f11dcf 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 incomplete

L U

preconditioning.

2 Specification

Fortran Interface

Subroutine f11dcf (

method, n, nnz, a, la, irow, icol, ipivp, ipivq, istr, idiag, b, m, tol, maxitn, x, rnorm, itn, work, lwork, ifail)

Integer, Intent (In)	::	n, nnz, la, irow(la), icol(la), istr(n+1), idiag(n), m, maxitn, lwork
Integer, Intent (Inout)	::	ipivp(n), ipivq(n), ifail
Integer, Intent (Out)	::	itn
Real (Kind=nag_wp), Intent (In)	::	a(la), b(n), tol
Real (Kind=nag_wp), Intent (Inout)	::	x(n)
Real (Kind=nag_wp), Intent (Out)	::	rnorm, work(lwork)
Character (*), Intent (In)	::	method

C Header Interface

#include <nag.h>

void

f11dcf_ (const char *method, const Integer *n, const Integer *nnz, const double a[], const Integer *la, const Integer irow[], const Integer icol[], Integer ipivp[], Integer ipivq[], const Integer istr[], const Integer idiag[], const double b[], const Integer *m, const double *tol, const Integer *maxitn, double x[], double *rnorm, Integer *itn, double work[], const Integer *lwork, Integer *ifail, const Charlen length_method)

The routine may be called by the names f11dcf or nagf_sparse_real_gen_solve_ilu.

3 Description

f11dcf 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.

f11dcf uses the incomplete

L U

factorization determined by f11daf as the preconditioning matrix. A call to f11dcf must always be preceded by a call to f11daf. Alternative preconditioners for the same storage scheme are available by calling 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 f11daf. The array a holds the nonzero entries in these matrices, while irow and icol hold the corresponding row and column indices.

f11dcf 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 Arguments

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: $n$ – 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 f11daf.

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 f11daf.

Constraint:

1 \leq nnz \leq n^{2}

4: $a (la)$ – Real (Kind=nag_wp) array Input

On entry: the values returned in the array a by a previous call to f11daf.

5: $la$ – Integer Input

On entry: the dimension of the arrays a, irow and icol as declared in the (sub)program from which f11dcf is called. This must be the same value as was supplied in the preceding call to f11daf.

Constraint:

la \geq 2 \times nnz

6: $irow (la)$ – Integer array Input

7: $icol (la)$ – Integer array Input

8: $ipivp (n)$ – Integer array Input

9: $ipivq (n)$ – Integer array Input

10: $istr (n + 1)$ – Integer array Input

11: $idiag (n)$ – Integer array Input

On entry: the values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to f11daf.

ipivp and ipivq are restored on exit.

12: $b (n)$ – Real (Kind=nag_wp) array Input

On entry: the right-hand side vector

b

13: $m$ – Integer Input

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 $method ='BICGSTAB'$ , $0 < m \leq \min (n, 10)$ .

14: $tol$ – 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}, 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: $x (n)$ – Real (Kind=nag_wp) array Input/Output

On entry: an initial approximation to the solution vector

x

On exit: an improved approximation to the solution vector

x

17: $rnorm$ – 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.

18: $itn$ – Integer Output

On exit: the number of iterations carried out.

19: $work (lwork)$ – Real (Kind=nag_wp) array Workspace

20: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f11dcf is called.

Constraints:

if $method ='RGMRES'$ , $lwork \geq 4 \times n + m \times (m + n + 5) + 101$ ;
if $method ='CGS'$ , $lwork \geq 8 \times n + 100$ ;
if $method ='BICGSTAB'$ , $lwork \geq 2 \times n \times (m + 3) + m \times (m + 2) + 100$ ;
if $method ='TFQMR'$ , $lwork \geq 11 \times n + 100$ .

21: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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, $la = ⟨ value ⟩$ and $nnz = ⟨ value ⟩$ .
Constraint: $la \geq 2 \times nnz$ .

On entry, lwork is too small: $lwork = ⟨ value ⟩$ . Minimum required value of $lwork = ⟨ value ⟩$ .

On entry, $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $m \geq 1$ and $m \leq \min (n, ⟨ value ⟩)$ .

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, $nnz = ⟨ value ⟩$ .
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, $a (i)$ is out of order: $i = ⟨ value ⟩$ .

On entry, $i = ⟨ value ⟩$ , $icol (i) = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $icol (i) \geq 1$ and $icol (i) \leq n$ .

On entry, $i = ⟨ value ⟩$ , $irow (i) = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $irow (i) \geq 1$ and $irow (i) \leq n$ .

On entry, the location ( $irow (i), icol (i)$ ) is a duplicate: $i = ⟨ value ⟩$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dcf and f11daf.

$ifail = 3$: 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 f11dcf and f11daf.

$ifail = 4$: 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.

$ifail = 7$: A serious error, code $⟨ value ⟩$ , has occurred in an internal call. Check all subroutine calls and array sizes. Seek expert help.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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.

f11dcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f11dcf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by f11dcf for each iteration is roughly proportional to the value of nnzc returned from the preceding call to f11daf.

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 f11dcf 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 solves a sparse linear system of equations using the CGS method, with incomplete

L U

preconditioning.

f11dc: FL CL CPP AD PY MB

NAG FL Interfacef11dcf (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 FL Interface
f11dcf (real_gen_solve_ilu)