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

Fortran Interface

Subroutine f11jdf (

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(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)	::	check

C Header Interface

#include <nag.h>

void	f11jdf_ (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_check)

The routine may be called by the names f11jdf or nagf_sparse_real_symm_precon_ssor_solve.

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 10.1. 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 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: $a (nnz)$ – Real (Kind=nag_wp) array 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 routine f11zbf may be used to order the elements in this way.

4: $irow (nnz)$ – Integer array Input

5: $icol (nnz)$ – Integer array 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 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) array Input

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

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

On entry, $nnz = 〈value〉$ .
Constraint: $nnz \geq 1$ .

On entry, $nnz = 〈value〉$ and $n = 〈value〉$ .
Constraint: $nnz \leq n \times (n + 1) / 2$

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 $irow (I) = 〈value〉$ .
Constraint: $icol (I) \geq 1$ and $icol (I) \leq irow (I)$ .

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: $I = 〈value〉$ .
Consider calling f11zbf to reorder and sum or remove duplicates.

$ifail = 4$: The matrix $A$ has no diagonal entry in row $〈value〉$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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

f11jdf is not threaded in any implementation.

9 Further Comments

9.1 Timing

The time taken for a call to f11jdf is proportional to nnz.

9.2 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.

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 routine f11gef, 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 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.

Interfaces: FL CL AD

NAG FL Interface Introduction

F11 (Sparse) Chapter Contents

F11 (Sparse) Chapter Introduction

f11jd: FL CL AD

NAG FL Interfacef11jdf (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 FL Interface
f11jdf (real_symm_precon_ssor_solve)