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For a numerical methods class I have to implement the secant method to find the roots of a 2nd order polynomial and create a cobweb diagram.

I'm a bit confused on the cobweb diagram. I read here that a cobweb diagram is used to "Investigate the qualitative behaviour of one-dimensional iterated functions"

but the secant method seems to be a 2D function depending on the starting points $x_{n-1}$ and $x_{n-2}$ $$x_n = x_{n-1} - f(x_{n-1})\frac {x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}$$

So how do you create a cobweb diagram for a 2D iterative function or is this even possible? or can you somehow reduce the Secant method iterative function into 1D function?

I feel like this is somewhat on the right track
First plot a surface of $y=F(x_{n-1},x_{n-2})$, where F is the secant method, and then given the 2 starting points find the intersection with the plane x = y(?) and then match this point on the surface and continue

However this seems way too complicated and I don't know whether this can still be considered a cobweb

void life
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1 Answers1

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You are completely right, the plot would have to be in 3D space to realize the memory effect of using the last two samples. Convergence would then be to the full space diagonal. The corresponding surface is $x_{n+2}=F(x_n,x_{n+1})$ with $$F(x,y)=\frac{xf(y)-yf(x)}{f(y)-f(x)}.$$

The cobweb plot in 2D could be adapted to the Newton method.

Lutz Lehmann
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