Homotopy Methods For Solving Multiparameter Eigenvalue Problems

Eigenvalue problem is an important research subject in numerical algebra.It has wide ap-plications in the fields of science(such as other mathematical subjects,physics,mechanics)and engineering(such as information,economics).Many important theoretical results and efficient numerical methods have been achieved in the past decades,however,there still exist many im-portant problems need to be investigated further,especially those coming from important prac-tical applications,such as eigenvalue problems from differential equations.In this dissertation,we mainly consider the multiparameter eigenvalue problems related closely to multiparameter Sturm-Liouville problems and delay differential equations.In Chapter 1,we give a brief introduction on the background,basic definitions,properties,and the existent numerical methods of the multiparameter eigenvalue problem.Additionally,we present some knowledge on the homotopy method,which is the main method applied in our research.There is close relationship between the multiparameter eigenvalue problem and the the polynomial system,we also introduce the numerical methods for solving polynomial systems.In chapter 2,we study the structured multiparameter eigenvalue problem.We show the transformed simultaneous eigenvalue problems is singular,therefore,not only the theoretical analysis but also the design of efficient numerical methods are both difficult.Based on the structure of the problem,we give the upper bound of the number of isolated solutions,which is far smaller than the existent bound.Furthermore,utilizing this upper bound,we construct the efficient homotopy methods,analyze the complexity of our method,compare our method with the method of transforming it to simultaneous eigenvalue problem.Numerical methods show our method is more efficient than the existent method for large problem.In chapter 3,we study the quadratic two-parameter eigenvalue problem.Transforming a multiparameter eigenvalue problem to a simultaneous eigenvalue problem will increase the di-mension of the problem,on the other hand,this transform is not always can be achieved,it deeply depends on the structure of the problem.For the general problem,we construct the homotopy method and prove the smoothness and accessibility of the homotopy path through introducing the multihomogeneous theory for polynomial systems.More importantly,for the problems with some coefficient matrices being zero,we can provide a tighter upper bound such that the num-ber of homotopy paths have to be traced is same as the actual number of the isolated solutions.Numerical results show our method can actually be applied to solve problems and our method is more efficient than the existent method for large problem.In chapter 4.we study the polynomial multiparameter eigenvalue problem.This problem is of high complexity and plays an important role in the analysis of the stability of the delay differential equation.Different from the quadratic problem in Chapter 3,the polynomial multi-parameter eigenvalue problem is hard to be linearized or transformed to simultaneous eigenvalue problem,and thus there is little work on this topic.The existent methods can find only one solu-tion and it can not be guaranteed to be purely imaginary,therefore,these methods can not solve the polynomial multiparameter eigenvalue problem satisfactorily.We reconsider the problem from the viewpoint of algebraic geometry,design an efficient numerical method combining the GBQ method for solving polynomial system.Our method can find all solutions of this problem and thus can be applied to analyze the stability of the delay differential equation.In chapter 5,we give the conclusion of this dissertation and present some existent problems and future work.