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October 8th, 2015, 07:45 AM  #1 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,150 Thanks: 730 Math Focus: Physics, mathematical modelling, numerical and computational solutions  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 $\displaystyle x_{n+1} = w f(x_n) + (1w) x_n$ 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 underrelaxation) 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, singlevalued 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 underrelaxation factor to compensate? 

Tags 
diagnostics, divergence, function, test, unknown 
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