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 October 8th, 2015, 07:45 AM #1 Senior Member   Joined: Apr 2014 From: Glasgow Posts: 2,157 Thanks: 732 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 $\displaystyle x_{n+1} = w f(x_n) + (1-w) x_n$ 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? Tags diagnostics, divergence, function, test, unknown Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post king.oslo Complex Analysis 6 May 22nd, 2015 02:00 PM mathdisciple Calculus 6 March 23rd, 2014 12:24 AM hatsjoe Linear Algebra 0 March 17th, 2014 06:28 AM shiseonji Calculus 1 February 21st, 2014 04:53 AM aaron-math Calculus 8 October 12th, 2011 10:03 PM

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