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

When a function f(x) is holomorphic on an interval x ∈ [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 fN (x) using adaptive Chebyshev interpolation. Second, form the Chebyshev– Frobenius companion matrix whose elements are trivial functions of the Chebyshev coefficients of the interpolant fN (x). Third, compute all the eigenvalues of the companion matrix. The eigenvalues λ which lie on the real interval λ ∈ [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.