NAG Library Routine Document

F11JDF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

+− 8 Further Comments

8.1 Timing

8.2 Use of CHECK

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F11JDF solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a real sparse symmetric matrix, represented in symmetric coordinate storage format.

2 Specification

SUBROUTINE F11JDF (

N, NNZ, A, IROW, ICOL, RDIAG, OMEGA, CHECK, Y, X, IWORK, IFAIL)

INTEGER	N, NNZ, IROW(NNZ), ICOL(NNZ), IWORK(N+1), IFAIL
REAL (KIND=nag_wp)	A(NNZ), RDIAG(N), OMEGA, Y(N), X(N)
CHARACTER(1)	CHECK

3 Description

F11JDF solves a system of equations

M x = y

involving the preconditioning matrix

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

corresponding to symmetric successive-over-relaxation (SSOR) (see Young (1971)) on a linear system

A x = b

, where

A

is a sparse symmetric matrix stored in symmetric coordinate storage (SCS) format (see Section 2.1.2 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

, and

ω

is a user-defined relaxation parameter.

It is envisaged that a common use of F11JDF will be to carry out the preconditioning step required in the application of F11GEF to sparse linear systems. For an illustration of this use of F11JDF see the example program given in Section 9.1. F11JDF is also used for this purpose by the Black Box routine F11JEF.

4 References

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

5 Parameters

1: N – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 1

2: NNZ – INTEGERInput

On entry: the number of nonzero elements in the lower triangular part of

A

Constraint:

1 \leq NNZ \leq N \times (N + 1) / 2

3: A(NNZ) – REAL (KIND=nag_wp) arrayInput

On entry: the nonzero elements in the lower triangular part of 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 F11ZBF may be used to order the elements in this way.

4: IROW(NNZ) – INTEGER arrayInput

5: 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 these constraints (which may be imposed by a call to F11ZBF):

$1 \leq IROW (i) \leq N$ and $1 \leq ICOL (i) \leq IROW (i)$ , for $i = 1, 2, \dots, NNZ$ ;
$IROW (i - 1) < IROW (i)$ or $IROW (i - 1) = IROW (i)$ and $ICOL (i - 1) < ICOL (i)$ , for $i = 2, 3, \dots, NNZ$ .

6: 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

7: OMEGA – REAL (KIND=nag_wp)Input

On entry: the relaxation parameter

ω

Constraint:

0.0 < OMEGA < 2.0

8: CHECK – CHARACTER(1)Input

On entry: specifies whether or not the input data should be checked.

$CHECK ='C'$: Checks are carried out 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'

'N'

$IFAIL = 2$

On entry,	$N < 1$ ,
or	$NNZ < 1$ ,
or	$NNZ > N \times (N + 1) / 2$ ,
or	OMEGA lies outside the interval $(0.0, 2.0)$ ,

$IFAIL = 3$

On entry, the arrays IROW and ICOL fail to satisfy the following constraints:

$1 \leq IROW (i) \leq N$ and $1 \leq ICOL (i) \leq IROW (i)$ , for $i = 1, 2, \dots, NNZ$ ;
$IROW (i - 1) < IROW (i)$ or $IROW (i - 1) = IROW (i)$ and $ICOL (i - 1) < ICOL (i)$ , for $i = 2, 3, \dots, NNZ$ .

Therefore a nonzero element has been supplied which does not lie in the lower triangular part of

A

, is out of order, or has duplicate row and column indices. Call F11ZBF to reorder and sum or remove duplicates.

7 Accuracy

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 L)}^{T}|,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

8 Further Comments

8.1 Timing

The time taken for a call to F11JDF is proportional to NNZ.

8.2 Use of CHECK

It is expected that a common use of F11JDF will be to carry out the preconditioning step required in the application of F11GEF to sparse symmetric linear systems. In this situation F11JDF 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 to set

CHECK ='N'

for all subsequent calls.

9 Example

This example solves a sparse symmetric linear system of equations

A x = b,

using the conjugate-gradient (CG) method with SSOR preconditioning.

The CG algorithm itself is implemented by the reverse communication routine F11GEF, which returns repeatedly to the calling program with various values of the parameter IREVCM. This parameter 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 F11XEF.
If $IREVCM = 2$ , a solution of the preconditioning equation $M v = u$ is required. This is achieved by a call to F11JDF.
If $IREVCM = 4$ , F11GEF has completed its tasks. Either the iteration has terminated, or an error condition has arisen.

For further details see the routine document for F11GEF.

NAG Library Routine DocumentF11JDF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

8.1 Timing

8.2 Use of CHECK

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F11JDF