Computing of Extremal Characteristic Values of Symmetric Matrices by Individual Homotopy Algorithm

In this paper, we develop a new Homotopy method called the individual Homotopy method to solve the symmetric eigenproblem. The individual Homotopy method overcomes notable drawbacks of the existing Homotopy method, namely, (i) the possibility of breakdown or having a slow rate of convergence in the presence of clustering of the eigenvalues and (ii) the absence of a definite criterion to choose a step size that guarantees the convergence of the method. On the other hand, we also have a good approximations of the largest eigenvalue of a symmetric matrix from Lanczos algorithm. We apply it for the extremal eigenproblem of a very large symmetric matrix with good initial points.