Modified Richardson iteration
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Modified Richardson iteration is an iterative method for solving a system of linear equations. It is similar to the Jacobi and Gauss-Seidel method.
We seek the solution to a set of linear equations, expressed in matrix terms as
The Richardson iteration is
where ω > 0 is a parameter that has to be chosen such that 
, where 
is the spectral radius of A.
It is easy to see that the method is correct, because if it converges, i.e if x(k + 1) = x(k), then x(k) has to be a solution of Ax = b.
[edit] Convergence
Suppose that A is diagonalizable and that (λj,vj) are the eigenvalues and eigenvectors of A. If we express b, x(k) and the correct solution 
in the eigenvector basis
then we can rewrite the iteration as
With 
we can get the following formula for the error of each component
This error converges to 0 if | 1 − ωλj | < 1 for all eigenvalues λj. This can be guaranteed if ω is chosen such that 
.
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