Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later
Abstract
In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed. In practice, consideration of computational stability and efficiency indicates that some of the methods are preferable to others but that none are completely satisfactory. Most of this paper was originally published in 1978. An update, with a separate bibliography, describes a few recent developments.
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In the interaction picture, the time evolution of the system is shared between the density matrix and other operators.
“In the interaction picture, the time evolution of the system is shared between the density matrix and other operators [37, 38].”
From Fundamentals of quantum mechanics in Liouville space (2020) · cites this paper as [37]
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Gyamfi, J. A. (2020) · European Journal of Physics · doi:10.1088/1361-6404/ab9fdd · Cited by 77 · see citing works
The Complexity of Relating Quantum Channels to Master Equations
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Wyburd, M. K., Dinsdale, N. K., Jenkinson, M., & Namburete, A. I. L. (2024) · Medical Image Analysis · doi:10.1016/j.media.2024.103222 · Cited by 11 · see citing works
- Authors
- Moler, C., & Van Loan, C.
- Affiliations
- MathWorks (United Kingdom) (GB)
- Venue
- SIAM Review
- Published
- 2003-01
- Volume
- 45
- Issue
- 1
- Pages
- 3-49
- ISSN
- 0036-1445 (ISSN-L)
- 1095-7200
- OpenAlex
- W2106565812
- Cited by
- 2,357
- References
- 127
- Citations by year
- 2026: 67 · 2025: 98 · 2024: 115 · 2023: 82 · 2022: 99 · 2021: 124 · 2020: 135 · 2019: 144 · 2018: 136 · 2017: 161 · 2016: 129 · 2015: 127 · 2014: 161 · 2013: 114 · 2012: 108
- FWCI
- 44.44
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- SIAM Review (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Numerical Methods and Algorithms
- Physical Sciences › Mathematics › Numerical Analysis › Numerical methods for differential equations
- Keywords
- matrix exponential, numerical stability, computational efficiency, approximation theory, differential equations, matrix eigenvalues, characteristic polynomial
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