nag_sparse_nsym_precon_ssor_solve (f11ddc) (PDF version)
f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_sparse_nsym_precon_ssor_solve (f11ddc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_sparse_nsym_precon_ssor_solve (f11ddc) solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a real sparse nonsymmetric matrix, represented in coordinate storage format.

2  Specification

#include <nag.h>
#include <nagf11.h>
void  nag_sparse_nsym_precon_ssor_solve (Nag_TransType trans, Integer n, Integer nnz, const double a[], const Integer irow[], const Integer icol[], const double rdiag[], double omega, Nag_SparseNsym_CheckData check, const double y[], double x[], NagError *fail)

3  Description

nag_sparse_nsym_precon_ssor_solve (f11ddc) solves a system of linear equations
Mx=y, or MTx=y,
according to the value of the argument trans, where the matrix
M=1ω2-ω D+ω L D-1 D+ω U
corresponds to symmetric successive-over-relaxation (SSOR) (see Young (1971)) applied to a linear system Ax=b, where A is a real sparse nonsymmetric matrix stored in coordinate storage (CS) format (see Section 2.1.1 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, U is the strictly upper triangular part of A, and ω is a user-defined relaxation argument.
It is envisaged that a common use of nag_sparse_nsym_precon_ssor_solve (f11ddc) will be to carry out the preconditioning step required in the application of nag_sparse_nsym_basic_solver (f11bec) to sparse linear systems. For an illustration of this use of nag_sparse_nsym_precon_ssor_solve (f11ddc) see the example program given in Section 9. nag_sparse_nsym_precon_ssor_solve (f11ddc) is also used for this purpose by the Black Box function nag_sparse_nsym_sol (f11dec).

4  References

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

5  Arguments

1:     transNag_TransTypeInput
On entry: specifies whether or not the matrix M is transposed.
trans=Nag_NoTrans
Mx=y is solved.
trans=Nag_Trans
MTx=y is solved.
Constraint: trans=Nag_NoTrans or Nag_Trans.
2:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n1.
3:     nnzIntegerInput
On entry: the number of nonzero elements in the matrix A.
Constraint: 1nnzn2.
4:     a[nnz]const doubleInput
On entry: the nonzero elements in 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_nsym_sort (f11zac) may be used to order the elements in this way.
5:     irow[nnz]const IntegerInput
6:     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 the following constraints (which may be imposed by a call to nag_sparse_nsym_sort (f11zac)):
  • 1irow[i]n and 1icol[i]n, for i=0,1,,nnz-1;
  • either irow[i-1]<irow[i] or both irow[i-1]=irow[i] and icol[i-1]<icol[i], for i=1,2,,nnz-1.
7:     rdiag[n]const doubleInput
On entry: the elements of the diagonal matrix D-1, where D is the diagonal part of A.
8:     omegadoubleInput
On entry: the relaxation argument ω.
Constraint: 0.0<omega<2.0.
9:     checkNag_SparseNsym_CheckDataInput
On entry: specifies whether or not the CS representation of the matrix M should be checked.
check=Nag_SparseNsym_Check
Checks are carried on the values of n, nnz, irow, icol and omega.
check=Nag_SparseNsym_NoCheck
None of these checks are carried out.
See also Section 8.2.
Constraint: check=Nag_SparseNsym_Check or Nag_SparseNsym_NoCheck.
10:   y[n]const doubleInput
On entry: the right-hand side vector y.
11:   x[n]doubleOutput
On exit: the solution vector x.
12:   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: 1nnzn2.
NE_INT_2
On entry, nnz=value and n=value.
Constraint: 1nnzn2.
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_CS
On entry, i=value, icol[i-1]=value, n=value.
Constraint: icol[i-1]1 and icol[i-1]n.
On entry, i=value, irow[i-1]=value, 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.
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

If trans=Nag_NoTrans the computed solution x is the exact solution of a perturbed system of equations M+δMx=y, where
δMcnεD+ωLD-1D+ωU,
cn is a modest linear function of n, and ε is the machine precision. An equivalent result holds when trans=Nag_Trans.

8  Further Comments

8.1  Timing

The time taken for a call to nag_sparse_nsym_precon_ssor_solve (f11ddc) is proportional to nnz.

8.2  Use of check

It is expected that a common use of nag_sparse_nsym_precon_ssor_solve (f11ddc) will be to carry out the preconditioning step required in the application of nag_sparse_nsym_basic_solver (f11bec) to sparse linear systems. In this situation nag_sparse_nsym_precon_ssor_solve (f11ddc) 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_SparseNsym_Check for the first of such calls, and for all subsequent calls set check=Nag_SparseNsym_NoCheck.

9  Example

This example solves a sparse linear system of equations:
Ax=b,
using RGMRES with SSOR preconditioning.
The RGMRES algorithm itself is implemented by the reverse communication function nag_sparse_nsym_basic_solver (f11bec), 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_nsym_basic_solver (f11bec).

9.1  Program Text

Program Text (f11ddce.c)

9.2  Program Data

Program Data (f11ddce.d)

9.3  Program Results

Program Results (f11ddce.r)


nag_sparse_nsym_precon_ssor_solve (f11ddc) (PDF version)
f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

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