research-article

Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later

Authors:

Charles Van LoanAuthors Info & Claims

SIAM Review, Volume 45, Issue 1

Pages 3 - 49

https://doi.org/10.1137/S00361445024180

Published: 01 January 2003 Publication History

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.

References

[1]

Richard Bellman, Introduction to matrix analysis, Second edition, McGraw‐Hill Book Co., 1970xxiii+403

[2]

Chandler Davis, Explicit functional calculus, Linear Algebra and Appl., 6 (1973), 193–199

[3]

V. Faddeeva, Computational methods of linear algebra, Dover Publications Inc., 1959xi+252

[4]

George Forsythe, Michael Malcolm, Cleve Moler, Computer methods for mathematical computations, Prentice‐Hall Inc., 1977xi+259, Prentice‐Hall Series in Automatic Computation

Digital Library

[5]

J. Frame, Matrix functions and applications. II. Functions of a matrix, IEEE Spectrum, 1 (1964), 102–108

[6]

J. Frame, Matrix functions and applications. IV. Matrix functions and constituent matrices, IEEE Spectrum, 1 (1964), 123–131

[7]

J. Frame, Matrix functions and applications. V. Similarity reductions by rational or orthogonal matrices, IEEE Spectrum, 1 (1964), 103–109

[8]

F. R. Gantmacher, The Theory of Matrices, Vols. I and II, Chelsea, New York, 1959.

[9]

C. C. MacDuffee, The Theory of Matrices, Chelsea, New York, 1956.

[10]

L. Mirsky, An introduction to linear algebra, Oxford, at the Clarendon Press, 1955xi+433

[11]

M. Reed and B. Simon, Functional Analysis, Academic Press, New York, 1972.

[12]

R. Rinehart, The equivalence of definitions of a matric function, Amer. Math. Monthly, 62 (1955), 395–414

[13]

P. Rosenbloom, Bounds on functions of matrices, Amer. Math. Monthly, 74 (1967), 920–926

[14]

J. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965xviii+662

Digital Library

[15]

T. M. Apostol, Some explicit formulas for the matrix exponential, Amer. Math. Monthly, 76 (1969), pp. 284–292.

[16]

R. Atherton, A. DeGance, On evaluation of the derivative of the matrix exponential function, IEEE Trans. Automatic Control, AC‐20 (1975), 707–708

[17]

A. Bradshaw, The eigenstructure of sampled‐data systems with confluent eigenvalues, Internat. J. Systems Sci., 5 (1974), 607–613

[18]

Richard Bellman, Perturbation techniques in mathematics, physics, and engineering, Holt, 1964viii+118

[19]

J. C. Cavendish, On the norm of a matrix exponential, SIAM Rev., 17 (1975), pp. 174–175.

[20]

C. Cullen, Remarks on computing

e^{At}

, IEEE Trans. Automatic Control, AC‐16 (1971), 94–95

[21]

F. Fer, Résolution de l’équation matricielle

dU / dt = pU

par produit infini d’exponentielles matricielles, Acad. Roy. Belg. Bull. Cl. Sci. (5), 44 (1958), 818–829

[22]

Edward Fulmer, Computation of the matrix exponential, Amer. Math. Monthly, 82 (1975), 156–159

[23]

Bo Kágström, Bounds and perturbation bounds for the matrix exponential, Nordisk Tidskr. Informationsbehandling (BIT), 17 (1977), 39–57

[24]

Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer‐Verlag New York, Inc., New York, 1966xix+592

[25]

R. Kirchner, An explicit formula for

e^{At}

, Amer. Math. Monthly, 74 (1967), 1200–1204

[26]

Heinz‐Otto Kreiss, Über Matrizen die beschränkte Halbgruppen erzeugen, Math. Scand., 7 (1959), 71–80

[27]

David Powers, On the eigenstructure of the matrix exponential, Internat. J. Systems Sci., 7 (1976), 723–725

[28]

E. Putzer, Avoiding the Jordan canonical form in the discussion of linear systems with constant coefficients, Amer. Math. Monthly, 73 (1966), 2–7

[29]

N. M. Rice, More Explicit Formulas for the Exponential Matrix, Queen’s Mathematical Reprints 1970–21, Queen’s University, Kingston, ON, Canada, 1970.

