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If when you are using Newton's method and your results are just going back and forth between two values, say $0$ and $1$. It is $f(x)=x^3 -2x+2$ starting with $x=1$. What is the reasoning behind this?

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  • $\begingroup$ This means simply that the method doesn't work here :-(. $\endgroup$ – Karl Mar 13 '15 at 18:01
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    $\begingroup$ Draw a graph for this scenario. $\endgroup$ – Simon S Mar 13 '15 at 18:02
  • $\begingroup$ It means that the hypotheses for the convergence of Newton's method do not hold for the function you are analyzing. Can you give us more information on this function? In any case, you could try the Secant Method $\endgroup$ – Lonidard Mar 13 '15 at 18:07
  • $\begingroup$ My problem specifically asks to use newton's method and explain what this means geometrically. It is f(x)=x^3 -2x+2 starting with x=1. I will try drawing a graph that will be a good start, thanks! $\endgroup$ – Nicole Mar 13 '15 at 18:09
  • $\begingroup$ Why you don't give the details beforehand, but only now piecemeal wise? YOU are asking for help, we can't guess what your exact problem is if you don't say it :-((. $\endgroup$ – Karl Mar 13 '15 at 18:20
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In terms of the theorem about convergence of Newton's method, this is simply one of the issues that can occur if your initial guess for the root is not good enough. We are only guaranteed convergence for initial guesses which are already sufficiently close to the root.

In terms of what it means about the function, it doesn't mean very much analytically. It can be interpreted geometrically, if you want. The tangent line to $f$ at $0$ is $y=-2x+2$ while the tangent line to $f$ at $1$ is $y=(x-1)+1=x$. These have their roots at $1$ and $0$ respectively. Newton's method finds the root of the tangent line, so starting at $0$ sends you to $1$, then back to $0$, etc. This geometric observation correctly suggests that we may see cycles if we start out relatively close to an extremum of $f$ (since the derivative changes sign between the two values that are being hit).

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  • $\begingroup$ Awesome this was extremely helpful! $\endgroup$ – Nicole Mar 13 '15 at 18:51
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For illustration, see https://en.wikipedia.org/wiki/Newton%27s_method#Starting_point_enters_a_cycle

with the picture

Graph of Newton method

A Newton fractal with discussion of this example can also be found in https://www.math.sunysb.edu/~scott/Newton.ps.gz

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