NAG CL Interface
f11drc (complex_​gen_​precon_​ssor_​solve)

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1 Purpose

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>
void  f11drc (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)
The function may be called by the names: f11drc, nag_sparse_complex_gen_precon_ssor_solve or nag_sparse_nherm_precon_ssor_solve.

3 Description

f11drc solves a system of linear equations
Mx=y,   or  MHx=y,  
according to the value of the argument trans, where the matrix
M=1ω(2-ω) (D+ωL) D-1 (D+ωU)  
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 f11drc will be to carry out the preconditioning step required in the application of f11bsc to sparse linear systems. f11drc is also used for this purpose by the Black Box function f11dsc.

4 References

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

5 Arguments

1: trans Nag_TransType Input
On entry: specifies whether or not the matrix M is transposed.
Mx=y is solved.
MHx=y is solved.
Constraint: trans=Nag_NoTrans or Nag_Trans.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n1.
3: nnz Integer Input
On entry: the number of nonzero elements in the matrix A.
Constraint: 1nnzn2.
4: a[nnz] const Complex Input
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 f11znc may be used to order the elements in this way.
5: irow[nnz] const Integer Input
6: icol[nnz] const Integer Input
On entry: the row and column indices of the nonzero elements supplied in a.
irow and icol must satisfy the following constraints (which may be imposed by a call to 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 Complex Input
On entry: the elements of the diagonal matrix D-1, where D is the diagonal part of A.
8: omega double Input
On entry: the relaxation parameter ω.
Constraint: 0.0<omega<2.0.
9: check Nag_SparseNsym_CheckData Input
On entry: specifies whether or not the CS representation of the matrix M should be checked.
Checks are carried on the values of n, nnz, irow, icol and omega.
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 Complex Input
On entry: the right-hand side vector y.
11: x[n] Complex Output
On exit: the solution vector x.
12: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

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 f11znc to reorder and sum or remove duplicates.
The SSOR preconditioner is not appropriate for this problem.
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n1.
On entry, nnz=value.
Constraint: nnz1.
On entry, nnz=value and n=value.
Constraint: nnzn2.
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.
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.
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.
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.
On entry, omega=value.
Constraint: 0.0<omega<2.0
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+δM)x=y, where
c(n) 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

Background information to multithreading can be found in the Multithreading documentation.
f11drc is not threaded in any implementation.

9 Further Comments

9.1 Timing

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

9.2 Use of check

It is expected that a common use of f11drc will be to carry out the preconditioning step required in the application of f11bsc to sparse linear systems. In this situation 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
using RGMRES with SSOR preconditioning.
The RGMRES algorithm itself is implemented by the reverse communication function 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 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)