Direct GBQ Algorithm for Solving Mixed Trigonometric Polynomial Systems

In many fields of science and engineering, it is needed to find all solutions of mixed trigonometric polynomial systems. Commonly, mixed trigonometric polynomial systems are transformed into polynomial systems by variable substitution and adding some quadratic equations, and then solved by some numerical methods. However, transformation of a mixed trigonometric polynomial system into a polynomial system will increase the dimension of the system and hence induces extra computational work. In this paper, we consider to solve the mixed trigonometric polynomial systems by homotopy method directly. Homotopy with the start system constructed by GBQ-algorithm is presented and homotopy theorems are proved. Preliminary numerical results show that our constructed direct homotopy method is more efficient than the existent direct homotopy methods.