nag_sparse_sym_precon_ssor_solve (f11jdc) 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

#include <nag.h>

#include <nagf11.h>

void	nag_sparse_sym_precon_ssor_solve (Integer n, Integer nnz, const double a[], const Integer irow[], const Integer icol[], const double rdiag[], double omega, Nag_SparseSym_CheckData check, const double y[], double x[], NagError *fail)

3

Description

nag_sparse_sym_precon_ssor_solve (f11jdc) 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 nag_sparse_sym_precon_ssor_solve (f11jdc) will be to carry out the preconditioning step required in the application of nag_sparse_sym_basic_solver (f11gec) to sparse linear systems. For an illustration of this use of nag_sparse_sym_precon_ssor_solve (f11jdc) see the example program given in Section 10.1. nag_sparse_sym_precon_ssor_solve (f11jdc) is also used for this purpose by the Black Box function nag_sparse_sym_sol (f11jec).

4

References

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

5

Arguments

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: $a [nnz]$ – const doubleInput

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 function nag_sparse_sym_sort (f11zbc) may be used to order the elements in this way.

4: $irow [nnz]$ – const IntegerInput

5: $icol [nnz]$ – const IntegerInput

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 nag_sparse_sym_sort (f11zbc)):

$1 \leq irow [i] \leq n$ and $1 \leq icol [i] \leq irow [i]$ , for $i = 0, 1, \dots, nnz - 1$ ;
$irow [i - 1] < irow [i]$ or $irow [i - 1] = irow [i]$ and $icol [i - 1] < icol [i]$ , for $i = 1, 2, \dots, nnz - 1$ .

6: $rdiag [n]$ – const doubleInput

On entry: the elements of the diagonal matrix

D^{- 1}

, where

D

is the diagonal part of

A

7: $omega$ – doubleInput

On entry: the relaxation parameter

ω

Constraint:

0.0 < omega < 2.0

8: $check$ – Nag_SparseSym_CheckDataInput

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

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

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

On entry, $nnz = 〈value〉$ .
Constraint: $nnz \geq 1$ .
NE_INT_2: On entry, $nnz = 〈value〉$ and $n = 〈value〉$ .
Constraint: $nnz \leq n \times (n + 1) / 2$
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_SCS: On entry, $I = 〈value〉$ , $icol [I - 1] = 〈value〉$ and $irow [I - 1] = 〈value〉$ .
Constraint: $icol [I - 1] \geq 1$ and $icol [I - 1] \leq irow [I - 1]$ .

On entry, $i = 〈value〉$ , $irow [i - 1] = 〈value〉$ and $n = 〈value〉$ .
Constraint: $irow [i - 1] \geq 1$ and $irow [i - 1] \leq n$ .
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NOT_STRICTLY_INCREASING: On entry, $a [i - 1]$ is out of order: $i = 〈value〉$ .

On entry, the location ( $irow [I - 1], icol [I - 1]$ ) is a duplicate: $I = 〈value〉$ . Consider calling nag_sparse_sym_sort (f11zbc) to reorder and sum or remove duplicates.
NE_REAL: On entry, $omega = 〈value〉$ .
Constraint: $0.0 < omega < 2.0$
NE_ZERO_DIAG_ELEM: The matrix $A$ has no diagonal entry in row $〈value〉$ .

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

Parallelism and Performance

nag_sparse_sym_precon_ssor_solve (f11jdc) is not threaded in any implementation.

9

Further Comments

9.1

Timing

The time taken for a call to nag_sparse_sym_precon_ssor_solve (f11jdc) is proportional to nnz.

9.2

Use of check

It is expected that a common use of nag_sparse_sym_precon_ssor_solve (f11jdc) will be to carry out the preconditioning step required in the application of nag_sparse_sym_basic_solver (f11gec) to sparse symmetric linear systems. In this situation nag_sparse_sym_precon_ssor_solve (f11jdc) 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 = Nag_SparseSym_Check

for the first of such calls, and to set

check = Nag_SparseSym_NoCheck

for all subsequent calls.

10

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 function nag_sparse_sym_basic_solver (f11gec), 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 nag_sparse_sym_matvec (f11xec).
If $irevcm = 2$ , a solution of the preconditioning equation $M v = u$ is required. This is achieved by a call to nag_sparse_sym_precon_ssor_solve (f11jdc).
If $irevcm = 4$ , nag_sparse_sym_basic_solver (f11gec) has completed its tasks. Either the iteration has terminated, or an error condition has arisen.

For further details see the function document for nag_sparse_sym_basic_solver (f11gec).

NAG Library Function Document

nag_sparse_sym_precon_ssor_solve (f11jdc)

▸▿ 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