Modified Richardson iteration

Modified Richardson iteration is an iterative method for solving a system of linear equations. Richardson iteration was proposed by Lewis Fry Richardson in his work dated 1910. 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

${\displaystyle Ax=b.\,}$

The Richardson iteration is

${\displaystyle x^{(k+1)}=x^{(k)}+\omega \left(b-Ax^{(k)}\right),}$

where ${\displaystyle \omega }$ is a scalar parameter that has to be chosen such that the sequence ${\displaystyle x^{(k)}}$ converges.

It is easy to see that the method has the correct fixed points, because if it converges, then ${\displaystyle x^{(k+1)}\approx x^{(k)}}$ and ${\displaystyle x^{(k)}}$ has to approximate a solution of ${\displaystyle Ax=b}$.