Root isolation method for analytic functions using cubic interpolation
Root isolation for analytic functions using cubic hermite interpolation
Symbolic Computation
Summary
Finding the exact points where complex math functions equal zero can be tricky. This paper shows a new way to do that by using a type of curve fitting called cubic Hermite interpolation. The method guesses roots by checking if the function closely matches these curves at some points. It doesn’t guarantee perfect accuracy but works very well in practice. The authors hope this idea will lead to even more precise methods later.
What this means in practice
- •For numerical software developers: Improve root-finding modules by integrating heuristic cubic interpolation for faster approximate root isolation in analytic functions.
- •For applied mathematicians: Use the method to quickly locate roots in analytic functions during modeling and simulation workflows.
Authors
Christophe Raffalli
Abstract
We present a root isolation algorithm for an analytic function. The method combines interval subdivision with a heuristic test based on local cubic Hermite interpolation. Whenever the function is sufficiently close to its cubic Hermite interpolant at a few sample points, we assume that the critical points of the cubic interpolant give a valid root isolation for the original function. Such an algorithm cannot, in general, provide a rigorous certification of the computed intervals. Nevertheless, extensive numerical experiments show that it is highly efficient and almost always produces the correct results. We hope that a probabilistic or exact algorithm can be derived from this work.