NAG C Library Function Document

nag_sparse_nherm_precon_ssor_solve (f11drc)

1
Purpose

nag_sparse_nherm_precon_ssor_solve (f11drc) solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a complex sparse non-Hermitian matrix, represented in coordinate storage format.

2
Specification

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

3
Description

nag_sparse_nherm_precon_ssor_solve (f11drc) solves a system of linear equations
Mx=y,   or  MHx=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) Young (1971) applied to a linear system Ax=b, where A is a complex sparse non-Hermitian 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 parameter.
It is envisaged that a common use of nag_sparse_nherm_precon_ssor_solve (f11drc) will be to carry out the preconditioning step required in the application of nag_sparse_nherm_basic_solver (f11bsc) to sparse linear systems. For an illustration of this use of nag_sparse_nherm_precon_ssor_solve (f11drc) see the example program given in Section 10. nag_sparse_nherm_precon_ssor_solve (f11drc) is also used for this purpose by the Black Box function nag_sparse_nherm_sol (f11dsc).

4
References

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

5
Arguments

1:     trans Nag_TransTypeInput
On entry: specifies whether or not the matrix M is transposed.
trans=Nag_NoTrans
Mx=y is solved.
trans=Nag_Trans
MHx=y is solved.
Constraint: trans=Nag_NoTrans or Nag_Trans.
2:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n1.
3:     nnz IntegerInput
On entry: the number of nonzero elements in the matrix A.
Constraint: 1nnzn2.
4:     a[nnz] const ComplexInput
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_nherm_sort (f11znc) 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 a.
Constraints:
irow and icol must satisfy the following constraints (which may be imposed by a call to nag_sparse_nherm_sort (f11znc)):
  • 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 ComplexInput
On entry: the elements of the diagonal matrix D-1, where D is the diagonal part of A.
8:     omega doubleInput
On entry: the relaxation parameter ω.
Constraint: 0.0<omega<2.0.
9:     check Nag_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 9.2.
Constraint: check=Nag_SparseNsym_Check or Nag_SparseNsym_NoCheck.
10:   y[n] const ComplexInput
On entry: the right-hand side vector y.
11:   x[n] ComplexOutput
On exit: the solution vector x.
12:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

A nonzero element has been supplied which does not lie in the matrix A, is out of order, or has duplicate row and column indices. Consider calling nag_sparse_nherm_sort (f11znc) to reorder and sum or remove duplicates.
The SSOR preconditioner is not appropriate for this problem.
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: n1.
On entry, nnz=value.
Constraint: nnz1.
NE_INT_2
On entry, nnz=value and n=value.
Constraint: nnzn2.
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_CS
On entry, i=value, icol[i-1]=value and n=value.
Constraint: icol[i-1]1 and icol[i-1]n.
On entry, i=value, irow[i-1]=value and n=value.
Constraint: irow[i-1]1 and irow[i-1]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.
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
Parallelism and Performance

nag_sparse_nherm_precon_ssor_solve (f11drc) is not threaded in any implementation.

9
Further Comments

9.1
Timing

The time taken for a call to nag_sparse_nherm_precon_ssor_solve (f11drc) is proportional to nnz.

9.2
Use of check

It is expected that a common use of nag_sparse_nherm_precon_ssor_solve (f11drc) will be to carry out the preconditioning step required in the application of nag_sparse_nherm_basic_solver (f11bsc) to sparse linear systems. In this situation nag_sparse_nherm_precon_ssor_solve (f11drc) 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 check=Nag_SparseNsym_NoCheck for all subsequent calls.

10
Example

This example solves a complex 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_nherm_basic_solver (f11bsc), 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_nherm_basic_solver (f11bsc).

10.1
Program Text

Program Text (f11drce.c)

10.2
Program Data

Program Data (f11drce.d)

10.3
Program Results

Program Results (f11drce.r)