[30]

H. Trotter, On the product of semi‐groups of operators, Proc. Amer. Math. Soc., 10 (1959), 545–551

[31]

C. F. Van Loan, A Study of the Matrix Exponential, Numerical Analysis Report 7, Department of Mathematics, University of Manchester, Manchester, UK, 1975.

[32]

Charles Van Loan, The sensitivity of the matrix exponential, SIAM J. Numer. Anal., 14 (1977), 971–981

Digital Library

[33]

G. Weiss, A. Maradudin, The Baker‐Hausdorff formula and a problem in crystal physics, J. Mathematical Phys., 3 (1962), 771–777

[34]

Allen Ziebur, On determining the structure of

A

by analyzing

e^{At}

, SIAM Rev., 12 (1970), 98–102

Digital Library

[35]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath, Boston, 1965.

[36]

Germund Dahlquist, Stability and error bounds in the numerical integration of ordinary differential equations, Kungl. Tekn. Högsk. Handl. Stockholm. No., 130 (1959), 87–0

[37]

Charles Desoer, Hiromasa Haneda, The measure of a matrix as a tool to analyze computer algorithms for circuit analysis, IEEE Trans. Circuit Theory, CT‐19 (1972), 480–486

[38]

Emeric Deutsch, On matrix norms and logarithmic norms, Numer. Math., 24 (1975), 49–51

Digital Library

[39]

C. Pao, Logarithmic derivates of a square matrix, Linear Algebra and Appl., 6 (1973), 159–164

[40]

C. Pao, A further remark on the logarithmic derivatives of a square matrix, Linear Algebra and Appl., 7 (1973), 275–278

[41]

T. Ström, Minimization of norms and logarithmic norms by diagonal similarities, Computing (Arch. Elektron. Rechnen), 10 (1972), 1–7

[42]

Torsten Ström, On logarithmic norms, SIAM J. Numer. Anal., 12 (1975), 741–753

Digital Library

[43]

M. Healey, Study of methods of computing transition matrices, Proc. IEEE, 120 (1973), pp. 905–912.

[44]

Cleve Moler, Difficulties on computing the exponential of a matrix, AFIPS Press, Montvale, N.J., 1975, 79–82

[45]

L. Y. Bahar and A. K. Sinha, Matrix exponential approach to dynamic response, Comput. & Structures, 5 (1975), pp. 159–165.

[46]

T. A. Bickart, Matrix exponential: Approximation by truncated power series, Proc. IEEE, 56 (1968), pp. 372–373.

[47]

G. Bierman, Power series evaluation of transition and covariance matrices, IEEE Trans. Automatic Control, AC‐17 (1972), 228–232

[48]

D. A. Calahan, Numerical solution of linear systems with widely separated time constants, Proc. IEEE, 55 (1967), pp. 2016–2017.

[49]

K. C. Daly, Evaluating the matrix exponential, Electron. Lett., 8 (1972), p. 390.

[50]

W. Everling, On the evaluation of

e^{At}

by power series, Proc. IEEE, 55 (1967), p. 413.

[51]

D. A. Gall, The solution of linear constant coefficient ordinary differential equations with APL, Comput. Methods Mechanics and Engrg., 1 (1972), pp. 189–196.

[52]

M. L. Liou, A novel method of evaluating transient response, Proc. IEEE, 54 (1966), pp. 20–23.

[53]

J. B. Mankin and J. C. Hung, On Roundoff Errors in Computation of Transition Matrices, Reprints of the Joint Automatic Control Conference, University of Colorado, Boulder, CO, 1969, pp. 60–64.

[54]

E. J. Mastascusa, A relation between Liou’s method and fourth order Runge–Kutta method for evaluation of transient response, Proc. IEEE, 57 (1969), pp. 803–804.

[55]

J. B. Plant, On the computation of transient matrices for time invariant systems, Proc. IEEE, 56 (1968), pp. 1397–1398.

[56]

M. M. Shah, On the Evaluation of $e^{At}$ , Cambridge Report CUED/B‐Control TR8, Cambridge University, Cambridge, UK, 1971.

