f11dgc 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, with block Jacobi or additive Schwarz preconditioning.

2 Specification

#include <nag.h>

void

f11dgc (Nag_SparseNsym_Method method, Integer n, Integer nnz, const double a[], Integer la, const Integer irow[], const Integer icol[], Integer nb, const Integer istb[], const Integer indb[], Integer lindb, 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, NagError *fail)

The function may be called by the names: f11dgc, nag_sparse_real_gen_solve_bdilu or nag_sparse_nsym_precon_bdilu_solve.

3 Description

f11dgc 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)), Bi-CGSTAB(

ℓ

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

f11dgc uses the incomplete (possibly overlapping) block

L U

factorization determined by f11dfc as the preconditioning matrix. A call to f11dgc must always be preceded by a call to f11dfc. Alternative preconditioners for the same storage scheme are available by calling f11dcc or f11dec.

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 f11dfc. The array a holds the nonzero entries in these matrices, while irow and icol hold the corresponding row and column indices.

f11dgc is a Black Box function which calls f11bdc, f11bec and f11bfc. If you wish to use an alternative storage scheme, preconditioner, or termination criterion, or require additional diagnostic information, you should call these underlying functions 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

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.

$method = Nag_SparseNsym_RGMRES$: Restarted generalized minimum residual method.
$method = Nag_SparseNsym_CGS$: Conjugate gradient squared method.
$method = Nag_SparseNsym_BiCGSTAB$: Bi-conjugate gradient stabilized ( $ℓ$ ) method.
$method = Nag_SparseNsym_TFQMR$: Transpose-free quasi-minimal residual method.

Constraint:

method = Nag_SparseNsym_RGMRES

Nag_SparseNsym_CGS

Nag_SparseNsym_BiCGSTAB

Nag_SparseNsym_TFQMR

2: $n$ – Integer Input

3: $nnz$ – Integer Input

4: $a [la]$ – const double Input

5: $la$ – Integer Input

6: $irow [la]$ – const Integer Input

7: $icol [la]$ – const Integer Input

8: $nb$ – Integer Input

9: $istb [nb + 1]$ – const Integer Input

10: $indb [lindb]$ – const Integer Input

11: $lindb$ – Integer Input

12: $ipivp [lindb]$ – const Integer Input

13: $ipivq [lindb]$ – const Integer Input

14: $istr [lindb + 1]$ – const Integer Input

15: $idiag [lindb]$ – const Integer Input

On entry: the values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to f11dfc.

The arrays istb, indb and a together with the scalars n, nnz, la, nb and lindb must be the same values that were supplied in the preceding call to f11dfc.

16: $b [n]$ – const double Input

On entry: the right-hand side vector

b

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

18: $tol$ – 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

{‖ 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

19: $maxitn$ – Integer Input

On entry: the maximum number of iterations allowed.

Constraint:

maxitn \geq 1

20: $x [n]$ – double Input/Output

On entry: an initial approximation to the solution vector

x

On exit: an improved approximation to the solution vector

x

21: $rnorm$ – double * Output

On exit: the final value of the residual norm

{‖ r_{k} ‖}_{\infty}

, where

k

is the output value of itn.

22: $itn$ – Integer * Output

On exit: the number of iterations carried out.

23: $fail$ – 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_ACCURACY: The required accuracy could not be obtained. However a reasonable accuracy may have been achieved. 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.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_CONVERGENCE: The solution has not converged after $⟨ value ⟩$ iterations.
NE_INT: On entry, $istb [0] = ⟨ value ⟩$ .
Constraint: $istb [0] \geq 1$ .

On entry, $maxitn = ⟨ value ⟩$ .
Constraint: $maxitn \geq 1$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

On entry, $nnz = ⟨ value ⟩$ .
Constraint: $nnz \geq 1$ .
NE_INT_2: On entry, $la = ⟨ value ⟩$ and $nnz = ⟨ value ⟩$ .
Constraint: $la \geq 2 \times nnz$ .

On entry, $nb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq nb \leq n$ .

On entry, $nnz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $nnz \leq n^{2}$ .
NE_INT_3: On entry, $lindb = ⟨ value ⟩$ , $istb [nb] - 1 = ⟨ value ⟩$ and $nb = ⟨ value ⟩$ .
Constraint: $lindb \geq istb [nb] - 1$ .

On entry, $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $method = Nag_SparseNsym_RGMRES$ , $1 \leq m \leq \min (n, ⟨ value ⟩)$ .
If $method = Nag_SparseNsym_BiCGSTAB$ , $1 \leq m \leq \min (n, ⟨ value ⟩)$ .
NE_INT_ARRAY: On entry, $indb [⟨ value ⟩] = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq indb [m - 1] \leq n$ , for $m = istb [0], istb [0] + 1, \dots, istb [nb] - 1$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_CS: On entry, $icol [⟨ value ⟩] = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq icol [i - 1] \leq n$ , for $i = 1, 2, \dots, nnz$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dfc and f11dgc.

On entry, $irow [⟨ value ⟩] = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq irow [i - 1] \leq n$ , for $i = 1, 2, \dots, nnz$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dfc and f11dgc.
NE_INVALID_CS_PRECOND: The CS representation of the preconditioner is invalid.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dfc and f11dgc.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_STRICTLY_INCREASING: On entry, element $⟨ value ⟩$ of a was out of order.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dfc and f11dgc.

On entry, for $b = ⟨ value ⟩$ , $istb [b] = ⟨ value ⟩$ and $istb [b - 1] = ⟨ value ⟩$ .
Constraint: $istb [b] > istb [b - 1]$ , for $b = 1, 2, \dots, nb$ .

On entry, location $⟨ value ⟩$ of $(irow, icol)$ was a duplicate.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dfc and f11dgc.
NE_REAL: On entry, $tol = ⟨ value ⟩$ .
Constraint: $tol < 1.0$ .

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

f11dgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f11dgc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by f11dgc for each iteration is roughly proportional to the value of nnzc returned from the preceding call to f11dfc.

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 f11dgc 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 reads in a sparse matrix

A

and a vector

b

. It calls f11dfc, with the array

lfill = 0

and the array

dtol = 0.0

, to compute an overlapping incomplete

L U

factorization. This is then used as an additive Schwarz preconditioner on a call to f11dgc which uses the Bi-CGSTAB method to solve

A x = b

f11dg: FL CL CPP AD

NAG CL Interfacef11dgc (real_​gen_​solve_​bdilu)

▸▿ 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
f11dgc (real_gen_solve_bdilu)