Finding all solutions to non-linear system numericaly

[charnley]

Hello world,

Is it possible to find all solutions to a system of non-linear equations, without solving them analytically?

I have implemented a python solution of Newton's Method, is there a method of getting good guesses without scanning all numbers?

 

[HallsofIvy]

What do you mean by "find all solutions"? You can, of course, approximate solutions. Is that what you mean by "finding" them? All numerical methods of solving equations will give one equation- typically, though not necessarily, the one closest to the chosen "starting value". So to find "all" solutions you would have to have some information about their possible values to begin with. And how would you know if you had found all solutions? What about equations that have an infinite number of solutions?

 

[charnley]

By "find all solutions" I mean finding the roots. Is there a way to, numerically, analyse the functions. If I *don't* have any information about their possible values to begin with, find all possible roots.

Hmm, I actually did no consider a system with infinite number of solutions. Hmm.

So there is no way to find all roots, without doing a analytical analysis of the system?

 

[AlephZero]

So there is no way to find all roots, without doing a analytical analysis of the system?

Correct. If you don't do some analysis, you don't even know how many roots there are.

 

[CellCoree]

I can say that it is possible to find all solutions to a system of non-linear equations numerically. There are several methods and algorithms that can be used, such as Newton's Method, gradient descent, and the secant method. These methods involve iteratively refining a guess or estimate of the solutions until a desired level of accuracy is reached.

In terms of getting good initial guesses without scanning all numbers, one approach is to use the initial guesses from previous iterations or solutions. Another approach is to use heuristics or problem-specific knowledge to generate reasonable initial guesses. Additionally, some methods allow for multiple initial guesses to be used simultaneously, which can improve the chances of finding all solutions. It is also important to consider the characteristics of the non-linear system and choose a method that is most suitable for it.