ODE Solvers for Computing Matrix Functions

Year 2026, Volume: 7 Issue: 1, 17 - 29, 31.01.2026

Bahar Arslan

Abstract

This study investigates the use of ordinary differential equation solvers to estimate the action of matrix functions on vectors. In addition, an evaluation is conducted to ascertain the degree of computational efficiency that is achieved by executing matrix factorisation prior to the implementation of ODE solvers. Three models are employed to analyse the computation of matrix functions acting on vectors, including fractional matrix powers, the matrix exponential, and matrix cosine functions. The performance of the improved Euler, Taylor, Runge–Kutta and Adams–Bashforth methods are compared within these models.

Keywords

Matrix functions , Euler method , Taylor method , Runge-Kutta method , Adams-Bashforth method

Ethical Statement

The author declares that the materials and methods used in her study do not require ethical committee and/or legal special permission.

Thanks

The author would like to express her sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

References

• Higham N.J., Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, 2008.
• Al-Mohy A.H., Higham N.J., The complex step approximation to the Fréchet derivative of a matrix function, Numerical Algorithms, 53(1), 133-148, 2010.
• Allen E.J., Baglama J., Boyd S.K., Numerical approximation of the product of the square root of a matrix with a vector, Linear Algebra and its Applications, 310(1), 167-181, 2000.
• Davies P.I., Higham N.J., Computing f(A)b for matrix functions f , QCD and Numerical Analysis III, 15-24, 2005.
• Edwards R.G., Heller U.M., Narayanan R., Chiral fermions on the lattice, Parallel Computing, 25, 1395-1407, 1999.
• Eshof J., Frommer A., Lippert T., Schilling K., van der Vorst H.A., Numerical methods for the QCD overlap operator. I. Sign-function and error bounds, Computer Physics Communications, 146, 203-224, 2002.
• Benzi M., Boito P., Matrix functions in network analysis, Gesellschaft für Angewandte Mathematik und Mechanik, 43(3), e202000012, 2020.
• Hughes T.J.R., Levit I., Winget J., Element-by-element implicit algorithms for heat conduction, Journal of Engineering Mechanics, 109(2), 576-585, 1983.
• Burden R.L., Faires J.D., Burden A.M., Numerical Analysis, Cengage Learning, 2015.
• Higham N.J., Lin L., An improved Schur–Padé algorithm for fractional powers of a matrix and their Fréchet derivatives, SIAM Journal on Matrix Analysis and Applications, 34(3), 1341-1360, 2013.
• Al-Mohy A.H., Higham N.J., A new scaling and squaring algorithm for the matrix exponential, SIAM Journal on Matrix Analysis and Applications, 31(3), 970-989, 2009.
• Higham N.J., The Matrix Computation Toolbox, http://www.maths.manchester.ac.uk/~higham/ mctoolbox, 2013.
• Sinan O., Baydak S., Duman A., Aydın K., Computation of the solutions of Lyapunov matrix equations with iterative decreasing dimension method, Fundamental Journal of Mathematics and Applications, 5(2), 98-105, 2022.