More Numerically Accurate Algorithm for Stiff Matrix Exponential
Abstract
:1. Introduction and Related Work
2. The LEXPM Algorithm
2.1. Algorithm
Algorithm 1 LEXPM 

2.2. Proof of Soundness and Completeness
3. Numerical Algorithm Evaluation
3.1. Artificial Stiff Matrices Analysis
 Matrices for which the difference between the eigenvalues of the matrix is small but not negligible: we randomly pick a value ($a>0$) and an amplitude ($\u03f5<<1$) and generate matrices with eigenvalues that are in the range ($a\pm \u03f5$).
 Matrices for which the eigenvalues are approaching 0: we generate matrices with eigenvalues that satisfy the following formula: $1\le i\le n,{\lambda}_{i}=\frac{1}{{(i+2)}^{2}}$.
 Matrices with large diameters: we generate matrices with eigenvalues that satisfy the formula$0\le i\le n,{\lambda}_{i}=a\frac{(ab)i}{n}$, where a and b are picked randomly such that $b>>a$.
 Matrices that have a large condition number: we generate matrices with eigenvalues that satisfy the formula $1\le i\le n,{\lambda}_{i}\in [a,b]$, such that $\frac{\leftb\right}{\lefta\right}>>1$.
 Matrices that have eigenvalues with significant algebraic multiplicity: we generate matrices with eigenvalues with an algebraic multiplicity of at least two.
 Matrices with a single eigenvalue: we generate matrices with a single eigenvalue picked at random.
 Matrices with complex eigenvalues with a large imaginary part: we generate matrices with eigenvalues that satisfy the formula $0\le j\le n,{\lambda}_{i}=a+10i\xb7a$, where $a\in [100,100]$ is a random number.
3.2. Control System’s Observability Use Case
4. Matrix Exponential Decision Tree
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
 Dunn, S.M.; Constantinides, A.; Moghe, P.V. Chapter 7—Dynamic Systems: Ordinary Differential Equations. In Numerical Methods in Biomedical Engineering; Academic Press: Cambridge, MA, USA, 2006; pp. 209–287. [Google Scholar]
 Van Loan, C. Computing integrals involving the matrix exponential. IEEE Trans. Autom. Control 1978, 23, 395–404. [Google Scholar] [CrossRef]
 AlMohy, A.H.; Higham, N.J. Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators. SIAM J. Sci. Comput. 2011, 33, 488–511. [Google Scholar] [CrossRef]
 Aboanber, A.E.; Nahla, A.A.; ElMhlawy, A.M.; Maher, O. An efficient exponential representation for solving the twoenergy group point telegraph kinetics model. Ann. Nucl. Energy 2022, 166, 108698. [Google Scholar] [CrossRef]
 Damgaard, P.; Hansen, E.; Plante, L.; Vanhove, P. Classical observables from the exponential representation of the gravitational Smatrix. J. High Energy Phys. 2023, 2023, 183. [Google Scholar] [CrossRef]
 Datta, B.N. Chapter 5—Linear StateSpace Models and Solutions of the State Equations. In Numerical Methods for Linear Control Systems: Design and Analysis; Academic Press: Cambridge, MA, USA, 2004; pp. 107–157. [Google Scholar]
 Fadali, M.S.; Visioli, A. State–space representation. In Digital Control Engineering: Analysis and Design; Academic Press: Cambridge, MA, USA, 2020; pp. 253–318. [Google Scholar]
