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

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Answer 1 · 2017-11-06T20:29:41

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

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.