# Matlab Secant Method

Recall that Newton's Method is an iterative way of approximating a zero of a function $$f$$. The idea is that, given a starting guess $$x_0$$ and an error tolerance $$\tau$$, we compute new estimates of the zero of $$f$$ using the formula $x_{n+1} = x_n-f(x_n)/f'(x_n)$ for $$n=0,1,\ldots$$ We use this iteration until $$\vert x_n-x_{n-1}\vert\lt\tau$$ or until we give up trying. Unfortunately, we do not know how to use Matlab to compute the derivative of $$f$$ (it can, we just have not done it). Instead, we can choose some small number $$h$$ and use the approximation $f'(x_n) \approx \frac{f(x_n)-f(x_n-h)}{h}$ instead of $$f'$$. If we use successive estimates of the root, then the formula becomes $x_{n+1}=\frac{x_{n-1}f(x_n)-x_nf(x_{n-1})}{f(x_n)-f(x_{n-1})}.$ This is called the secant rule. Write a Matlab function secant(f,init_guess,tolerance) that finds the zero of a function using this secant formulation.

A solution for the final is available.

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