 Moler, C.; Van Loan, C. Nineteen dubious ways to compute the exponential of a matrix, twentyfive years later. SIAM Rev. 2003, 45, 3–49. [Google Scholar] [CrossRef]
 Ward, R.C. Numerical Computation of the Matrix Exponential with Accuracy Estimate. SIAM J. Numer. Anal. 1977, 14, 600–610. [Google Scholar] [CrossRef]
 Zhou, C.; Wang, Z.; Chen, Y.; Xu, J.; Li, R. Benchmark Buckling Solutions of Truncated Conical Shells by Multiplicative Perturbation With Precise Matrix Exponential Computation. J. Appl. Mech. 2022, 89, 081004. [Google Scholar] [CrossRef]
 Wan, M.; Zhang, Y.; Yang, G.; Guo, H. TwoDimensional Exponential Sparse Discriminant Local Preserving Projections. Mathematics 2023, 11, 1722. [Google Scholar] [CrossRef]
 Najfeld, I.; Havel, T. Derivatives of the Matrix Exponential and Their Computation. Adv. Appl. Math. 1995, 16, 321–375. [Google Scholar] [CrossRef]
 Genocchi, A.; Peano, G. Calcolo Differenziale e Principii di Calcolo Integrale; Fratelli Bocca: Rome, Italy, 1884; Volume 67, pp. XVII–XIX. (In Italian) [Google Scholar]
 Biswas, B.N.; Chatterjee, S.; Mukherjee, S.P.; Pal, S. A Discussion on Euler Method: A Review. Electron. J. Math. Anal. Appl. 2013, 1, 294–317. [Google Scholar]
 Hochbruck, M.; Ostermann, A. Exponential Integrators; Cambridge University Press: Cambridge, UK, 2010; pp. 209–286. [Google Scholar]
 Butcher, J. A history of RungeKutta methods. Appl. Numer. Math. 1996, 20, 247–260. [Google Scholar] [CrossRef]
 Wang, H. The Krylov Subspace Methods for the Computation of Matrix Exponentials. Ph.D. Thesis, University of Kentucky, Lexington, KY, USA, 2015. [Google Scholar]
 Dinh, K.N.; Sidje, R.B. Analysis of inexact Krylov subspace methods for approximating the matrix exponential. Math. Comput. Simul. 2017, 1038, 1–13. [Google Scholar] [CrossRef]
 Druskin, V.; Greenbaum, A.; Knizhnerman, L. Using nonorthogonal Lanczos vectors in the computation of matrix functions. SIAM J. Sci. Comput. 1998, 19, 38–54. [Google Scholar] [CrossRef]
 Druskin, V.L.; Knizhnerman, L.A. Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithemetic. Numer. Linear Algebra Appl. 1995, 2, 205–217. [Google Scholar] [CrossRef]
 Ye, Q. Error bounds for the Lanczos methods for approximating matrix exponentials. SIAM J. Numer. Anal. 2013, 51, 66–87. [Google Scholar] [CrossRef]
 Pulungan, R.; Hermanns, H. Transient Analysis of CTMCs: Uniformization or Matrix Exponential. Int. J. Comput. Sci. 2018, 45, 267–274. [Google Scholar]
 Reibman, A.; Trivedi, K. Numerical transient analysis of markov models. Comput. Oper. Res. 1988, 15, 19–36. [Google Scholar] [CrossRef]
 Wu, W.; Li, P.; Fu, X.; Wang, Z.; Wu, J.; Wang, C. GPUbased power converter transient simulation with matrix exponential integration and memory management. Int. J. Electr. Power Energy Syst. 2020, 122, 106186. [Google Scholar] [CrossRef]
 Dogan, O.; Yang, Y.; Taspınar, S. Information criteria for matrix exponential spatial specifications. Spat. Stat. 2023, 57, 100776. [Google Scholar] [CrossRef]
 Wahln, E. Alternative Proof of Putzer’s Algorithm. 2013. Available online: http://www.ctr.maths.lu.se/media11/MATM14/2013vt2013/putzer.pdf (accessed on 17 February 2021).
