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Solving Polynomial Equations Using Linear Algebra

Quadric intersection is a common class of nonlinear systems of equations. Quadrics, which are the class of all degree-two polynomials in three or more variables, appear in many engineering problems, such as multilateration. Typically, numerical methods are used to solve such problems. Unfortunately, these methods require on initial guess and, although rates of convergence are well understood, convergence is not necessarily certain. The method discussed in this article transforms the problem of simultaneously solving a system of polynomials into a linear algebra problem that, unlike other root-finding methods, does not require an initial guess. Additionally, iterative methods only give one solution at a time. The method outlined here gives all solutions (including complex ones).