NAG Library Routine Document

f11drf solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a complex sparse non-Hermitian matrix, represented in coordinate storage format.

2

Specification

Fortran Interface

Subroutine f11drf (

trans, n, nnz, a, irow, icol, rdiag, omega, check, y, x, iwork, ifail)

Integer, Intent (In)	::	n, nnz, irow(nnz), icol(nnz)
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	iwork(2*n+1)
Real (Kind=nag_wp), Intent (In)	::	omega
Complex (Kind=nag_wp), Intent (In)	::	a(nnz), rdiag(n), y(n)
Complex (Kind=nag_wp), Intent (Out)	::	x(n)
Character (1), Intent (In)	::	trans, check

C Header Interface

#include <nagmk26.h>

void

f11drf_ (const char *trans, const Integer *n, const Integer *nnz, const Complex a[], const Integer irow[], const Integer icol[], const Complex rdiag[], const double *omega, const char *check, const Complex y[], Complex x[], Integer iwork[], Integer *ifail, const Charlen length_trans, const Charlen length_check)

3

Description

f11drf solves a system of linear equations

M x = y,   or M^{H} x = y,

according to the value of the argument trans, where the matrix

M = \frac{1}{ω (2 - ω)} (D + ω L) D^{- 1} (D + ω U)

corresponds to symmetric successive-over-relaxation (SSOR) Young (1971) applied to a linear system

A x = b

, where

A

is a complex sparse non-Hermitian matrix stored in coordinate storage (CS) format (see Section 2.1.1 in the F11 Chapter Introduction).

In the definition of

M

given above

D

is the diagonal part of

A

L

is the strictly lower triangular part of

A

U

is the strictly upper triangular part of

A

, and

ω

is a user-defined relaxation parameter.

It is envisaged that a common use of f11drf will be to carry out the preconditioning step required in the application of f11bsf to sparse linear systems. For an illustration of this use of f11drf see the example program given in Section 10. f11drf is also used for this purpose by the Black Box routine f11dsf.

4

References

Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York

5

Arguments

1: $trans$ – Character(1)Input

On entry: specifies whether or not the matrix

M

is transposed.

$trans ='N'$: $M x = y$ is solved.
$trans ='T'$: $M^{H} x = y$ is solved.

Constraint:

trans ='N'

'T'

2: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 1

3: $nnz$ – IntegerInput

On entry: the number of nonzero elements in the matrix

A

Constraint:

1 \leq nnz \leq n^{2}

4: $a (nnz)$ – Complex (Kind=nag_wp) arrayInput

On entry: the nonzero elements in the matrix

A

, ordered by increasing row index, and by increasing column index within each row. Multiple entries for the same row and column indices are not permitted. The routine f11znf may be used to order the elements in this way.

5: $irow (nnz)$ – Integer arrayInput

6: $icol (nnz)$ – Integer arrayInput

On entry: the row and column indices of the nonzero elements supplied in a.

Constraints:

irow and icol must satisfy the following constraints (which may be imposed by a call to f11znf):

$1 \leq irow (i) \leq n$ and $1 \leq icol (i) \leq n$ , for $i = 1, 2, \dots, nnz$ ;
either $irow (i - 1) < irow (i)$ or both $irow (i - 1) = irow (i)$ and $icol (i - 1) < icol (i)$ , for $i = 2, 3, \dots, nnz$ .

7: $rdiag (n)$ – Complex (Kind=nag_wp) arrayInput

On entry: the elements of the diagonal matrix

D^{- 1}

, where

D

is the diagonal part of

A

8: $omega$ – Real (Kind=nag_wp)Input

On entry: the relaxation parameter

ω

Constraint:

0.0 < omega < 2.0

9: $check$ – Character(1)Input

On entry: specifies whether or not the CS representation of the matrix

M

should be checked.

$check ='C'$: Checks are carried on the values of n, nnz, irow, icol and omega.
$check ='N'$: None of these checks are carried out.

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, $check \neq'C'$ or $'N'$ : $check = 〈value〉$ .

On entry, $trans \neq'N'$ or $'T'$ : $trans = 〈value〉$ .

$ifail = 2$: 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, $omega = 〈value〉$ .
Constraint: $0.0 < omega < 2.0$

$ifail = 3$: 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: $I = 〈value〉$ .
A nonzero element has been supplied which does not lie in the matrix $A$ , is out of order, or has duplicate row and column indices. Consider calling f11znf to reorder and sum or remove duplicates.

$ifail = 4$: The matrix $A$ has no diagonal entry in row $〈value〉$ .
The SSOR preconditioner is not appropriate for this problem.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

trans ='N'

the computed solution

x

is the exact solution of a perturbed system of equations

(M + δ M) x = y

, where

|δ M| \leq c (n) ε |D + ω L| |D^{- 1}| |D + ω U|,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision. An equivalent result holds when

trans ='T'

8

Parallelism and Performance

f11drf is not threaded in any implementation.

9

Further Comments

9.1

Timing

The time taken for a call to f11drf is proportional to nnz.

9.2

Use of check

It is expected that a common use of f11drf will be to carry out the preconditioning step required in the application of f11bsf to sparse linear systems. In this situation f11drf is likely to be called many times with the same matrix

M

. In the interests of both reliability and efficiency, you are recommended to set

check ='C'

for the first of such calls, and

check ='N'

for all subsequent calls.

10

Example

This example solves a complex sparse linear system of equations

A x = b,

using RGMRES with SSOR preconditioning.

The RGMRES algorithm itself is implemented by the reverse communication routine f11bsf, which returns repeatedly to the calling program with various values of the argument irevcm. This argument indicates the action to be taken by the calling program.

If $irevcm = 1$ , a matrix-vector product $v = A u$ is required. This is implemented by a call to f11xnf.
If $irevcm = - 1$ , a conjugate transposed matrix-vector product $v = A^{H} u$ is required in the estimation of the norm of $A$ . This is implemented by a call to f11xnf.
If $irevcm = 2$ , a solution of the preconditioning equation $M v = u$ is required. This is achieved by a call to f11drf.
If $irevcm = 4$ , f11bsf has completed its tasks. Either the iteration has terminated, or an error condition has arisen.

For further details see the routine document for f11bsf.

NAG Library Routine Document

f11drf (complex_gen_precon_ssor_solve)

▸▿ 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

9.1 Timing

9.2 Use of check

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

9.1

Timing

9.2

Use of check

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results