NAG CL Interface
f11jrc (complex_​herm_​precon_​ssor_​solve)

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1 Purpose

f11jrc solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a complex sparse Hermitian matrix, represented in symmetric coordinate storage format.

2 Specification

#include <nag.h>
void  f11jrc (Integer n, Integer nnz, const Complex a[], const Integer irow[], const Integer icol[], const double rdiag[], double omega, Nag_SparseSym_CheckData check, const Complex y[], Complex x[], NagError *fail)
The function may be called by the names: f11jrc, nag_sparse_complex_herm_precon_ssor_solve or nag_sparse_herm_precon_ssor_solve.

3 Description

f11jrc solves a system of equations
involving the preconditioning matrix
M=1ω(2-ω) (D+ωL) D-1 (D+ωL)H  
corresponding to symmetric successive-over-relaxation (SSOR) (see Young (1971)) on a linear system Ax=b, where A is a sparse complex Hermitian 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 parameter. Note that since A is Hermitian the matrix D is necessarily real.

4 References

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

5 Arguments

1: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n1.
2: nnz Integer Input
On entry: the number of nonzero elements in the lower triangular part of the matrix A.
Constraint: 1nnzn×(n+1)/2.
3: a[nnz] const Complex Input
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 f11zpc may be used to order the elements in this way.
4: irow[nnz] const Integer Input
5: 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 f11zpc):
  • 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 double Input
On entry: the elements of the diagonal matrix D-1, where D is the diagonal part of A. Note that since A is Hermitian the elements of D-1 are necessarily real.
7: omega double Input
On entry: the relaxation parameter ω.
Constraint: 0.0<omega<2.0.
8: check Nag_SparseSym_CheckData Input
On entry: specifies whether or not the input data should be checked.
Checks are carried out on the values of n, nnz, irow, icol and omega.
None of these checks are carried out.
Constraint: check=Nag_SparseSym_Check or Nag_SparseSym_NoCheck.
9: y[n] const Complex Input
On entry: the right-hand side vector y.
10: x[n] Complex Output
On exit: the solution vector x.
11: 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 lower triangular part of A, is out of order, or has duplicate row and column indices. Consider calling f11zpc to reorder and sum or remove duplicates.
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: nnzn×(n+1)/2.
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, irow[I-1]=value.
Constraint: 1icol[i-1]irow[i-1].
On entry, I=value, irow[I-1]=value and n=value.
Constraint: 1irow[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

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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f11jrc is not threaded in any implementation.

9 Further Comments

9.1 Timing

The time taken for a call to f11jrc is proportional to nnz.

10 Example

This example program solves the preconditioning equation Mx=y for a 9×9 sparse complex Hermitian matrix A, given in symmetric coordinate storage (SCS) format.

10.1 Program Text

Program Text (f11jrce.c)

10.2 Program Data

Program Data (f11jrce.d)

10.3 Program Results

Program Results (f11jrce.r)