Numerical Study of Eigenvalue Problem Using Iterative Methods

Abstract

This study focuses on solving generalized eigenvalue problems by iterative method which are power method and Rayleigh quotient iteration. The primary purpose is to compare the performance of these two methods with respect to convergence speed, accuracy and computational efficiency when applied to a workforce transition model among three sectors which are information technology (IT), education, and media. The transition matrix used is a symmetric, stochastic with dominant eigenvalue corresponds to the steady-state distribution of workers. The power method algorithm is used to approximate the largest eigenvalue and corresponding eigenvector through repeated matrix-vector multiplication and normalization, while the Rayleigh quotient iteration used inverse iteration with dynamic shifting for better convergence. Rayleigh quotient iteration achieves faster convergence with 2 to 3 iterations than the power method that up to 250 iterations. Both methods yield the same result with a dominant eigenvalue of 1.0027 and consistent steady-state eigenvectors. The results are verified with eigen function which comes in MATLAB.