| Mixed Trigonometric polynomial systems, whose terms consists of some variables and the sine and cosine functions of other variables, are common in various fields of science and engin-eering.There are two types of methods, indirect methods and direct methods, to find all isolated solutions of the mixed trigonometric polynomial systems. Indirect methods find all isolated so-lutions of a mixed trigonometric polynomial system through solving the transformed polynom-ial systems. However, the transformation introduces new variables and hence increases the di-mension of the problems. Direct methods do not increase the dimension of the problems, how-ever,they are only suitable to be applied to solve the dense or specially sparse systems.In this paper, we consider to solve the mixed trigonometric polynomial systems by direct polyhedral homotopy method. We give a linear homotpy map, and use it to find all isolated sol-utions of the target equation. Numerical experiments show that our method has the advantage over the existent numerical methods. Precisely, this paper consists of the following contents.In the first chapter, we introduce the concept of mixed trigonometric polynomial systems, some examples and the solving methods, also the analysis of advantages and disadvantages of these two methods are studied.In the second chapter, homotopy methods for solving polynomial systems are presented. The fundamental knowledge of homotopy method and its application in solving polynomial sy-stems and the numerical path following technique is introduced. Specially, we precisely descri-be the polyhedral homotopy method.In the third chapter, we give the construction of the polyhedral homotopy method for solv-ing the mixed trigonometric polynomial systems and the convergence theorem of the method. The construction and the solving method of the start system are presented. Some numerical ex-periments are showed to prove the efficiency of our direct polyhedral homotpy method. |
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