f11def 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, without preconditioning, with Jacobi, or with SSOR preconditioning.

2 Specification

Fortran Interface

Subroutine f11def (

method, precon, n, nnz, a, irow, icol, omega, b, m, tol, maxitn, x, rnorm, itn, work, lwork, iwork, ifail)

Integer, Intent (In)	::	n, nnz, irow(nnz), icol(nnz), m, maxitn, lwork
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	itn, iwork(2*n+1)
Real (Kind=nag_wp), Intent (In)	::	a(nnz), omega, b(n), tol
Real (Kind=nag_wp), Intent (Inout)	::	x(n)
Real (Kind=nag_wp), Intent (Out)	::	rnorm, work(lwork)
Character (*), Intent (In)	::	method
Character (1), Intent (In)	::	precon

C Header Interface

#include <nag.h>

void

f11def_ (const char *method, const char *precon, const Integer *n, const Integer *nnz, const double a[], const Integer irow[], const Integer icol[], const double *omega, const double b[], const Integer *m, const double *tol, const Integer *maxitn, double x[], double *rnorm, Integer *itn, double work[], const Integer *lwork, Integer iwork[], Integer *ifail, const Charlen length_method, const Charlen length_precon)

The routine may be called by the names f11def or nagf_sparse_real_gen_solve_jacssor.

3 Description

f11def solves a real sparse nonsymmetric system of linear equations

A x = b,

using an RGMRES (see Saad and Schultz (1986)), CGS (see Sonneveld (1989)), Bi-CGSTAB(

ℓ

) (see Van der Vorst (1989) and Sleijpen and Fokkema (1993)), or TFQMR (see Freund and Nachtigal (1991) and Freund (1993)) method.

The routine 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

L U

(ILU) preconditioning see f11dcf.

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.

f11def 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

Sleijpen G L G and Fokkema D R (1993) BiCGSTAB

(ℓ)

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 Arguments

1: $method$ – Character(*) Input

On entry: the iterative method to be used.

$method ='RGMRES'$: Restarted generalized minimum residual method.
$method ='CGS'$: Conjugate gradient squared method.
$method ='BICGSTAB'$: Bi-conjugate gradient stabilized ( $ℓ$ ) method.
$method ='TFQMR'$: Transpose-free quasi-minimal residual method.

Constraint:

method ='RGMRES'

'CGS'

'BICGSTAB'

'TFQMR'

2: $precon$ – Character(1) Input

On entry: specifies the type of preconditioning to be used.

$precon ='N'$: No preconditioning.
$precon ='J'$: Jacobi.
$precon ='S'$: Symmetric successive-over-relaxation.

Constraint:

precon ='N'

'J'

'S'

3: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 1

4: $nnz$ – Integer Input

On entry: the number of nonzero elements in the matrix

A

Constraint:

1 \leq nnz \leq n^{2}

5: $a (nnz)$ – Real (Kind=nag_wp) array Input

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 f11zaf may be used to order the elements in this way.

6: $irow (nnz)$ – Integer array Input

7: $icol (nnz)$ – Integer array Input

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 f11zaf):

$1 \leq irow (i) \leq n$ and $1 \leq icol (i) \leq n$ , for $i = 1, 2, \dots, nnz$ ;
$irow (i - 1) < irow (i)$ or $irow (i - 1) = irow (i)$ and $icol (i - 1) < icol (i)$ , for $i = 2, 3, \dots, nnz$ .

8: $omega$ – Real (Kind=nag_wp) Input

On entry: if

precon ='S'

, omega is the relaxation parameter

ω

to be used in the SSOR method. Otherwise omega need not be initialized and is not referenced.

Constraint:

0.0 < omega < 2.0

9: $b (n)$ – Real (Kind=nag_wp) array Input

On entry: the right-hand side vector

b

10: $m$ – Integer Input

On entry: if

method ='RGMRES'

, m is the dimension of the restart subspace.

method ='BICGSTAB'

, m is the order

ℓ

of the polynomial Bi-CGSTAB method.

Otherwise, m is not referenced.

Constraints:

if $method ='RGMRES'$ , $0 < m \leq \min (n, 50)$ ;
if $method ='BICGSTAB'$ , $0 < m \leq \min (n, 10)$ .

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

{‖r_{k}‖}_{\infty} \leq τ \times ({‖b‖}_{\infty} + {‖A‖}_{\infty} {‖x_{k}‖}_{\infty}) .

tol \leq 0.0

τ = \max \sqrt{ε}, 10 ε, \sqrt{n} ε

is used, where

ε

is the machine precision. Otherwise

τ = \max (tol, 10 ε, \sqrt{n} ε)

is used.

