NAG FL Interface
f11jdf (real_​symm_​precon_​ssor_​solve)

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1 Purpose

f11jdf solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a real sparse symmetric matrix, represented in symmetric coordinate storage format.

2 Specification

Fortran Interface
Subroutine f11jdf ( n, nnz, a, irow, icol, rdiag, omega, check, y, x, iwork, ifail)
Integer, Intent (In) :: n, nnz, irow(nnz), icol(nnz)
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: iwork(n+1)
Real (Kind=nag_wp), Intent (In) :: a(nnz), rdiag(n), omega, y(n)
Real (Kind=nag_wp), Intent (Out) :: x(n)
Character (1), Intent (In) :: check
C Header Interface
#include <nag.h>
void  f11jdf_ (const Integer *n, const Integer *nnz, const double a[], const Integer irow[], const Integer icol[], const double rdiag[], const double *omega, const char *check, const double y[], double x[], Integer iwork[], Integer *ifail, const Charlen length_check)
The routine may be called by the names f11jdf or nagf_sparse_real_symm_precon_ssor_solve.

3 Description

f11jdf solves a system of equations
involving the preconditioning matrix
M=1ω(2-ω) (D+ωL) D-1 (D+ωL)T  
corresponding to symmetric successive-over-relaxation (SSOR) (see Young (1971)) on a linear system Ax=b, where A is a sparse symmetric 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.
It is envisaged that a common use of f11jdf will be to carry out the preconditioning step required in the application of f11gef to sparse linear systems. f11jdf is also used for this purpose by the Black Box routine f11jef.

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 A.
Constraint: 1nnzn×(n+1)/2.
3: a(nnz) Real (Kind=nag_wp) array 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 routine f11zbf may be used to order the elements in this way.
4: irow(nnz) Integer array Input
5: icol(nnz) Integer array Input
On entry: the row and column indices of the nonzero elements supplied in array a.
irow and icol must satisfy these constraints (which may be imposed by a call to f11zbf):
  • 1irow(i)n and 1icol(i)irow(i), for i=1,2,,nnz;
  • irow(i-1)<irow(i) or irow(i-1)=irow(i) and icol(i-1)<icol(i), for i=2,3,,nnz.
6: rdiag(n) Real (Kind=nag_wp) array Input
On entry: the elements of the diagonal matrix D-1, where D is the diagonal part of A.
7: omega Real (Kind=nag_wp) Input
On entry: the relaxation parameter ω.
Constraint: 0.0<omega<2.0.
8: check Character(1) 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.
See also Section 9.2.
Constraint: check='C' or 'N'.
9: y(n) Real (Kind=nag_wp) array Input
On entry: the right-hand side vector y.
10: x(n) Real (Kind=nag_wp) array Output
On exit: the solution vector x.
11: iwork(n+1) Integer array Workspace
12: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, check'C' or 'N': check=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
On entry, omega=value.
Constraint: 0.0<omega<2.0
On entry, a(i) is out of order: i=value.
On entry, I=value, icol(I)=value and irow(I)=value.
Constraint: icol(I)1 and icol(I)irow(I).
On entry, i=value, irow(i)=value and n=value.
Constraint: irow(i)1 and irow(i)n.
On entry, the location (irow(I),icol(I)) is a duplicate: I=value.
Consider calling f11zbf to reorder and sum or remove duplicates.
The matrix A has no diagonal entry in row value.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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.
f11jdf is not threaded in any implementation.

9 Further Comments

9.1 Timing

The time taken for a call to f11jdf is proportional to nnz.

9.2 Use of check

It is expected that a common use of f11jdf will be to carry out the preconditioning step required in the application of f11gef to sparse symmetric linear systems. In this situation f11jdf 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='C' for the first of such calls, and to set check='N' for all subsequent calls.

10 Example

This example solves a sparse symmetric linear system of equations
using the conjugate-gradient (CG) method with SSOR preconditioning.
The CG algorithm itself is implemented by the reverse communication routine f11gef, 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 routine document for f11gef.

10.1 Program Text

Program Text (f11jdfe.f90)

10.2 Program Data

Program Data (f11jdfe.d)

10.3 Program Results

Program Results (f11jdfe.r)