Homotopy Methods For Mixed Trigonometric Polynomial Systems

Solving nonlinear systems is a major task of computational mathematics. Finding all solutions to a nonlinear system is a challenging problem and has practical applications in many fields of science and engineering. Homotopy method is an efficient numerical method for finding all isolated solutions to some special kinds of nonlinear systems, e.g., polynomial systems. In this dissertation, we consider to solve mixed trigonometric polynomial systems and polynomial systems transformed from them more efficiently. We present two kinds of methods for such problems:1. Direct homotoy methods to solve mixed trigonometric polynomial systems;2. For the polynomial systems transformed from the mixed trigonometric polynomial systems, we utilize its special structure to construct more efficient homotopies.In Chapter 1, we give an introduction of the homotopy method and its applications in the field of science and engineering, especially homotopy methods for solving polynomial systems. Also we formulate the general form of mixed trigonometric polynomial systemsas as well as transformations between a mixed trigonometric polynomial system and a polynomial system. Some practical examples are also listed.In Chapter 2, we present some direct homotopy methods for mixed trigonometric polynomial systems, that is, we construct homotopy directly for mixed trigonometric polynomial systems and do not transform them into polynomial systems. By doing like this, no additional variables is introduced and hence can solve the problem more efficiently. For general mixed trigonometric polynomial systems, we present standard homotopies. Furthermore, because mixed trigonometric polynomial systems arising in practice are mainly deficient, we present two efficient random linear product homotopies: multi-homogeneous homotopy and product homotopy based on the generalized Bézout number to solve this class of systems, and in the latter method, a new and more efficient variable partition method is presented. We give some theoretical results, implement the methods by applying Matlab programming language, and make a comparison between our direct homotopy methods and the existing methods to show their effectiveness.In Chapter 3, efficient methods for solving polynomial systems transformed from mixed trigonometric polynomial systems are given. This class of polynomial systems have a special structure. Applying this special structure, an efficient hybrid method is presented. It combines the homotopy method, in which the homotopy is a combination of coefficient parameter homotopy and the random product homotopy, with symbolic computation methods, such as decomposition, variable substitution and reduction techniques. Furthermore, based on the symmetric structure of the lower part of the target system, a symmetric homotopy and hybrid method are presented. This method can keep the symmetric structure of the target system, which can save computational work greatly. We prove some theoretical results, implement our method by applying C++ programming language and make a comparison between our methods and the existing methods to show their effectiveness.In Chapter 4, by further numerical experiments, we discuss the advantages and disadvantages of direct homotopy methods and hybrid methods in solving different classes of mixed trigonometric polynomial systems. Then, we turn to a challenging practical problem, which arises in signal processing of sonar and radar and is hard to be solved by existing solving methods. We give a fast solving method which is the combination of the symmetric hybrid method and the coefficient-parameter homotopy method.