Which numerical method for solving $f(x) = 0$ if the solution of the nonlinear equation is unique? We only assume that function $f$ is continuous.
2026-04-01 01:35:15.1775007315
which numerical method for solving $f(x) = 0$ if the solution is unique?
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You actually need a bit more than just uniqueness of the solution of $f(x) = 0$ for this to be guaranteed to work: if $f$ takes both positive and negative values, you can search for $x_1$ and $x_2$ such that $f(x_1)$ and $f(x_2)$ have opposite signs, then use bisection.
On the other hand, if all you know is that $f$ is continuous and the solution to $f(x) = 0$ is unique, then it's hopeless: there is no way for your numerical method to detect whether the solution is in a given interval.