[57]

M. M. Shah, Analysis of Roundoff and Truncation Errors in the Computation of Transition Matrices, Cambridge Report CUED/B‐Control TR12, Cambridge University, Cambridge, UK, 1971.

[58]

C. Standish, Truncated Taylor series approximation to the state transition matrix of a continuous parameter finite Markov chain, Linear Algebra and Appl., 12 (1975), 179–183

[59]

D. E. Whitney, Propagated error bounds for numerical solution of transient response, Proc. IEEE, 54 (1966), pp. 1084–1085.

[60]

D. E. Whitney, More similarities between Runge–Kutta and matrix exponential methods for evaluating transient response, Proc. IEEE, 57 (1969), pp. 2053–2054.

Rational approximation.

[61]

James Blue, Hermann Gummel, Rational approximations to matrix exponential for systems of stiff differential equations, J. Computational Phys., 5 (1970), 70–83

[62]

W. Cody, G. Meinardus, R. Varga, Chebyshev rational approximations to

e^{- x}

in

[0, + \infty)

and applications to heat‐conduction problems, J. Approximation Theory, 2 (1969), 50–65

[63]

Wyman Fair, Yudell Luke, Padé approximations to the operator exponential, Numer. Math., 14 (1969/1970), 379–382

Digital Library

[64]

Syvert Nørsett,

C

‐polynomials for rational approximation to the exponential function, Numer. Math., 25 (1975/76), 39–56

Digital Library

[65]

E. Saff, On the degree of best rational approximation to the exponential function, J. Approximation Theory, 9 (1973), 97–101

[66]

E. Saff, R. Varga, On the zeros and poles of Padé approximants to

e^{z}

, Numer. Math., 25 (1975/76), 1–14

Digital Library

[67]

R. E. Scraton, Comment on rational approximants to the matrix exponential, Electron. Lett., 7 (1971), pp. 260–261.

[68]

J. Siemieniuch, Properties of certain rational approximations to

e^{‐ z}

, Nordisk Tidskr. Informationsbehandling (BIT), 16 (1976), 172–191

[69]

G. Siemieniuch and I. Gladwell, On Time Discretizations for Linear Time Dependent Partial Differential Equations, Numerical Analysis Report 5, Department of Mathematics, University of Manchester, Manchester, UK, 1974.

[70]

D. Trujillo, The direct numerical integration of linear matrix differential equations using Padé approximations, Internat. J. Numer. Methods Engrg., 9 (1975), 259–270

[71]

Richard Varga, On higher order stable implicit methods for solving parabolic partial differential equations, J. Math. and Phys., 40 (1961), 220–231

[72]

R. C. Ward, Numerical computation of the matrix exponential with accuracy estimate, SIAM J. Numer. Anal., 14 (1977), pp. 600–610.

Digital Library

[73]

A. Wragg and C. Davies, Evaluation of the matrix exponential, Electron. Lett., 9 (1973), pp. 525–526.

[74]

A. Wragg, C. Davies, Computation of the exponential of a matrix. I. Theoretical considerations, J. Inst. Math. Appl., 11 (1973), 369–375

[75]

A. Wragg, C. Davies, Computation of the exponential of a matrix. II. Practical considerations, J. Inst. Math. Appl., 15 (1975), 273–278

[76]

V. Zakian, Rational approximants to the matrix exponential, Electron. Lett., 6 (1970), pp. 814–815.

[77]

V. Zakian and R. E. Scraton, Comments on rational approximations to the matrix exponential, Electron. Lett., 7 (1971), pp. 260–262.

Polynomial methods.

[78]

G. J. Bierman, Finite series solutions for the transition matrix and covariance of a time‐invariant system, IEEE Trans. Automat. Control, AC‐16 (1971), pp. 173–175.

[79]

J. A. Boehm and J. A. Thurman, An algorithm for generating constituent matrices, IEEE Trans. Circuit Theory, CT‐18 (1971), pp. 178–179.

[80]

C. F. Chen and R. R. Parker, Generalization of Heaviside’s expansion technique to transition matrix evaluation, IEEE Trans. Educ., E‐9 (1966), pp. 209–212.

