Research Of Singular Function Method In Partial Eigenvalue Problem

Due to a lot of scientific problems causes an algebraic eigenvalue problem(for example, solving partial differential equations leads to large scale sparse matrix), therefore, researching into methods of solving eigenvalue problem is very important in numerical mathematics. Because science is continually evolving and developing, the eigenvalue problem becomes more and more difficult, so matrices, which appear in some scientific problems, is not numeric matrix. For example, the model of some physical problems can be produced by using second order ordinary differential equations with matrix coefficients. Solving such differential equations produces second order polynomial matrices.There are many different methods for solving eigenvalue problem of the numerical matrix in computational mathematics. But if the eigenvalue problem is problem of polynomial matrix, required, the effectiveness of these methods can be reduced. Therefore, solving eigenvalue problem of a polynomial matrix requires another methods and approach. In this regard, computational mathematics still need to develop, improve existing methods and give new in field of eigenvalue problem.This paper describes and analyzes a method of singular function, which is given by the Russian mathematician Y. M. Nechepurenko in 20 th century, but this method is not widespread. The singular function method is an iterative method, which is based on calculation a singular vector of polynomial matrix. We conducted some numerical experiments using Matlab system; these experiments show that, in partial eigenvalue problem, if power of the polynomial matrix and the matrix are large, the singular function method is better use than some other methods. However, if the power and matrix size is small, the singular function method has no effect. In addition to the analysis of the singular function method, this thesis also considers a variant of its modification. Using this modification, we can compute eigenvalues on a given line. It is possible that the optimization of the calculation program can get better results.