# Overrelaxation with w < 0

Are there any circumstances under which using a value $$w < 0$$ would help us find a solution in over-relaxation faster than we can with the ordinary relaxation method?

Over Relaxation Method:

$$x'= [1 + w]f(x) - wx$$

### Example

Calculating $$x = 1-e^{-3x}$$

Take x = 1 as initial value, and w as 0.2

x' = (1+0.2)f(1)-0.2(1) = 0.94025551795

x' = (1+0.2)f(0.94025551795)-0.2(0.94025551795) = 0.94047657354

x' = (1+0.2)f(0.94047657354)-0.2(0.94047657354) = 0.94047974478

x' = (1+0.2)f(0.94047974478)=0.2(0.94047974478) = 0.94047979005


We stop until the value get to a certain accuracy

Why would the over-relaxation reach the solution faster if we consider $$w < 0$$ in non-linear function such as $$x = 1 - e^{(1 - x^2)}$$?

• Cross posted on Physics: physics.stackexchange.com/q/468753/25301 – Kyle Kanos Mar 26 at 14:31
• It seems like you are trying to find the roots of a nonlinear equation using fixed-point iteration. It also seems that you are using the opposite signs for the $w$, compared with the usual convention. I would say that the other sign looks more natural (to me) because it resembles a convex linear combination. – nicoguaro Apr 1 at 15:39