Polynomial-Based Approach for Efficient Function Computation of Symmetric Matrices Using Restarted Heavy Ball Method

Abstract

This paper presents a polynomial-based approach for efficiently computing functions ofsymmetric matrices by leveraging the Restarted Heavy Ball (RHB) method. The RHB method is employedto overcome the slow convergence issue commonly encountered when computing functions of symmetricmatrices. The key idea of our approach is to approximate the desired function using a polynomial. Byrepresenting the function as a polynomial, we can leverage the efficient computation of polynomials toaccelerate the overall function computation process. We introduce a systematic methodology forconstructing an optimal polynomial approximation that minimizes the approximation error. To furtherenhance the convergence speed, we incorporate the Restarted Heavy Ball method into our polynomialbased approach. The Restarted Heavy Ball iteration is applied after a certain number of iterations to resetthe computation process and mitigate the slow convergence behavior. The experimental results and analysisvalidate the effectiveness and practicality of our approach, highlighting its potential for various applicationsinvolving function computations of symmetric matrices. Overall, our polynomial-based approach,integrated with the Restarted Heavy Ball method, offers an efficient and accurate solution for computingfunctions of symmetric matrices. The experimental results and analysis validate the effectiveness andpracticality of our approach, highlighting its potential for various applications involving functioncomputations of symmetric matrices.