 Lanczos, C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Natl. Bur. Stand. 1950, 45, 225–282. [Google Scholar] [CrossRef]
 Ojalvo, I.U.; Newman, M. Vibration modes of large structures by an automatic matrixreduction methods. AIAA J. 1970, 8, 1234–1239. [Google Scholar] [CrossRef]
 Barnard, S.T.; Simon, H.D. Fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems. Concurr. Comput. Pract. Exp. 1994, 6, 101–117. [Google Scholar] [CrossRef]
 Liu, R.; Liu, E.; Yang, J.; Li, M.; Wang, F. Optimizing the Hyperparameters for SVM by Combining Evolution Strategies with a Grid Search. In Intelligent Control and Automation; Springer: Berlin/Heidelberg, Germany, 2006; Volume 344. [Google Scholar]
 Almohy, A.H.; Higham, N.J. A New Scaling and Squaring Algorithm for the Matrix Exponential. SIAM J. Matrix Anal. Appl. 2009, 31, 970–989. [Google Scholar] [CrossRef]
 Higham, N.J. The Scaling and Squaring Method for the Matrix Exponential Revisited. SIAM J. Matrix Anal. Appl. 2005, 26, 1179–1193. [Google Scholar] [CrossRef]
 Demmel, J.; Dumitriu, I.; Holtz, O.; Kleinberg, R. Fast matrix multiplication is stable. Numer. Math. 2007, 106, 199–224. [Google Scholar] [CrossRef]
 Poulsen, N.K. The Matrix Exponential, Dynamic Systems and Control; DTU Compute: Kongens Lyngby, Denmark, 2004. [Google Scholar]
 Kailath, T. Linear Systems; Prentice Hall: Upper Saddle River, NJ, USA, 1980. [Google Scholar]
 Farman, M.; Saleem, M.U.; Tabassum, M.F.; Ahmad, A.; Ahmad, M.O. A linear control of composite model for glucose insulin glucagon pump. Ain Shams Eng. J. 2019, 10, 867–872. [Google Scholar] [CrossRef]
 Swain, P.H.; Hauska, H. The decision tree classifier: Design and potential. IEEE Trans. Geosci. Electron. 1977, 15, 142–147. [Google Scholar] [CrossRef]
 Stiglic, G.; Kocbek, S.; Pernek, I.; Kokol, P. Comprehensive Decision Tree Models in Bioinformatics. PLoS ONE 2012, 7, e33812. [Google Scholar] [CrossRef] [PubMed]
 Kohavi, R. A Study of Cross Validation and Bootstrap for Accuracy Estimation and Model Select. In Proceedings of the International Joint Conference on Artificial Intelligence, Montreal, QC, Canada, 20–25 August 1995. [Google Scholar]
 Lu, Y.Y. Exponentials of symmetric matrices through tridiagonal reductions. Linear Algerba Its Appl. 1998, 279, 317–324. [Google Scholar] [CrossRef]
 Koza, J.R.; Poli, R. Genetic Programming. In Search Methodologies: Introductory Tutorials in Optimization and Decision Support Techniques; Springer: New York, NY, USA, 2005; pp. 127–164. [Google Scholar]
 Alexi, A.; Lazebnik, T.; Shami, L. Microfounded Tax Revenue Forecast Model with Heterogeneous Population and Genetic Algorithm Approach. Comput. Econ. 2023. [Google Scholar] [CrossRef]