[81]

W. C. Davidon, Exponential Function of a 2‐by‐2 Matrix, Hewlett‐Packard HP65 Library Program.

[82]

S. Deards, On the evaluation of

\exp (tA)

, Matrix Tensor Quart., 23 (1973), pp. 141–142.

[83]

S. Ganapathy and R. S. Rao, Transient response evaluation from the state transition matrix, Proc. IEEE, 57 (1969), pp. 347–349.

[84]

İzzet Göknar, On the evaluation of constituent matrices, Internat. J. Systems Sci., 5 (1974), 213–218

[85]

Ignace Kolodner, On

\exp (tA)

with

A

satisfying a polynomial, J. Math. Anal. Appl., 52 (1975), 514–524

[86]

Y. L. Kuo and M. L. Liou, Comments on “A novel method of evaluating

e^{At}

in closed form,” IEEE Trans. Automat. Control, AC‐16 (1971), p. 521.

[87]

E. J. Mastascusa, A method of calculating

e^{At}

based on the Cayley–Hamilton theorem, Proc. IEEE, 57 (1969), pp. 1328–1329.

[88]

K. R. Rao and N. Ahmed, Heaviside expansion of transition matrices, Proc. IEEE, 56 (1968), pp. 884–886.

[89]

K. R. Rao and N. Ahmed, Evaluation of transition matrices, IEEE Trans. Automat. Control, AC‐14 (1969), pp. 779–780.

[90]

B. Roy, A. K. Mandal, D. Roy Choudhury, and A. K. Choudhury, On the evaluation of the state transition matrix, Proc. IEEE, 57 (1969), pp. 234–235.

[91]

M. N. S. Swamy, On a formula for evaluating

e^{At}

when the eigenvalues are not necessarily distinct, Matrix Tensor Quart., 23 (1972), pp. 67–72.

[92]

Mathukumalli Vidyasagar, A novel method of evaluating

e^{At}

in closed form, IEEE Trans. Automatic Control, AC‐15 (1970), 600–601

[93]

V. Zakian, Solution of homogeneous ordinary linear differential systems by numerical inversion of Laplace transforms, Electron. Lett., 7 (1971), 546–548

[94]

A. K. Choudhury, D. R. Choudhury, B. Roy, and A. K. Mandal, On the evaluation of

e^{At}

, Proc. IEEE, 56 (1968), pp. 1110–1111.

[95]

L. Falcidieno and A. Luvinson, A Numerical Approach to Computing the Companion Matrix Exponential, CSELT technical report 4, 1975, pp. 69–71.

[96]

C. Harris, Evaluation of matrix polynomials in the state companion matrix of linear time invariant systems, Internat. J. Systems Sci., 4 (1973), 301–307

[97]

D. W. Kammler, Numerical Evaluation of $\exp (At)$ When $A$ is a Companion Matrix, manuscript, University of Southern Illinois, Carbondale, IL, 1976.

[98]

I. Kaufman, Evaluation of an analytical function of a companion matrix with distinct eigenvalues, Proc. IEEE, 57 (1969), 1180–1181

[99]

I. Kaufman, A note on the “Evaluation of an analytical function of a companion matrix with distinct eigenvalues,” Proc. IEEE, 57 (1969), pp. 2083–2084.

[100]

I. Kaufman and P. H. Roe, On systems described by a companion matrix, IEEE Trans. Automat. Control, AC‐15 (1970), pp. 692–693.

[101]

I. Kaufman, H. Mann, and J. Vlach, A fast procedure for the analysis of linear time invariant networks, IEEE Trans. Circuit Theory, CT‐18 (1971), pp. 739–741.

[102]

M. L. Liou, Evaluation of the transition matrix, Proc. IEEE, 55 (1967), pp. 228–229.

[103]

A. K. Mandal, et al., Numerical computation method for the evaluation of the transition matrix, Proc. IEEE, 116 (1969), pp. 500–502.

[104]

W. E. Thomson, Evaluation of transient response, Proc. IEEE, 54 (1966), p. 1584.

Ordinary differential equations.

[105]

B. Ehle, J. Lawson, Generalized Runge‐Kutta processes for stiff initial‐value problems, J. Inst. Math. Appl., 16 (1975), 11–21

[106]

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice–Hall, Englewood Cliffs, NJ, 1971.

