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Lets say I have found out the fixed point for a given function.

I'm only given a point that is basically mapped to itself, but how do I find the root of the function and thus any solutions using this fixed point I just found?

Adrian MK
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  • What can you say about the function? e.g. Is it differentiable? – Badoe Jun 19 '14 at 18:09
  • I'm just trying to understand the basic concept. Say we have an extremely simple function such as (6-x^2)/5, and just found out that the fixed point equals 1. How to I find out the root with this information? – Adrian MK Jun 19 '14 at 18:14
  • I don't think there are any general relations between fixed point and roots. They are two very different concepts. Wikipedia page http://en.wikipedia.org/wiki/Root-finding_algorithm gives a nice intro to numerical methods for finding roots. The only relation I can think of between roots and fixed points is that both numerical root-finding algorithms and numerical fixed-point-finding algorithms use iteration. – Badoe Jun 19 '14 at 18:26

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A fixed point of $f$ is $x$ such that $x = f(x)$. A zero is $x$ such that $f(x) = 0$. Plotting such means that a zero is where the graph for $f$ cuts the $X$ axis, a fixed point is where it cuts the line $y = x$, i.e., the line bisecting the angle between the $X$ and $Y$ axes. Do a few sketches, you will see that there isn't any necesary relation between both values. They might be the same ($x = 0$), the fixed point might be before or after the zero, there might be several zeros and fixed points, or none whatsoever.

vonbrand
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  • So basically fixed-point iteration doesn't necessarily help me finding the root of a function? How does it help solving the function then if I know the fixed point? – Adrian MK Jun 19 '14 at 18:27
  • Oh, now I see. To solve $f(x) = 0$ you can rewrite that equation as e.g. $x = x - f(x) / f'(x) = g(x)$ (this is Newton's method). Then $x^*$ is a fixed point of $g$ if it is a zero of $f$. A way of getting there is to go $x_{k + 1} = g(x_k)$ with a suitable $x_0$. – vonbrand Jun 19 '14 at 19:24