Question

How can you find approximations to the zeros of a function?

Answer 1

Use Newton's method to recursively define sequences whose limits are zeros...

Explanation:

If `f(x)` is a continuous, differentiable function then we can usually find its zeros using Newton's method:

Given an approximation `a_i` to a zero of `f(x)`, a better one is given by the formula:

`a_(i+1) = a_i - f(a_i)/(f'(a_i))`

We can use this formula to recursively define a sequence:

`a_0, a_1, a_2,...`

Then the limit of the sequence is a zero of `f(x)`.

By choosing different values for the initial term `a_0`, the resulting sequence will tend to other zeros of `f(x)`.

This method is both easy to apply and generally quite effective with polynomial functions.

It also works with both real and complex zeros.