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Algorithms to approximate numerically a root of a nonlinear equation or system: for instance, Newton's method, secant method, bisection, etc.
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Rigorous estimates on roots of function
Let
$$a_i=\sin ^2\left(\frac{\pi i}{N}\right)\qquad \text{and} \qquad b_i=1+\sin ^2\left(\frac{\pi i}{2 N}\right)$$ and consider
$$f(x)=1-\frac{1}{N} \sum_{i=1}^N \frac{a_i}{b_i-x}$$ For the root be …