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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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root solving without analytic derivative

I have two related questions about numerical methods for root solving: 1) $f: R \to R$ is continuous and piece-wise smooth, with $f(a)f(b) < 0$. $f$ has very high number of knot-points and computing …
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