Efficient algorithm for the oscillatory matrix functions

TL;DR

This paper presents a new efficient and stable algorithm for computing oscillatory matrix functions, essential for solving second-order semi-linear initial value problems, using scaling, Taylor series, and error analysis.

Contribution

The paper introduces a novel algorithm combining scaling, Taylor series, and error analysis for oscillatory matrix functions, improving stability and efficiency.

Findings

Algorithm is stable and accurate in numerical experiments

Performs well in terms of efficiency and precision

Effective for second-order semi-linear initial value problems

Abstract

This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and restoring technique based on a quadruple angle formula in conjunction with a truncated Taylor series. The choice of the scaling parameter and the degree of the Taylor polynomial relies on a forward error analysis. Numerical experiments show that the new algorithm behaves in a stable fashion and performs well in both accuracy and efficiency.