A Deterministic Algorithm for Finding the Entire Roots or Extrema of Finitely Smooth Transcendental Univariates Arising in Mechanical Vibrations With Application to Nonlinear Eigenvalue Problems

Abstract

Abstract. Vibration and control problems often lead to solving intricate transcendental univariates. However, finding the entire solution set to such functions is an open problem. Existing methods employ (i) approximations to convert the problem into a simpler, or generally a proxy, function amenable to the entire solution set, and then (ii) refinements to remove the approximation error. However, the former could be hampered by the degree of the nonlinearity of the problem, and the latter could miss solutions if converged to the same solution for distinct approximate ones. In this article, a deterministic algorithm is proposed for finding the entire solution set to any finitely smooth univariate for a given interval. The algorithm, utilizing no proxy functions, sweeps the solutions to the univariate through the application of the proposed Principle of Next Solution and a localized solver, the restrained Newton’s method. The next solution is the endpoint of the solutions to the cascade of higher order derivatives of the function. This is useful for parametric studies and eigenvalue problems in vibrations and control, where solutions within particular ranges are of interest. The algorithm also solves the global optimization problem for transcendental univariates. It finds the entire extrema of the problem, among which it extracts the global optimizer. The algorithm is naturally parallelizable, allowing the independent processing of subdivisions of the input interval, thus speeding up the computations. Examples from nonlinear eigenvalue problems to a suite of univariate global optimization test problems are presented to demonstrate the superb performance of the proposed algorithm.