The Contour Integral Algorithm For Solving Two-parameter Eigenvalue Problem

The two-parameter eigenvalue problem is an important research problem in numerical algebra,which often appears in the fields of physics,meteorology,aerospace,structural dynamics.There are two types of methods for solving two-parameter eigenvalue problems.The first type is to transform the two-parameter eigenvalue problem into a generalized eigenvalue problem,including the minimum residual quotient iteration(MRQI)method and QZ algorithm for solving all eigenvalues,and Jacobi-Davidson method for solving the nearby eigenvalues of the given value.The second type is the homotopy method,which solves all the eigenvalues.Because the contour integration method has good parallelism and the advantages of solving the eigenvalues in a given area,In this paper,we try to apply the contour integration method for solving a two-parameter eigenvalue problem with a structure.In this paper,we transform the two-parameter eigenvalue problem into a larger generalized eigenvalue problem,whose coefficient matrix is composed of the Kronecker product of two matrices.Then apply the contour integral method to solve the generalized eigenvalue problem,where the Kronecker product linear system is the part of the maximum computation costs.Considering the structure of the coefficient matrix,the properties of the Kronecker product and the Sherman-Morrison-Woodbury formula,we get the solution by solving the ordinary linear equations and the shifted linear equations,instead of expressing the coefficient matrix explicitly.Therefore,we propose an effective way to solve the two-parameter eigenvalue problem by using the Sakurai-Suguira method.Finally,numerical experiments show its effectiveness for the two-parameter eigenvalue problem with structure in this paper.