Pixel Arrays: A fast and elementary method for solving nonlinear systems

TL;DR

The pixel array method is a fast, elementary approach for approximating all solutions of nonlinear systems within a bounding box, leveraging array multiplications and unexposed variables to outperform traditional Newton-based solvers.

Contribution

This paper introduces the pixel array method, a novel technique that efficiently finds all solutions of nonlinear systems by exploiting unexposed variables and array operations.

Findings

More than 10x faster than Julia's NLsolve with even a single unexposed variable

Requires no false negatives in solution approximation

Applicable to a broader class of systems than Newton-based methods

Abstract

We present a new method, called the pixel array method, for approximating all solutions in a bounding box for an arbitrary nonlinear system of relations. In contrast with other solvers, our approach requires that the user must specify which variables are to be exposed, and which are to be left latent. The entire solution set is then obtained---in terms of these exposed variables---by performing a series of array multiplications on the $n_i$-dimensional plots of the individual relations $R_i$. This procedure introduces no false negatives and is much faster than Newton-based solvers. The key is the unexposed variables, which Newton methods can make no use of. In fact, we found that with even a single unexposed variable our method was more than 10x faster than Julia's NLsolve. Due to its relative simplicity, the pixel array method is also applicable to a broader class of systems than Newton-based solvers are. The purpose of this article is to give an account of this new method.