Homotopy Method For The Inhomogeneous Symmetric Eigenvalue Problem

The inhomogeneous eigenvalue problem has many applications in mathematics and other areas, such as the study of the stability of linear differential equations and asymptotic estimation, the solution of the stable point of quadratic function on a sphere, the problem of constrained eigenvalue and so on. Homotopy method is an effective method to solve nonlinear problem, which overcomes the disadvantage of the local convergence of traditional iterative method, and has no strict limitation of initial value selection. What's more, it can make global convergence possible, and implement parallel computing easily.Following results were obtained: Firstly, we analyzed the influence of the inhomogeneous term on the existence of the inhomogeneous symmetric eigenvalue, and obtained the sensitivity theorem of the solution to the inhomogeneous term and validated it with some examples. Secondly, we proposed the homotopy method to solve all solutions of the inhomogeneous symmetric eigenvalue. According to the properties of the eigenvalue of irreducible tri-diagonal matrix, we used orthogonal similarity transformation to turn symmetric matrix into some low order irreducible tri-diagonal matrices, and solved the inhomogeneous eigenvalue of each matrix with homotopy method, by which all the solutions can be got.