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Let $f(x)$ be $x^2-x-2$. I want to find the root using FPI in an interval where it will converge. I have chosen $g(x)=x^2-2$ and so $g'(x)=2x$. The convergence condition, $|g'(x)|<1$ is obviously satisfied in $-0.5<x<0.5$.

Problem: I have failed to find a consistent interval outside of this convergence interval up to $\pm 1$ for which the iteration consistently diverges.

Question: Does this convergence condition only guarantee convergence (in the bounds) but not divergence (outside the bounds)?

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  • $\begingroup$ Is that fixed-point iteration fixed? From $x^2=2+x$ one finds the better iteration $x_{n+1}=\sqrt{2+x_n}$ for the positive root. $\endgroup$ – LutzL Apr 16 at 16:25
  • $\begingroup$ Yes, but I thought the reason it’s ‘better’ is because it satisfies abs(g’(x))<1 in some interval. But g(x) in op works just fine up to -+1. $\endgroup$ – AKubilay Apr 16 at 18:10
  • $\begingroup$ $g(x)=\sqrt{2+x}$ has $g'(x)=\frac1{2\sqrt{2+x}}\le\frac25$ for all $x>0$, $g(x)=1+\frac2x$ has $g'(x)=-\frac{2}{x^2}>-1$ for $x>\frac32$, so you get intervals with $|g'(x)|<1$ for many fixed-point functions. $\endgroup$ – LutzL Apr 16 at 18:53
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Let us consider the fixed point iterations associated to the function $g: x \mapsto x^2-2$, defined by the quadratic map $$ x_{n+1} = {x_n}^2 - 2, \qquad x_0 \in \Bbb R . $$ This map has many periodic points, even with large period. The period-one fixed points $-1$, $2$ are both repelling fixed points (indices $2>1$ and $4>1$, respectively). Thus, fixed-point iterations will not converge towards these values unless the starting value $x_0$ is exactly equal to $-1$ or $2$. Setting $y_n = -\frac{1}{4} x_n + \frac{1}{2}$, the logistic map $y_{n+1} = r y_n (1-y_n)$ with parameter $r=4$ is obtained, which exact solution is bounded and exhibits chaotic behavior (see also this article). Therefore, $$ x_n = 2\cos\left(2^n\cos^{-1}(x_0/2)\right) $$ is bounded and exhibits chaotic behavior too (the sequence does not diverge to infinity).

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