Next: 2.3.3 Gauss-Seidel Relaxation (GSR) Up: 2.3 Iterative Methods Previous: 2.3.1 Iterative Improvement

## 2.3.2 Jacobi Relaxation

Divide the given matrix according to where contains only the diagonal elements of , while and are the left and right parts of , respectively.

Choose and write the iteration formula as

or

EXAMPLE: In let

Starting from the estimated solution

and using the diagonal part of ,

in the iteration we find the increasingly more accurate solutions

Convergence rate:

Writing the Jacobi scheme in the form

with the Jacobi block matrix

convergence requires that all eigenvalues of be smaller than one (by absolute value). Denoting the largest eigenvalue (the spectral radius) of by , we have for the asymptotic rate of convergence

In the above example and .

The electrostatic problem shown before was treated by Jacobi iteration:

Next: 2.3.3 Gauss-Seidel Relaxation (GSR) Up: 2.3 Iterative Methods Previous: 2.3.1 Iterative Improvement
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001