The High Precision Algorithms For Isospectral Flows

The isospectral flows are a kind of special matrix dynamic systems. The efficient algorithms for computing them have been a research topic for decades at home and abroad. Many researchers, such as Arieh Iserles and Antonella Zanna, have proposed several fast efficient algorithms, based on, for example, Runge-Kutta methods,modified Guass-Legendre Runge-Kutta methods, Semi-explicit methods and Lie-group algorithms.In this paper, we will put forward several new algorithms based on finite integration methods and Taylor methods to deal with isospectral flows. What's more, our numerical results show that our new algorithms are superior to the finite difference methods.The structure of this thesis consist of the following five chapters:In Chapter 1, we will give a brief introduction for research backgrounds, current situation, research contents of such systems, and the innovation of the paper.In Chapter 2, we introduce some definitions, theorems, lemmas and basic properties,which will be used in sequel.In Chapter 3, finite integral method(ordinary linear approximate and radial basis function) will be introduced, and we develop algorithms for computing isospectral flows, based on finite integration methods.In Chapter 4, Taylor methods will be introduced, and we similarly develop a algorithm for computing isospectral flows, based on Taylor methods.In Chapter 5, we test some numerical examples and present some comparisons between our algorithms and finite difference method(second order Runge-Kutta).