[107]

L. Shampine, M. Gordon, Computer solution of ordinary differential equations, W. H. Freeman and Co., 1975x+318, The initial value problem

[108]

L. F. Shampine and H. A. Watts, Practical Solution of Ordinary Differential Equations by Runge–Kutta Methods, Sandia Lab Report SAND 76‐0585, Albuquerque, NM, 1976.

[109]

J. Starner, Numerical Solution of Implicit Differential‐Algebraic Equations, Ph.D. Thesis, University of New Mexico, Albuquerque, NM, 1976.

Matrix decomposition methods.

[110]

G. Golub, J. Wilkinson, Ill‐conditioned eigensystems and the computation of the Jordan canonical form, SIAM Rev., 18 (1976), 578–619

Digital Library

[111]

Bo K

\dot{a}

gström, Axel Ruhe, An algorithm for numerical computation of the Jordan normal form of a complex matrix, ACM Trans. Math. Software, 6 (1980), 398–419

Digital Library

[112]

B. Parlett, A recurrence among the elements of functions of triangular matrices, Linear Algebra and Appl., 14 (1976), 117–121

[113]

B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, and C. B. Moler, Matrix Eigensystem Routines: EISPACK Guide, 2nd ed., Lecture Notes in Comput. Sci. 6, Springer‐Verlag, New York, 1976.

Integrals involving $e^{At}$ .

[114]

E. S. Armstrong and A. K. Caglayan, An Algorithm for the Weighting Matrices in the Sampled‐Data Optimal Linear Regulator Problem, NASA technical note, TN D‐8372, 1976.

[115]

J. Johnson and C. L. Phillips, An algorithm for the computation of the integral of the state transition matrix, IEEE Trans. Automat. Control, AC‐16 (1971), pp. 204–205.

[116]

C. Kallstrom, Computing

\exp (A)

and

\int \exp (As) ds,

Report 7309, Division of Automatic Control, Lund Institute of Technology, Lund, Sweden, 1973.

[117]

Alexander Levis, Some computational aspects of the matrix exponential, IEEE Trans. Automatic Control, AC‐14 (1969), 410–411

[118]

Charles Van Loan, Computing integrals involving the matrix exponential, IEEE Trans. Automat. Control, 23 (1978), 395–404

[119]

F. H. Branin, Computer methods of network analysis, Proc. IEEE, 55 (1967), pp. 1787–1801.

[120]

R. Brockett, Finite Dimensional Linear Systems, John Wiley, New York, 1970.

[121]

K. F. Hansen, B. V. Koen, and W. W. Little, Stable numerical solutions of the reactor kinetics equations, Nuclear Sci. Engrg., 22 (1965), pp. 51–59.

[122]

J. A. W. da Nobrega, A new solution of the point kinetics equations, Nuclear Sci. Engrg., 46 (1971), pp. 366–375.

Cited By

Bladt MPeralta O(2025)Strongly Convergent Homogeneous Approximations to Inhomogeneous Markov Jump Processes and ApplicationsMathematics of Operations Research10.1287/moor.2022.015350:1(334-355)Online publication date: 1-Feb-2025
https://dl.acm.org/doi/10.1287/moor.2022.0153
Skau EHollis AEidenbenz SRasmussen KAlexandrov B(2024)Generating Hidden Markov Models from Process Models Through Nonnegative Tensor FactorizationACM Transactions on Modeling and Computer Simulation10.1145/366481334:4(1-19)Online publication date: 12-Jul-2024
https://dl.acm.org/doi/10.1145/3664813
Zeng ZLiu JYuan Y(2024)A Generalized Nyquist-Shannon Sampling Theorem Using the Koopman OperatorIEEE Transactions on Signal Processing10.1109/TSP.2024.343661072(3595-3610)Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1109/TSP.2024.3436610
Show More Cited By

Index Terms

Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later

Index terms have been assigned to the content through auto-classification.

