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October 8th, 2015, 07:45 AM   #1
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Diagnostics to test divergence of an unknown function

I have a problem associated with convergence in a computer program. Consider a numerical solution based on iteration with the form

$\displaystyle x_{n+1} = w f(x_n) + (1-w) x_n$

where $\displaystyle w$ is known as the relaxation factor. Basically, you mix a portion of the original answer together with the new result using a weighting factor. By setting $\displaystyle 0 \lt w \le 1$ (called under-relaxation) you can improve the stability of the solution at the cost of convergence rate.

The following properties are true:

1. $\displaystyle f(x_n)$ is unknown, as are its derivatives. However, we can be sure that $\displaystyle f(x_n)$ does not depend on $\displaystyle n$ and is "sensible" (no discontinuities, single-valued over the reals, etc.)
2. A solution indeed exists such that after some threshold number of iterations, $\displaystyle n > N$, and convergent iteration,$\displaystyle \frac{x_{N+1} - x_N}{x_N} < \delta$ where $\displaystyle \delta$ is some sufficiently small number (such as 0.01) is true.

Does anyone know of a decent algorithm for detecting divergence and, if so, how to set the under-relaxation factor to compensate?
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