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I was just told the following:

The stopping criterion in the bisection method is $|b − a| < \epsilon$. However for most other algorithms it is is $|f(x)| < \epsilon$. These two stopping criteria are within a factor of approximately $|f'(x^*)|$ assuming $f$ is smooth and $\epsilon$ is small.

I'm confused by what "within a factor of approximately $|f'(x^*)|$" means and how to go about showing this.

John Smith
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1 Answers1

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Near 0 we can say that $f(\frac{b+a}{2})\approx df(x)$ for $dx=b-a$. So the bigger $|f'(x*)|$ (where $x*$ is true solution) is, the lesser $dx$ has to be.

lazba
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