 Grabmeier, J.L.; Lambe, L.A. Decision trees for binary classification variables grow equally with the Gini impurity measure and Pearson’s chisquare test. Int. J. Bus. Intell. Data Min. 2007, 2, 213–226. [Google Scholar] [CrossRef]
 Lazebnik, T.; Bahouth, Z.; BunimovichMendrazitsky, S.; Halachmi, S. Predicting acute kidney injury following open partial nephrectomy treatment using SATpruned explainable machine learning model. BMC Med. Inform. Decis. Mak. 2022, 22, 133. [Google Scholar] [CrossRef] [PubMed]
 Moore, E.F. The shortest path through a maze. In Proceedings of the International Symposium on the Theory of Switching; Harvard University Press: Cambridge, MA, USA, 1959; pp. 285–292. [Google Scholar]
Matrix Size  Algorithm  Type 1  Type 2  Type 3  Type 4  Type 5  Type 6  Type 7 

$3\times 3$  Naive  $2\times {10}^{3}$  $3\times {10}^{3}$  $6\times {10}^{2}$  $8\times {10}^{1}$  $3\times {10}^{3}$  $8\times {10}^{4}$  $1\times {10}^{3}$ 
$3\times 3$  Pade  $\mathbf{5}\times {\mathbf{10}}^{\mathbf{5}}$  $6\times {10}^{5}$  $\mathbf{6}\times {\mathbf{10}}^{\mathbf{4}}$  $9\times {10}^{3}$  $2\times {10}^{5}$  $7\times {10}^{5}$  $1\times {10}^{4}$ 
$3\times 3$  Newton  $8\times {10}^{3}$  $5\times {10}^{2}$  $3\times {10}^{2}$  $9\times {10}^{2}$  $5\times {10}^{5}$  $\mathbf{9}\times {\mathbf{10}}^{\mathbf{7}}$  $7\times {10}^{6}$ 
$3\times 3$  Lagrange  $3\times {10}^{2}$  $1\times {10}^{2}$  $4\times {10}^{2}$  $8\times {10}^{2}$  $1\times {10}^{6}$  $1\times {10}^{6}$  $\mathbf{8}\times {\mathbf{10}}^{\mathbf{7}}$ 
$3\times 3$  LEXPM  $1\times {10}^{4}$  $\mathbf{8}\times {\mathbf{10}}^{\mathbf{6}}$  $8\times {10}^{4}$  $\mathbf{8}\times {\mathbf{10}}^{\mathbf{3}}$  $\mathbf{9}\times {\mathbf{10}}^{\mathbf{7}}$  $1\times {10}^{6}$  $3\times {10}^{7}$ 
$10\times 10$  Naive  $4\times {10}^{1}$  $3\times {10}^{1}$  $1\times {10}^{0}$  $8\times {10}^{0}$  $4\times {10}^{1}$  $8\times {10}^{1}$  $1\times {10}^{0}$ 
$10\times 10$  Pade  $\mathbf{1}\times {\mathbf{10}}^{\mathbf{4}}$  $5\times {10}^{5}$  $\mathbf{2}\times {\mathbf{10}}^{\mathbf{3}}$  $2\times {10}^{1}$  $8\times {10}^{2}$  $8\times {10}^{2}$  $1\times {10}^{2}$ 
$10\times 10$  Newton  $2\times {10}^{2}$  $5\times {10}^{1}$  $8\times {10}^{1}$  $3\times {10}^{0}$  $5\times {10}^{1}$  $\mathbf{1}\times {\mathbf{10}}^{\mathbf{3}}$  $3\times {10}^{3}$ 
$10\times 10$  Lagrange  $2\times {10}^{2}$  $3\times {10}^{1}$  $1\times {10}^{0}$  $4\times {10}^{0}$  $7\times {10}^{1}$  $5\times {10}^{3}$  $\mathbf{2}\times {\mathbf{10}}^{\mathbf{3}}$ 
