nag_sparse_sym_precon_ssor_solve (f11jdc) (PDF version)
f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_sparse_sym_precon_ssor_solve (f11jdc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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
Mx=y
involving the preconditioning matrix
M=1ω2-ω D+ω L D-1 D+ω LT
corresponding to symmetric successive-over-relaxation (SSOR) (see Young (1971)) on a linear system Ax=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 argument.
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 9.1. 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:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n1.
2:     nnzIntegerInput
On entry: the number of nonzero elements in the lower triangular part of A.
Constraint: 1nnzn×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)):
  • 1irow[i]n and 1icol[i]irow[i], for i=0,1,,nnz-1;
  • irow[i-1]<irow[i] or irow[i-1]=irow[i] and icol[i-1]<icol[i], for i=1,2,,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:     omegadoubleInput
On entry: the relaxation argument ω.
Constraint: 0.0<omega<2.0.
8:     checkNag_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.
See also Section 8.2.
Constraint: check=Nag_SparseSym_Check or Nag_SparseSym_NoCheck.
9:     y[n]const doubleInput
On entry: the right-hand side vector y.
10:   x[n]doubleOutput
On exit: the solution vector x.
11:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n1.
On entry, nnz=value.
Constraint: nnz1.
NE_INT_2
On entry, nnz=value and n=value.
Constraint: nnzn×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.
NE_INVALID_SCS
On entry, i=value, icol[i-1]=value, irow[i-1]=value.
Constraint: icol[i-1]1 and icol[i-1]irow[i-1].
On entry, i=value, irow[i-1]=value and n=value.
Constraint: irow[i-1]1 and irow[i-1]n.
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+δMx=y, where
δMcnεD+ωLD-1D+ωLT,
cn 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 nag_sparse_sym_precon_ssor_solve (f11jdc) is proportional to nnz.

8.2  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.

9  Example

This example solves a sparse symmetric linear system of equations
Ax=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.
For further details see the function document for nag_sparse_sym_basic_solver (f11gec).

9.1  Program Text

Program Text (f11jdce.c)

9.2  Program Data

Program Data (f11jdce.d)

9.3  Program Results

Program Results (f11jdce.r)


nag_sparse_sym_precon_ssor_solve (f11jdc) (PDF version)
f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012