# Counterexamples to Newton's method and to fixed point method

I am teaching calculus 1, and soon I will have to discuss some numerical methods. Among those, there are Newton's method and the fixed point method. While I feel there are easy ways to convey that "close enough depends on the function" (e.g. one fist considers $f(x)$, then $10f(10^{-1}x)$), I do not have good examples of functions with pathologies on the top of my head (e.g. functions for which the method loops around, or for which the sequence diverges, etc.).

Since I am not an artist either (my graphs are not great, and it would be very hard to sketch something on the spot), I would like to print out the graph (and maybe some iterations using those functions) of some functions that best illustrate the caveats around these methods.

Could you provide me with some examples?

• – Moo Oct 24 '16 at 16:23
• The newton-method can already fail for cubic functions. There are even examples for which the values oscillate between two real numbers. You should find such an example in Moo's link. – Peter Oct 24 '16 at 16:44
• The fixpoint-method is even weaker than newton's method. – Peter Oct 24 '16 at 16:47
• For Newton: $x^{-2}-5$ starting at $x=1$ or $x^3-5x$ starting at $x=1$. – WimC Oct 24 '16 at 16:48
• But to be fair when treating polynomials: please mention the good and excellent methods of Weierstrass-Durand-Kerner and Aberth-Ehrlich. Both are only a slight variation on Newton's method. – WimC Oct 24 '16 at 17:01

## 2 Answers

This is a (mostly english with only a tiny bit of german) reference for for a lecture about counter examples by Prof. Kummer at Humboldt University Berlin. He introduced the "Blitz-Funktion" (bolt/flash/lightning-function, I dont know how he called it in english) which is a uniformly strictly monotonic Lipschitz-continous function (quite nice properties!) on which newtons method still fails everywhere (except 0) such that it cycles/oscillates. For details see page 37 in the script (there is also a low quality picture :-))

In the low quality picture the blue lines target to the opposite side of zero such that newtons method oscillates, while the black lines target to any blue line away from 0 such that oscillation must start.

Also, the rest of the pdf might be useful for you, depending on what exactly you are planning to teach.

A good example where Newton's method does not work is $$\sqrt{x}$$ Specifically, attempting to find the derivative at $x=0$ You'll find that the values get further and further away from 0.