# 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.

The "final exam" for this course will take place
at 8:00 AM on Tuesday, 12 December. This will be an ordinary
50 minute test. It will be comprehensive, but weighted toward
the latter half of the semester. As always, paper notes will
be permitted, but no electronic devices will be allowed.
A sample exam is available.

A
Solution example is available
for the quiz. The solution to
Test 1 is still available too.

The ultimate assignment is posted.