NAG Library Routine Document

F11DGF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F11DGF 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

SUBROUTINE F11DGF (

METHOD, N, NNZ, A, LA, IROW, ICOL, NB, ISTB, INDB, LINDB, IPIVP, IPIVQ, ISTR, IDIAG, B, M, TOL, MAXITN, X, RNORM, ITN, WORK, LWORK, IFAIL)

INTEGER	N, NNZ, LA, IROW(LA), ICOL(LA), NB, ISTB(NB+1), INDB(LINDB), LINDB, IPIVP(LINDB), IPIVQ(LINDB), ISTR(LINDB+1), IDIAG(LINDB), M, MAXITN, ITN, LWORK, IFAIL
REAL (KIND=nag_wp)	A(LA), B(N), TOL, X(N), RNORM, WORK(LWORK)
CHARACTER(*)	METHOD

3 Description

F11DGF 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.

F11DGF uses the incomplete (possibly overlapping) block

L U

factorization determined by F11DFF as the preconditioning matrix. A call to F11DGF must always be preceded by a call to F11DFF. Alternative preconditioners for the same storage scheme are available by calling F11DCF or F11DEF.

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 F11DFF. The array A holds the nonzero entries in these matrices, while IROW and ICOL hold the corresponding row and column indices.

F11DGF 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

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 Parameters

1: METHOD – CHARACTER(*)Input

On entry: specifies 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: N – INTEGERInput

On entry:

n

, the order of the matrix

A

. This must be the same value as was supplied in the preceding call to F11DFF.

Constraint:

N \geq 1

3: NNZ – INTEGERInput

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 F11DFF.

Constraint:

1 \leq NNZ \leq N^{2}

4: A(LA) – REAL (KIND=nag_wp) arrayInput

On entry: the values returned in the array A by a previous call to F11DFF.

5: LA – INTEGERInput

On entry: the dimension of the arrays A, IROW and ICOL as declared in the (sub)program from which F11DGF is called. This must be the same value as was supplied in the preceding call to F11DFF.

Constraint:

LA \geq 2 \times NNZ

6: IROW(LA) – INTEGER arrayInput

7: ICOL(LA) – INTEGER arrayInput

8: NB – INTEGERInput

9: ISTB( $NB + 1$ ) – INTEGER arrayInput

10: INDB(LINDB) – INTEGER arrayInput

11: LINDB – INTEGERInput

12: IPIVP(LINDB) – INTEGER arrayInput

13: IPIVQ(LINDB) – INTEGER arrayInput

14: ISTR( $LINDB + 1$ ) – INTEGER arrayInput

15: IDIAG(LINDB) – INTEGER arrayInput

On entry: the values returned in arrays IROW, ICOL, IPIVP, IPIVQ, ISTR and IDIAG by a previous call to F11DFF.

The arrays ISTB, INDB and the scalars NB and LINDB must be the same values that were supplied in the preceding call to F11DFF.

16: B(N) – REAL (KIND=nag_wp) arrayInput

On entry: the right-hand side vector

b

17: M – INTEGERInput

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 $METHOD ='BICGSTAB'$ , $0 < M \leq \min (N, 10)$ .

18: 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}, \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 – INTEGERInput

On entry: the maximum number of iterations allowed.

Constraint:

MAXITN \geq 1

20: X(N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: an initial approximation to the solution vector

x

On exit: an improved approximation to the solution vector

x

21: 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.

22: ITN – INTEGEROutput

On exit: the number of iterations carried out.

23: WORK(LWORK) – REAL (KIND=nag_wp) arrayWorkspace

24: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F11DGF is called.

Constraints:

if $METHOD ='RGMRES'$ , $LWORK \geq 6 \times N + M \times (M + N + 5) + 101$ ;
if $METHOD ='CGS'$ , $LWORK \geq 10 \times N + 100$ ;
if $METHOD ='BICGSTAB'$ , $LWORK \geq 2 \times N \times M + 8 \times N + M \times (M + 2) + 100$ ;
if $METHOD ='TFQMR'$ , $LWORK \geq 13 \times N + 100$ .

25: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. 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, for $b = ⟨value⟩$ , $ISTB (b + 1) = ⟨value⟩$ and $ISTB (b) = ⟨value⟩$ .
Constraint: $ISTB (b + 1) > ISTB (b)$ for all $b$ .

On entry, $INDB (⟨value⟩) = ⟨value⟩$ and $N = ⟨value⟩$ .
Constraint: $1 \leq INDB (b) \leq N$ for all $b$ .

On entry, $ISTB (1) = ⟨value⟩$ .
Constraint: $ISTB (1) \geq 1$ .

On entry, $LA = ⟨value⟩$ and $NNZ = ⟨value⟩$ .
Constraint: $LA \geq 2 \times NNZ$ .

On entry, $LINDB = ⟨value⟩$ , $ISTB (NB + 1) - 1 = ⟨value⟩$ and $NB = ⟨value⟩$ .
Constraint: $LINDB \geq ISTB (NB + 1) - 1$ .

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

On entry, $M = ⟨value⟩$ and $N = ⟨value⟩$ .
Constraint: $1 \leq M \leq \min (N, ⟨value⟩)$ .

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, $NB = ⟨value⟩$ and $N = ⟨value⟩$ .
Constraint: $1 \leq NB \leq N$ .

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

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

On entry, $TOL = ⟨value⟩$ .
Constraint: $TOL < 1.0$ .

$IFAIL = 2$: 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 F11DFF and F11DGF.

On entry, $ICOL (⟨value⟩) = ⟨value⟩$ and $N = ⟨value⟩$ .
Constraint: $1 \leq ICOL (i) \leq N$ for all $i$ .
Check that A, IROW, ICOL, IPIVP, IPIVQ, ISTR and IDIAG have not been corrupted between calls to F11DFF and F11DGF.

On entry, $IROW (⟨value⟩) = ⟨value⟩$ and $N = ⟨value⟩$ .
Constraint: $1 \leq IROW (i) \leq N$ for all $i$ .
Check that A, IROW, ICOL, IPIVP, IPIVQ, ISTR and IDIAG have not been corrupted between calls to F11DFF and F11DGF.

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 F11DFF and F11DGF.

$IFAIL = 3$: 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 F11DFF and F11DGF.

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

$IFAIL = 5$: 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.

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 Further Comments

The time taken by F11DGF for each iteration is roughly proportional to the value of NNZC returned from the preceding call to F11DFF.

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 F11DGF 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).

9 Example

This example program reads in a sparse matrix

A

and a vector

b

. It calls F11DFF, 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 F11DGF which uses the Bi-CGSTAB method to solve

A x = b

NAG Library Routine DocumentF11DGF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F11DGF