Homotopy Method For Generalized Eigenvalue Problem

Homotopy algorithm is developed in the 1970s to solve nonlinear problems of nu-merical methods.Because of its intrinsic parallelism and the characteristics of large-scaleconvergence,Thus easy to implement parallel computing.In recent years homotopy algo-rithm development take along two main lines start, the simplex method and continuousmethod. Homotopy algorithm used in algebra eigenvalue problem began in the mid-1980s,It opened up a new way for solving generalized eigenvalue problem. For generalized eigen-value problems have classic QR algorithm, due to the complexity of the practical problems,it is sometimes diffcult to achieve the corresponding eigenvalue problem, we can structurean ordinary problem, Starting from the eigenvalue of ordinary problem through the track-ing path to achieve complex eigenvalue problem. First this paper analyzed the the devel-opment of homotopy algorithm and the application, the study pointed out the directionof the homotopy algorithm,second followed by a list of generalized eigenvalue problemsolving methods,then advance a path-following algorithm to the generalized eigenvalueproblem and applied to example ,thus verifying the correctness of the algorithm, and higheffciency; In order to overcome the disadvantage of homotopy algorithms computation,we using Matlab language and it contains a function of programming, compared to For-tran language , Simple and effcient so as to reduce the strength, improve programmingeffciency. In this paper, the homotopy tracking algorithm particularly suitable for sparseproblem and pathological problems, and eigenvalue problems from bad. It has to find ameans Eigenvalues for the convenience, thus it plays an important role in the field of themechanical vibration, theoretical physics, finite element method.