
4:     NNZ – INTEGERInput
On entry: the number of nonzero elements in the matrix $A$.
Constraint: $1\le {\mathbf{NNZ}}\le {{\mathbf{N}}}^{2}$.
5:     A(NNZ) – COMPLEX (KIND=nag_wp) arrayInput
On entry: the nonzero elements of the matrix $A$, 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:     IROW(NNZ) – INTEGER arrayInput
7:     ICOL(NNZ) – 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):
• $1\le {\mathbf{IROW}}\left(\mathit{i}\right)\le {\mathbf{N}}$ and $1\le {\mathbf{ICOL}}\left(\mathit{i}\right)\le {\mathbf{N}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{NNZ}}$;
• either ${\mathbf{IROW}}\left(\mathit{i}-1\right)<{\mathbf{IROW}}\left(\mathit{i}\right)$ or both ${\mathbf{IROW}}\left(\mathit{i}-1\right)={\mathbf{IROW}}\left(\mathit{i}\right)$ and ${\mathbf{ICOL}}\left(\mathit{i}-1\right)<{\mathbf{ICOL}}\left(\mathit{i}\right)$, for $\mathit{i}=2,3,\dots ,{\mathbf{NNZ}}$.
8:     OMEGA – REAL (KIND=nag_wp)Input
On entry: if ${\mathbf{PRECON}}=\text{'S'}$, OMEGA is the relaxation parameter $\omega$ to be used in the SSOR method. Otherwise OMEGA need not be initialized and is not referenced.
Constraint: $0.0<{\mathbf{OMEGA}}<2.0$.
9:     B(N) – COMPLEX (KIND=nag_wp) arrayInput
On entry: the right-hand side vector $b$.
10:   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 Bi-CGSTAB 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 ${\mathbf{METHOD}}=\text{'BICGSTAB'}$, $0<{\mathbf{M}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{N}},10\right)$.
11:   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
 $rk∞≤τ×b∞+A∞xk∞.$
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: ${\mathbf{TOL}}<1.0$.
12:   MAXITN – INTEGERInput
On entry: the maximum number of iterations allowed.
Constraint: ${\mathbf{MAXITN}}\ge 1$.
13:   X(N) – COMPLEX (KIND=nag_wp) arrayInput/Output
On entry: an initial approximation to the solution vector $x$.
On exit: an improved approximation to the solution vector $x$.
14:   RNORM – REAL (KIND=nag_wp)Output
On exit: the final value of the residual norm ${‖{r}_{k}‖}_{\infty }$, where $k$ is the output value of ITN.
15:   ITN – INTEGEROutput
On exit: the number of iterations carried out.
16:   WORK(LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
17:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F11DSF is called.
Constraints:
• if ${\mathbf{METHOD}}=\text{'RGMRES'}$, ${\mathbf{LWORK}}\ge 4×{\mathbf{N}}+{\mathbf{M}}×\left({\mathbf{M}}+{\mathbf{N}}+5\right)+\mathit{nu}+121$;
• if ${\mathbf{METHOD}}=\text{'CGS'}$, ${\mathbf{LWORK}}\ge 8×{\mathbf{N}}+\mathit{nu}+120$;
• if ${\mathbf{METHOD}}=\text{'BICGSTAB'}$, ${\mathbf{LWORK}}\ge 2×{\mathbf{N}}×\left({\mathbf{M}}+3\right)+{\mathbf{M}}×\left({\mathbf{M}}+2\right)+\mathit{nu}+120$;
• if ${\mathbf{METHOD}}=\text{'TFQMR'}$, ${\mathbf{LWORK}}\ge 11×{\mathbf{N}}+\mathit{nu}+120$.
Where $\mathit{nu}={\mathbf{N}}$ for ${\mathbf{PRECON}}=\text{'J'}$ or $\text{'S'}$ and $\mathit{nu}=0$ otherwise.
18:   IWORK($2×{\mathbf{N}}+1$) – INTEGER arrayWorkspace
19:   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 $\text{'TFQMR'}$, or ${\mathbf{PRECON}}\ne \text{'N'}$, $\text{'J'}$ or $\text{'S'}$, or ${\mathbf{N}}<1$, or ${\mathbf{NNZ}}<1$, or ${\mathbf{NNZ}}>{{\mathbf{N}}}^{2}$, or ${\mathbf{PRECON}}=\text{'S'}$ and OMEGA lies outside the interval $\left(0.0,2.0\right)$, or ${\mathbf{M}}<1$, or ${\mathbf{M}}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{N}},50\right)$, when ${\mathbf{METHOD}}=\text{'RGMRES'}$, or ${\mathbf{M}}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{N}},10\right)$, when ${\mathbf{METHOD}}=\text{'BICGSTAB'}$, or ${\mathbf{TOL}}\ge 1.0$, or ${\mathbf{MAXITN}}<1$, or LWORK is too small.
${\mathbf{IFAIL}}=2$
On entry, the arrays IROW and ICOL fail to satisfy the following constraints:
• $1\le {\mathbf{IROW}}\left(i\right)\le {\mathbf{N}}$ and $1\le {\mathbf{ICOL}}\left(i\right)\le {\mathbf{N}}$, for $i=1,2,\dots ,{\mathbf{NNZ}}$;
• ${\mathbf{IROW}}\left(i-1\right)<{\mathbf{IROW}}\left(i\right)$, or ${\mathbf{IROW}}\left(i-1\right)={\mathbf{IROW}}\left(i\right)$ and ${\mathbf{ICOL}}\left(i-1\right)<{\mathbf{ICOL}}\left(i\right)$, for $i=2,3,\dots ,{\mathbf{NNZ}}$.
Therefore a nonzero element has been supplied which does not lie within the matrix $A$, is out of order, or has duplicate row and column indices. Call F11ZNF to reorder and sum or remove duplicates.
${\mathbf{IFAIL}}=3$
On entry, the matrix $A$ has a zero diagonal element. Jacobi and SSOR preconditioners are therefore not appropriate for this problem.
${\mathbf{IFAIL}}=4$
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.
${\mathbf{IFAIL}}=7$ (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.

## 7  Accuracy

On successful termination, the final residual ${r}_{k}=b-A{x}_{k}$, where $k={\mathbf{ITN}}$, satisfies the termination criterion
 $rk∞≤τ×b∞+A∞xk∞.$
The value of the final residual norm is returned in RNORM.

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 $\stackrel{-}{A}={M}^{-1}A$, for some preconditioning matrix $M$.

## 9  Example

This example solves a complex sparse non-Hermitian system of equations using the CGS method, with no preconditioning.

### 9.1  Program Text

Program Text (f11dsfe.f90)

### 9.2  Program Data

Program Data (f11dsfe.d)

### 9.3  Program Results

Program Results (f11dsfe.r)