f11dcc 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, with incomplete

L U

preconditioning.

2 Specification

#include <nag.h>

void

f11dcc (Nag_SparseNsym_Method method, Integer n, Integer nnz, const double a[], Integer la, const Integer irow[], const Integer icol[], const Integer ipivp[], const Integer ipivq[], const Integer istr[], const Integer idiag[], 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: f11dcc, nag_sparse_real_gen_solve_ilu or nag_sparse_nsym_fac_sol.

3 Description

f11dcc solves a real sparse nonsymmetric linear system of equations:

A x = b,

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

(ℓ)

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

f11dcc uses the incomplete

L U

factorization determined by f11dac as the preconditioning matrix. A call to f11dcc must always be preceded by a call to f11dac. Alternative preconditioners for the same storage scheme are available by calling f11dec.

The matrix

A

, and the preconditioning matrix

M

, are represented in coordinate storage (CS) format (see the F11 Chapter Introduction) in the arrays a, irow and icol, as returned from f11dac. The array a holds the nonzero entries in these matrices, 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

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

(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

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.
$method = Nag_SparseNsym_BiCGSTAB$: Then the bi-conjugate gradient stabilised $(ℓ)$ method is used.

Constraint:

method = Nag_SparseNsym_RGMRES

Nag_SparseNsym_CGS

Nag_SparseNsym_BiCGSTAB

2: $n$ – Integer Input

On entry: the order of the matrix

A

. This must be the same value as was supplied in the preceding call to f11dac.

Constraint:

n \geq 1

3: $nnz$ – Integer Input

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 f11dac.

Constraint:

1 \leq nnz \leq n^{2}

4: $a [la]$ – const double Input

On entry: the values returned in the array a by a previous call to f11dac.

5: $la$ – Integer Input

On entry: the second dimension of the arrays a, irow and icol.This must be the same value as returned by a previous call to f11dac.

Constraint:

la \geq 2 \times nnz

6: $irow [la]$ – const Integer Input

7: $icol [la]$ – const Integer Input

8: $ipivp [n]$ – const Integer Input

9: $ipivq [n]$ – const Integer Input

10: $istr [n + 1]$ – const Integer Input

11: $idiag [n]$ – const Integer Input

On entry: the values returned in the arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to f11dac.

12: $b [n]$ – const double Input

On entry: the right-hand side vector

b

13: $m$ – 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 $method = Nag_SparseNsym_BiCGSTAB$ , $0 < m \leq \min (n, 10)$ .

14: $tol$ – 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

15: $maxitn$ – Integer Input

On entry: the maximum number of iterations allowed.

Constraint:

maxitn \geq 1

16: $x [n]$ – double Input/Output

On entry: an initial approximation to the solution vector

x

On exit: an improved approximation to the solution vector

x

17: $rnorm$ – double * Output

On exit: the final value of the residual norm

{‖ r_{k} ‖}_{\infty}

, where

k

is the output value of itn.

18: $itn$ – Integer * Output

On exit: the number of iterations carried out.

19: $comm$ – 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.

20: $fail$ – 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_2_INT_ARG_LT: On entry, $la = ⟨ value ⟩$ while $nnz = ⟨ value ⟩$ . These arguments must satisfy $la \geq 2 \times nnz$ .
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_ALG_FAIL: Algorithmic breakdown. A solution is returned, although it is possible that it is completely inaccurate.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument method 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_INVALID_CS: On entry, the CS representation of $A$ is invalid. Check that the call to f11dcc has been preceded by a valid call to f11dac, and that the arrays a, irow and icol have not been corrupted between the two calls.
NE_INVALID_CS_PRECOND: On entry, the CS representation of the preconditioning matrix M is invalid. Check that the call to f11dcc has been preceded by a valid call to f11dac, and that the arrays a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between the two calls.
NE_NOT_REQ_ACC: The required accuracy has not been obtained in maxitn iterations.
NE_REAL_ARG_GE: On entry, tol must not be greater than or equal to 1.0: $tol = ⟨ value ⟩$ .

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

f11dcc is not threaded in any implementation.

9 Further Comments

The time taken by f11dcc for each iteration is roughly proportional to the value of nnzc returned from the preceding call to f11dac.

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

Some illustrations of the application of f11dcc 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).

10 Example

This example program solves a sparse linear system of equations using the CGS method, with incomplete

L U

preconditioning.

f11dc: FL CL CPP AD

NAG CL Interfacef11dcc (real_​gen_​solve_​ilu)

▸▿ 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
f11dcc (real_gen_solve_ilu)