NAG Library Routine Document

f11ddf solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a real sparse nonsymmetric matrix, represented in coordinate storage format.

2

Specification

Fortran Interface

Subroutine f11ddf (

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)	::	a(nnz), rdiag(n), omega, y(n)
Real (Kind=nag_wp), Intent (Out)	::	x(n)
Character (1), Intent (In)	::	trans, check

C Header Interface

#include <nagmk26.h>

void

f11ddf_ (const char *trans, const Integer *n, const Integer *nnz, const double a[], const Integer irow[], const Integer icol[], const double rdiag[], const double *omega, const char *check, const double y[], double x[], Integer iwork[], Integer *ifail, const Charlen length_trans, const Charlen length_check)

3

Description

f11ddf solves a system of linear equations

\begin{matrix} M x = y, & or & M^{T} x = y, \end{matrix}

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) (see Young (1971)) applied to a linear system

A x = b

, where

A

is a real sparse nonsymmetric 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 f11ddf will be to carry out the preconditioning step required in the application of f11bef to sparse linear systems. For an illustration of this use of f11ddf see the example program given in Section 10. f11ddf is also used for this purpose by the Black Box routine f11def.

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.
$trans ='T'$: $M^{T} x = y$ 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: $a (nnz)$ – Real (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 f11zaf 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 array a.

Constraints:

irow and icol must satisfy the following constraints (which may be imposed by a call to f11zaf):

$1 \leq irow (i) \leq n$ 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: $rdiag (n)$ – Real (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, $check = 〈value〉$ .
Constraint: $check ='C'$ or $'N'$ .

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

$ifail = 2$: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

On entry, $nnz = 〈value〉$ .
Constraint: $1 \leq nnz \leq n^{2}$ .

On entry, $nnz = 〈value〉$ and $n = 〈value〉$ .
Constraint: $1 \leq nnz \leq n^{2}$ .

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: $I = 〈value〉$ .

$ifail = 4$: 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

f11ddf is not threaded in any implementation.

9

Further Comments

9.1

Timing

The time taken for a call to f11ddf is proportional to nnz.

9.2

Use of check

It is expected that a common use of f11ddf will be to carry out the preconditioning step required in the application of f11bef to sparse linear systems. In this situation f11ddf 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 for all subsequent calls set

check ='N'

10

Example

This example solves a sparse linear system of equations:

A x = b,

using RGMRES with SSOR preconditioning.

The RGMRES algorithm itself is implemented by the reverse communication routine f11bef, 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 f11xaf.
If $irevcm = - 1$ , a transposed matrix-vector product $v = A^{T} u$ is required in the estimation of the norm of $A$ . This is implemented by a call to f11xaf.
If $irevcm = 2$ , a solution of the preconditioning equation $M v = u$ is required. This is achieved by a call to f11ddf.
If $irevcm = 4$ , f11bef has completed its tasks. Either the iteration has terminated, or an error condition has arisen.

For further details see the routine document for f11bef.

NAG Library Routine Document

f11ddf (real_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