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We know that for a single equation root-finding, we can use the Newton's method, or a combination of Newton with bisection to guarantee convergence. Can we use Newton+bisection for a system of nonlinear equations? I can't figure out how bisection could work on multi-dimensional equations.

For example, if Newton iteration causes one equation to have the $x$-value out of the bisection range, then in the next iteration, does this one equation performs bisection while other equations perform Newton iteration?

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  • $\begingroup$ Bisection is based on the intermediate-value theorem, which has no analogue in higher dimensions. There is in general no way to know for certain that a given volume contains a zero. $\endgroup$
    – gmvh
    Commented Jun 22, 2022 at 12:09
  • $\begingroup$ @gmvh For a high dimension system, what other relatively simple algorithm can accompany Newton's method to guarantee the root-finding? I want to implement it in C++. $\endgroup$
    – Jen
    Commented Jun 22, 2022 at 15:00
  • $\begingroup$ @DavidKetcheson: assuming the function whose zero is thought is scalar. But here, the OP is asking for a system of equations, so he's looking for a zero of a function $\mathbb{R}^n\to\mathbb{R}^m$. I know of no equivalent of bisection in that context. $\endgroup$
    – gmvh
    Commented Jun 23, 2022 at 12:45

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