f11dec 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), or stabilized bi-conjugate gradient (Bi-CGSTAB) method, without preconditioning, with Jacobi, or with SSOR preconditioning.

2 Specification

#include <nag.h>

void

f11dec (Nag_SparseNsym_Method method, Nag_SparseNsym_PrecType precon, Integer n, Integer nnz, const double a[], const Integer irow[], const Integer icol[], double omega, const double b[], Integer m, double tol, Integer maxitn, double x[], double *rnorm, Integer *itn, Nag_Sparse_Comm *comm, NagError *fail)

The function may be called by the names: f11dec, nag_sparse_real_gen_solve_jacssor or nag_sparse_nsym_sol.

3 Description

f11dec solves a real sparse nonsymmetric system of linear equations:

A x = b,

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

(ℓ)

method (see Van der Vorst (1989), Sleijpen and Fokkema (1993)).

The function 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 f11dcc.

The matrix

A

is represented in coordinate storage (CS) format (see 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.

4 References

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: $method$ – Nag_SparseNsym_Method Input

On entry: specifies the iterative method to be used.

$method = Nag_SparseNsym_RGMRES$: The restarted generalized minimum residual method is used.
$method = Nag_SparseNsym_CGS$: The conjugate gradient squared method is used.
$method = Nag_SparseNsym_BiCGSTAB$: The bi-conjugate gradient stabilised $(ℓ)$ method is used.

Constraint:

method = Nag_SparseNsym_RGMRES

Nag_SparseNsym_CGS

Nag_SparseNsym_BiCGSTAB

2: $precon$ – Nag_SparseNsym_PrecType Input

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

$precon = Nag_SparseNsym_NoPrec$: No preconditioning.
$precon = Nag_SparseNsym_SSORPrec$: Symmetric successive-over-relaxation.
$precon = Nag_SparseNsym_JacPrec$: Jacobi.

Constraint:

precon = Nag_SparseNsym_NoPrec

Nag_SparseNsym_SSORPrec

Nag_SparseNsym_JacPrec

3: $n$ – Integer Input

On entry: 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: $a [nnz]$ – const double 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 function f11zac may be used to order the elements in this way.

6: $irow [nnz]$ – const Integer Input

7: $icol [nnz]$ – const Integer 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 f11zac):;
$1 \leq irow [i] \leq n$ and $1 \leq icol [i] \leq n$ , for $i = 0, 1, \dots, nnz - 1$ ;
$irow [i - 1] < irow [i]$ or $irow [i - 1] = irow [i]$ and $icol [i - 1] < icol [i]$ , for $i = 1, 2, \dots, nnz - 1$ .

8: $omega$ – double Input

On entry: if

precon = Nag_SparseNsym_SSORPrec

, omega is the relaxation argument

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: $b [n]$ – const double Input

On entry: the right-hand side vector

b

10: $m$ – Integer Input

On entry: if

method = Nag_SparseNsym_RGMRES

, m is the dimension of the restart subspace.

method = Nag_SparseNsym_BiCGSTAB

, m is the order

ℓ

of the polynomial Bi-CGSTAB method; otherwise m is not referenced.

Constraints:

if $method = Nag_SparseNsym_RGMRES$ , $0 < m \leq \min (n, 50)$ ;
if $method = Nag_SparseNsym_BiCGSTAB$ , $0 < m \leq \min (n, 10)$ .

11: $tol$ – double 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:

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

tol \leq 0.0

τ = \max (\sqrt{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: $x [n]$ – double Input/Output

On entry: an initial approximation to the solution vector

x

On exit: an improved approximation to the solution vector

x

14: $rnorm$ – double * 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: $comm$ – Nag_Sparse_Comm * Input/Output

On entry/exit: a pointer to a structure of type Nag_Sparse_Comm whose members are used by the iterative solver.

17: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ACC_LIMIT

The required accuracy could not be obtained. However, a reasonable accuracy has been obtained and further iterations cannot 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.

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument method had an illegal value.

On entry, argument precon had an illegal value.

NE_INT_2

On entry,

m = 〈value〉

\min (n, 10) = 〈value〉

.
Constraint:

0 < m \leq \min (n, 10)

when

method = Nag_SparseNsym_BiCGSTAB

On entry,

m = 〈value〉

\min (n, 50) = 〈value〉

.
Constraint:

0 < m \leq \min (n, 50)

when

method = Nag_SparseNsym_RGMRES

On entry,

nnz = 〈value〉

n = 〈value〉

.
Constraint:

1 \leq nnz \leq n^{2}

NE_INT_ARG_LT

On entry,

maxitn = 〈value〉

.
Constraint:

maxitn \geq 1

On entry,

n = 〈value〉

.
Constraint:

n \geq 1

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

NE_NONSYMM_MATRIX_DUP

A nonzero matrix element has been supplied which does not lie within the matrix

A

, is out of order or has duplicate row and column indices, i.e., one or more of the following constraints has been violated:

$1 \leq irow [i] \leq n$ and $1 \leq icol [i] \leq n$ , for $i = 0, 1, \dots, nnz - 1$ .
$irow [i - 1] < irow [i]$ , or
$irow [i - 1] = irow [i]$ and $icol [i - 1] < icol [i]$ , for $i = 1, 2, \dots, nnz - 1$ .

Call f11zac to reorder and sum or remove duplicates.

NE_NOT_REQ_ACC

The required accuracy has not been obtained in maxitn iterations.

NE_REAL

On entry,

omega = 〈value〉

.
Constraint:

0.0 < omega < 2.0

when

precon = Nag_SparseNsym_SSORPrec

NE_REAL_ARG_GE

On entry, tol must not be greater than or equal to 1:

tol = 〈value〉

NE_ZERO_DIAGONAL_ELEM

On entry, the matrix a has a zero diagonal element. Jacobi and SSOR preconditioners are not appropriate for this problem.

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

f11dec is not threaded in any implementation.

9 Further Comments

The time taken by f11dec 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 matrix of the coefficients

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

10 Example

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

Interfaces: FL CL AD

NAG CL Interface Introduction

F11 (Sparse) Chapter Contents

F11 (Sparse) Chapter Introduction

f11de: FL CL AD

NAG CL Interfacef11dec (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 CL Interface
f11dec (real_gen_solve_jacssor)