Finding the Zeros of a Univariate Equation: Proxy Rootfinders, Chebyshev Interpolation, and the Companion Matrix

Abstract

When a function $f(x)$ is holomorphic on an interval $x \in [a, b]$, its roots on the interval can be computed by the following three-step procedure. First, approximate $f(x)$ on $[a, b]$ by a polynomial $f_{N}(x)$ using adaptive Chebyshev interpolation. Second, form the Chebyshev--Frobenius companion matrix whose elements are trivial functions of the Chebyshev coefficients of the interpolant $f_{N}(x)$. Third, compute all the eigenvalues of the companion matrix. The eigenvalues $\lambda$ which lie on the real interval $\lambda \in [a, b]$ are very accurate approximations to the zeros of $f(x)$ on the target interval. (To minimize cost, the adaptive phase can automatically subdivide the interval, applying the Chebyshev rootfinder separately on each subinterval, to keep $N$ bounded or to solve rare “dynamic range” complications.) We also discuss generalizations to compute roots on an infinite interval, zeros of functions singular on the interval $[a, b]$, and slightly complex roots. The underlying ideas are undergraduate-friendly, but link the disparate fields of algebraic geometry, linear algebra, and approximation theory.