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 The Richardson iteration is where
is a scalar parameter that has to be chosen such that the sequence
It is easy to see that the method has the correct fixed points, because if it converges, then
has to approximate a solution of
Subtracting the exact solution
, and introducing the notation for the error
, we get the equality for the errors Thus, for any vector norm and the corresponding induced matrix norm.
, the method converges.
is symmetric positive definite and that
The error converges to
If, e.g., all eigenvalues are positive, this can be guaranteed if
ω
0 < ω <
ω
The optimal choice, minimizing all
{\displaystyle \omega _{\text{opt}}:=2/(\lambda _{\text{min}}(A)+\lambda _{\text{max}}(A))}
, which gives the simplest Chebyshev iteration.
This optimal choice yields a spectral radius of where
is the condition number.
If there are both positive and negative eigenvalues, the method will diverge for any
if the initial error
has nonzero components in the corresponding eigenvectors.
Consider minimizing the function
Since this is a convex function, a sufficient condition for optimality is that the gradient is zero (
) which gives rise to the equation Define
Because of the form of A, it is a positive semi-definite matrix, so it has no negative eigenvalues.
A step of gradient descent is which is equivalent to the Richardson iteration by making