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>

void	f11jdc (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)

The function may be called by the names: f11jdc, nag_sparse_real_symm_precon_ssor_solve or nag_sparse_sym_precon_ssor_solve.

3 Description

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

4 References

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

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 1

2: $nnz$ – Integer Input

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 double Input

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 f11zbc may be used to order the elements in this way.

4: $irow [nnz]$ – const Integer Input

5: $icol [nnz]$ – const Integer Input

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 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 double Input

On entry: the elements of the diagonal matrix

D^{- 1}

, where

D

is the diagonal part of

A

7: $omega$ – double Input

On entry: the relaxation parameter

ω

Constraint:

0.0 < omega < 2.0

8: $check$ – Nag_SparseSym_CheckData Input

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

Consider calling f11zbc to reorder and sum or remove duplicates.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface 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 8 in the Introduction to the NAG Library CL Interface 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: $I = 〈value〉$ .
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

f11jdc is not threaded in any implementation.

9 Further Comments

9.1 Timing

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

9.2 Use of check

It is expected that a common use of f11jdc will be to carry out the preconditioning step required in the application of f11gec to sparse symmetric linear systems. In this situation 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 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 f11xec.
If $irevcm = 2$ , a solution of the preconditioning equation $M v = u$ is required. This is achieved by a call to f11jdc.
If $irevcm = 4$ , f11gec has completed its tasks. Either the iteration has terminated, or an error condition has arisen.

For further details see the function document for f11gec.

Interfaces: FL CL AD

NAG CL Interface Introduction

F11 (Sparse) Chapter Contents

F11 (Sparse) Chapter Introduction

f11jd: FL CL AD

NAG CL Interfacef11jdc (real_​symm_​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

NAG CL Interface
f11jdc (real_symm_precon_ssor_solve)