Constraint:

tol < 1.0

12: $maxitn$ – Integer Input

On entry: the maximum number of iterations allowed.

Constraint:

maxitn \geq 1

13: $x (n)$ – Real (Kind=nag_wp) array Input/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: $itn$ – Integer Output

On exit: the number of iterations carried out.

16: $work (lwork)$ – Real (Kind=nag_wp) array Workspace

17: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f11def is called.

Constraints:

if $method ='RGMRES'$ , $lwork \geq 4 \times n + m \times (m + n + 5) + ν + 101$ ;
if $method ='CGS'$ , $lwork \geq 8 \times n + ν + 100$ ;
if $method ='BICGSTAB'$ , $lwork \geq 2 \times n \times (m + 3) + m \times (m + 2) + ν + 100$ ;
if $method ='TFQMR'$ , $lwork \geq 11 \times n + ν + 100$ .

where

ν = n

for

precon ='J'

'S'

, and

0

otherwise

18: $iwork (2 \times n + 1)$ – Integer array Workspace

19: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

- 1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

- 1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

- 1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, lwork is too small: $lwork = 〈value〉$ . Minimum required value of $lwork = 〈value〉$ .

On entry, $m = 〈value〉$ and $n = 〈value〉$ .
Constraint: $m \geq 1$ and $m \leq MIN (n, 〈value〉)$ .

On entry, $maxitn = 〈value〉$ .
Constraint: $maxitn \geq 1$ .

On entry, $method = 〈value〉$ .
Constraint: $method ='RGMRES'$ , $'CGS'$ or $'BICGSTAB'$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

On entry, $nnz = 〈value〉$ .
Constraint: $nnz \geq 1$ .

On entry, $nnz = 〈value〉$ and $n = 〈value〉$ .
Constraint: $nnz \leq n^{2}$ .

On entry, $omega = 〈value〉$ .
Constraint: $omega > 0.0$ and $omega < 2.0$ .

On entry, $precon \neq'N'$ , $'J'$ or $'S'$ : $precon = 〈value〉$ .

On entry, $tol = 〈value〉$ .
Constraint: $tol < 1.0$ .

$ifail = 2$: On entry, $a (i)$ is out of order: $i = 〈value〉$ .

On entry, $i = 〈value〉$ , $icol (i) = 〈value〉$ and $n = 〈value〉$ .
Constraint: $icol (i) \geq 1$ and $icol (i) \leq n$ .

On entry, $i = 〈value〉$ , $irow (i) = 〈value〉$ and $n = 〈value〉$ .
Constraint: $irow (i) \geq 1$ and $irow (i) \leq n$ .

On entry, the location ( $irow (I), icol (I)$ ) is a duplicate: $I = 〈value〉$ .

$ifail = 3$: The matrix $A$ has a zero diagonal entry in row $〈value〉$ .

The matrix $A$ has no diagonal entry in row $〈value〉$ .
Jacobi and SSOR preconditioners are not appropriate for this problem.

$ifail = 4$: The required accuracy could not be obtained. However, a reasonable accuracy may have been achieved.
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.

$ifail = 5$: The solution has not converged after $〈value〉$ iterations.

$ifail = 6$: Algorithmic breakdown. A solution is returned, although it is possible that it is completely inaccurate.

$ifail = 7$: A serious error has occurred in an internal call: $ifail = 〈value〉$ . Check all subroutine calls and array sizes. Seek expert help.

A serious error has occurred in an internal call: $IREVCM = 〈value〉$ . Check all subroutine calls and array sizes. Seek expert help.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

On successful termination, the final residual

r_{k} = b - A x_{k}

, where

k = itn

, satisfies the termination criterion

{‖r_{k}‖}_{\infty} \leq τ \times ({‖b‖}_{\infty} + {‖A‖}_{\infty} {‖x_{k}‖}_{\infty}) .

The value of the final residual norm is returned in rnorm.

8 Parallelism and Performance

f11def is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f11def 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.

9 Further Comments

The time taken by f11def for each iteration is roughly proportional to nnz.

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

\bar{A} = m^{- 1} A

10 Example

This example solves a sparse nonsymmetric system of equations using the RGMRES method, with SSOR preconditioning.

f11de: FL CL CPP AD

NAG FL Interfacef11def (real_​gen_​solve_​jacssor)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f11def (real_gen_solve_jacssor)