Bi-parameter Exponential Homotopy Method And Its Application For Solving Nonlinear Equations

Homotopy method is a kind of method with the wide range of convergence in the numerical methods for solving nonlinear equations. Although homotopy method expands the range of the initial value, its convergence is affected by the construction of homotopy oper-ator. Jacobi singularity is, on the other hand, hard to be overcome in the extension process. So,homotopy method is often divergent when solving some complex equations. For that reason,a new kind of bi-parameter exponential homotopy operator is constructed and two new ho-motopy methods(bi-parameter numerical continuation method and bi-parameter differentiation method) are proposed on the basis of bi-parameter exponential homotopy operator.First of all, the prominent position of nonlinear problems in scientific computing is an-alyzed, and the important role of numerical methods for solving nonlinear equations is stud-ied. Secondly, the development of homotopy method is reviewed, and the homotopy method shortcomings depending on the initial value and overcoming Jacobi singularity are discussed.Thirdly, the basic idea of the homotopy operator is introduced, and a new bi-parameter ex-ponential homotopy operator is proposed on the basis of homotopy operator. Finally, based on numerical continuation method and parameter differentiation method, convergence of bi-paramete numerical continuation method and bi-parameter differentiation method is analyzed respectively.The feasibility and effectiveness of bi-parameter numerical continuation method and bi-parameter differential method are verified in numerical experiments. Compared with numerical continuation method, parameter differentiation method and Newton method, controllable pa-rameter value can be changed by bi-parameter numerical continuation method and bi-parameter differential method to modify homotopy operator. Simultaneously, convergence range of bi-parameter numerical continuation method and bi-parameter differential method is obviously enlarged. Therefore, bi-parameter numerical continuation method and bi-parameter differen-tial method not only solve the problem that numerical continuation method and parameter dif-ferentiation method is heavily dependent on the initial value but also overcome Jacobi singu-larity. Furthermore, bi-parameter numerical continuation method and bi-parameter differential method can provide a new way for solving all roots of nonlinear equations because of con-vergence range of bi-parameter numerical continuation method and bi-parameter differential method can be changed with the modification of the controllable parameters.