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Efficient algorithm for the oscillatory matrix functions

Dongping Li, Xue Wang, Xiuying Zhang

TL;DR

The paper addresses efficient computation of general oscillatory $φ$-functions of a matrix, $φ_l(A)$, which arise in second-order IVPs. It introduces a quadruple-angle formula and a scaling/restoring framework, where $s$ and the Taylor degree $m$ are selected via forward error analysis to guarantee a prescribed tolerance. Core contributions include a stable recurrence-based algorithm, an error-bound analysis, and integration with the Paterson–Stockmeyer method for efficient matrix polynomials. Empirical results demonstrate both numerical stability and improved efficiency over a standard ODE solver in various test scenarios, with publicly available code for replication and further exploration.

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.

Efficient algorithm for the oscillatory matrix functions

TL;DR

The paper addresses efficient computation of general oscillatory -functions of a matrix, , which arise in second-order IVPs. It introduces a quadruple-angle formula and a scaling/restoring framework, where and the Taylor degree are selected via forward error analysis to guarantee a prescribed tolerance. Core contributions include a stable recurrence-based algorithm, an error-bound analysis, and integration with the Paterson–Stockmeyer method for efficient matrix polynomials. Empirical results demonstrate both numerical stability and improved efficiency over a standard ODE solver in various test scenarios, with publicly available code for replication and further exploration.

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.
Paper Structure (5 sections, 2 theorems, 39 equations, 2 figures, 2 tables, 3 algorithms)

This paper contains 5 sections, 2 theorems, 39 equations, 2 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Quadruple angle algorithm for $\phi_l(A)$
  3. Determination of the scaling parameter $s$ and the Taylor degree $m$
  4. Numerical experiments
  5. Conclusion

Key Result

Lemma 1

Given $A\in \mathbb{R}^{N\times N}$ and an integer $l\geq 2,$ then for any $a, b \in \mathbb{R},$ we have

Figures (2)

  • Figure 1: Relative errors of quadphi for solving $\varphi_l(A)$ for $l=0, . . . , 7$ (Left to Right) of Experiment \ref{['exa1']}.
  • Figure 2: Relative errors of quadphi and ode45 for solving $\phi_l(A)b$ for $l=0, . . . , 7$ (Left to Right) of Experiment \ref{['exa2']}.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2