f11dqc 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, with incomplete

L U

preconditioning.

2 Specification

#include <nag.h>

void

f11dqc (Nag_SparseNsym_Method method, Integer n, Integer nnz, const Complex a[], Integer la, const Integer irow[], const Integer icol[], const Integer ipivp[], const Integer ipivq[], const Integer istr[], const Integer idiag[], const Complex b[], Integer m, double tol, Integer maxitn, Complex x[], double *rnorm, Integer *itn, NagError *fail)

The function may be called by the names: f11dqc, nag_sparse_complex_gen_solve_ilu or nag_sparse_nherm_fac_sol.

3 Description

f11dqc solves a complex sparse non-Hermitian 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.

f11dqc uses the incomplete

L U

factorization determined by f11dnc as the preconditioning matrix. A call to f11dqc must always be preceded by a call to f11dnc. Alternative preconditioners for the same storage scheme are available by calling f11dsc.

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 f11dnc. The array a holds the nonzero entries in these matrices, while irow and icol hold the corresponding row and column indices.

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

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

On entry:

n

, the order of the matrix

A

. This must be the same value as was supplied in the preceding call to f11dnc.

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 f11dnc.

Constraint:

1 \leq nnz \leq n^{2}

4: $a [la]$ – const Complex Input

On entry: the values returned in the array a by a previous call to f11dnc.

5: $la$ – Integer Input

On entry: the dimension of the arrays a, irow and icol. This must be the same value as was supplied in the preceding call to f11dnc.

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 arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to f11dnc.

ipivp and ipivq are restored on exit.

12: $b [n]$ – const Complex 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

{‖ 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

15: $maxitn$ – Integer Input

On entry: the maximum number of iterations allowed.

Constraint:

maxitn \geq 1

16: $x [n]$ – Complex 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: $fail$ – NagError * Input/Output

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

6 Error Indicators and Warnings

Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dqc and f11dnc.; Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dnc and f11dqc.
NE_ACCURACY: The required accuracy could not be obtained. However, a reasonable accuracy may have been achieved.
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, $maxitn = ⟨ value ⟩$ .
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, $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $m \geq 1$ and $m \leq \min (n, ⟨ value ⟩)$ .

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

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, $i = ⟨ value ⟩$ , $icol [i - 1] = ⟨ value ⟩$ , and $n = ⟨ value ⟩$ .
Constraint: $icol [i - 1] \geq 1$ and $icol [i - 1] \leq n$ .

On entry, $i = ⟨ value ⟩$ , $irow [i - 1] = ⟨ value ⟩$ , $n = ⟨ value ⟩$ .
Constraint: $irow [i - 1] \geq 1$ and $irow [i - 1] \leq n$ .
NE_INVALID_CS_PRECOND: The CS representation of the preconditioner is invalid.
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, $a [i - 1]$ is out of order: $i = ⟨ value ⟩$ .

On entry, the location ( $irow [i - 1], icol [i - 1]$ ) is a duplicate: $i = ⟨ value ⟩$ .
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

Background information to multithreading can be found in the Multithreading documentation.

f11dqc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f11dqc 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 f11dqc for each iteration is roughly proportional to the value of nnzc returned from the preceding call to f11dnc.

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 complex sparse non-Hermitian linear system of equations using the CGS method, with incomplete

L U

preconditioning.

f11dq: FL CL CPP AD PY MB

NAG CL Interfacef11dqc (complex_​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
f11dqc (complex_gen_solve_ilu)