Question

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

Answer 1

Answer:

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

Explanation:

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

Given an approximation a_iai to a zero of f(x)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.