Recommendations

Distance Problems for Hermitian Matrix Pencils with Eigenvalues of Definite Type

Given a Hermitian matrix pencil $L(z) = zA - B$ with only real eigenvalues that are either of positive or negative type, the distance to a nearest Hermitian pencil outside the class is considered with respect to a specified norm. These problems are ...
Rational Krylov: A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils

The rational Krylov algorithm computes eigenvalues and eigenvectors of a regular not necessarily symmetric matrix pencil. It is a generalization of the shifted and inverted Arnoldi algorithm, where several factorizations with different shifts are used in ...
Circulant preconditioners for discrete ill-posed Toeplitz systems

Circulant preconditioners are commonly used to accelerate the rate of convergence of iterative methods when solving linear systems of equations with a Toeplitz matrix. Block extensions that can be applied when the system has a block Toeplitz matrix with ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image SIAM Review

SIAM Review Volume 45, Issue 1

2003

156 pages

ISSN:0036-1445

DOI:10.1137/siread.2003.45.issue-1

Issue’s Table of Contents

Copyright © 2003 Society for Industrial and Applied Mathematics.

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2003

Author Tags

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

148
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 15 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Bladt MPeralta O(2025)Strongly Convergent Homogeneous Approximations to Inhomogeneous Markov Jump Processes and ApplicationsMathematics of Operations Research10.1287/moor.2022.015350:1(334-355)Online publication date: 1-Feb-2025
https://dl.acm.org/doi/10.1287/moor.2022.0153
Skau EHollis AEidenbenz SRasmussen KAlexandrov B(2024)Generating Hidden Markov Models from Process Models Through Nonnegative Tensor FactorizationACM Transactions on Modeling and Computer Simulation10.1145/366481334:4(1-19)Online publication date: 12-Jul-2024
https://dl.acm.org/doi/10.1145/3664813
Zeng ZLiu JYuan Y(2024)A Generalized Nyquist-Shannon Sampling Theorem Using the Koopman OperatorIEEE Transactions on Signal Processing10.1109/TSP.2024.343661072(3595-3610)Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1109/TSP.2024.3436610
Liao HWang XWen C(2024)Original energy dissipation preserving corrections of integrating factor Runge-Kutta methods for gradient flow problemsJournal of Computational Physics10.1016/j.jcp.2024.113456519:COnline publication date: 15-Dec-2024
https://dl.acm.org/doi/10.1016/j.jcp.2024.113456
Wang HShi L(2024)A multi-direction guided mutation-driven stable swarm intelligence algorithm with translation and rotation invariance for global optimizationApplied Soft Computing10.1016/j.asoc.2024.111614159:COnline publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1016/j.asoc.2024.111614
Zhan RFang J(2024)Stability analysis of explicit exponential Rosenbrock methods for stiff differential equations with constant delayApplied Mathematics and Computation10.1016/j.amc.2024.128978482:COnline publication date: 1-Dec-2024
https://dl.acm.org/doi/10.1016/j.amc.2024.128978
Bouhamidi AElbouyahyaoui LHeyouni M(2024)The constant solution method for solving large-scale differential Sylvester matrix equations with time invariant coefficientsNumerical Algorithms10.1007/s11075-023-01653-396:1(449-488)Online publication date: 1-May-2024
https://dl.acm.org/doi/10.1007/s11075-023-01653-3
Cao YLu J(2024)Structure-Preserving Numerical Schemes for Lindblad EquationsJournal of Scientific Computing10.1007/s10915-024-02707-x102:1Online publication date: 7-Dec-2024
https://dl.acm.org/doi/10.1007/s10915-024-02707-x
Ravelo FPeixoto PSchreiber M(2024)An Explicit Exponential Integrator Based on Faber Polynomials and its Application to Seismic Wave ModelingJournal of Scientific Computing10.1007/s10915-023-02413-098:2Online publication date: 3-Jan-2024
https://dl.acm.org/doi/10.1007/s10915-023-02413-0
Rupp KSchill RSüskind JGeorg PKlever MLösch AGrasedyck LWettig TSpang R(2024)Differentiated uniformization: a new method for inferring Markov chains on combinatorial state spaces including stochastic epidemic modelsComputational Statistics10.1007/s00180-024-01454-939:7(3643-3663)Online publication date: 1-Dec-2024
https://dl.acm.org/doi/10.1007/s00180-024-01454-9
Show More Cited By

View Options

View options

Figures

Tables

Media

View Issue’s Table of Contents