NAG CL Interface
f11ddc (real_​gen_​precon_​ssor_​solve)

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1 Purpose

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>
void  f11ddc (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)
The function may be called by the names: f11ddc, nag_sparse_real_gen_precon_ssor_solve or nag_sparse_nsym_precon_ssor_solve.

3 Description

f11ddc solves a system of linear equations
Mx=y, or MTx=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) (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 parameter.
It is envisaged that a common use of f11ddc will be to carry out the preconditioning step required in the application of f11bec to sparse linear systems. f11ddc is also used for this purpose by the Black Box function f11dec.

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.
MTx=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 double 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 f11zac 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 array a.
irow and icol must satisfy the following constraints (which may be imposed by a call to 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 double 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 double Input
On entry: the right-hand side vector y.
11: x[n] double 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

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: 1nnzn2.
On entry, nnz=value and n=value.
Constraint: 1nnzn2.
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.
The SSOR preconditioner is not appropriate for this problem.

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

f11ddc is not threaded in any implementation.

9 Further Comments

9.1 Timing

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

9.2 Use of check

It is expected that a common use of f11ddc will be to carry out the preconditioning step required in the application of f11bec to sparse linear systems. In this situation 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.

10 Example

This example solves a sparse linear system of equations:
using RGMRES with SSOR preconditioning.
The RGMRES algorithm itself is implemented by the reverse communication function 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 f11bec.

10.1 Program Text

Program Text (f11ddce.c)

10.2 Program Data

Program Data (f11ddce.d)

10.3 Program Results

Program Results (f11ddce.r)