$10\times 10$  LEXPM  $4\times {10}^{4}$  $\mathbf{1}\times {\mathbf{10}}^{\mathbf{6}}$  $\mathbf{2}\times {\mathbf{10}}^{\mathbf{3}}$  $\mathbf{8}\times {\mathbf{10}}^{\mathbf{2}}$  $\mathbf{6}\times {\mathbf{10}}^{\mathbf{2}}$  $7\times {10}^{2}$  $5\times {10}^{3}$ 
$100\times 100$  Naive  $2\times {10}^{0}$  $8\times {10}^{1}$  $4\times {10}^{1}$  $6\times {10}^{2}$  $3\times {10}^{1}$  $2\times {10}^{1}$  $8\times {10}^{1}$ 
$100\times 100$  Pade  $7\times {10}^{2}$  $5\times {10}^{4}$  $\mathbf{9}\times {\mathbf{10}}^{\mathbf{1}}$  $\mathbf{3}\times {\mathbf{10}}^{\mathbf{0}}$  $\mathbf{2}\times {\mathbf{10}}^{\mathbf{0}}$  $1\times {10}^{1}$  $5\times {10}^{1}$ 
$100\times 100$  Newton  $4\times {10}^{0}$  $1\times {10}^{3}$  $5\times {10}^{1}$  $5\times {10}^{1}$  $3\times {10}^{2}$  $\mathbf{2}\times {\mathbf{10}}^{\mathbf{0}}$  $\mathbf{7}\times {\mathbf{10}}^{\mathbf{0}}$ 
$100\times 100$  Lagrange  $3\times {10}^{0}$  $1\times {10}^{3}$  $7\times {10}^{1}$  $1\times {10}^{2}$  $8\times {10}^{2}$  $\mathbf{2}\times {\mathbf{10}}^{\mathbf{0}}$  $8\times {10}^{0}$ 
$100\times 100$  LEXPM  $\mathbf{3}\times {\mathbf{10}}^{\mathbf{2}}$  $\mathbf{3}\times {\mathbf{10}}^{\mathbf{4}}$  $1\times {10}^{0}$  $5\times {10}^{0}$  $8\times {10}^{0}$  $4\times {10}^{0}$  $1\times {10}^{1}$ 
$1000\times 1000$  Naive  $2\times {10}^{5}$  $7\times {10}^{6}$  Err  Err  Err  $7\times {10}^{6}$  Err 
$1000\times 1000$  Pade  $8\times {10}^{2}$  $\mathbf{2}\times {\mathbf{10}}^{\mathbf{3}}$  Err  Err  $\mathbf{3}\times {\mathbf{10}}^{\mathbf{3}}$  $6\times {10}^{2}$  $\mathbf{1}\times {\mathbf{10}}^{\mathbf{4}}$ 
$1000\times 1000$  Newton  $3\times {10}^{4}$  Err  Err  Err  Err  $1\times {10}^{3}$  $3\times {10}^{4}$ 
$1000\times 1000$  Lagrange  $2\times {10}^{4}$  Err  Err  Err  Err  $3\times {10}^{3}$  $6\times {10}^{4}$ 
$1000\times 1000$  LEXPM  $\mathbf{4}\times {\mathbf{10}}^{\mathbf{1}}$  $4\times {10}^{5}$  Err  Err  $5\times {10}^{3}$  $\mathbf{7}\times {\mathbf{10}}^{\mathbf{1}}$  $2\times {10}^{4}$ 
Algorithm  Metric  3 × 3  10 × 10  100 × 100  1000 × 1000 

Pade  Error  $5.2\times {10}^{3}\pm 2.8\times {10}^{4}$  $9.8\times {10}^{3}\pm 1.0\times {10}^{3}$  $6.5\times {10}^{1}\pm 8.3\times {10}^{2}$  $8.3\times {10}^{0}\pm 5.1\times {10}^{1}$ 
Time  $3.1\times {10}^{3}\pm 0.2\times {10}^{3}$  $7.0\times {10}^{1}\pm 0.4\times {10}^{1}$  $5.8\times {10}^{1}\pm 5.9\times {10}^{0}$  $3.2\times {10}^{4}\pm 0.7\times {10}^{4}$  
LEXPM  Error  $4.3\times {10}^{3}\pm 7.5\times {10}^{5}$  $6.9\times {10}^{3}\pm 7.7\times {10}^{4}$  $2.1\times {10}^{2}\pm 4.1\times {10}^{3}$  $7.6\times {10}^{1}\pm 1.3\times {10}^{1}$ 
Time  $5.2\times {10}^{1}\pm 0.9\times {10}^{1}$  $1.4\times {10}^{1}\pm 0.2\times {10}^{1}$  $2.2\times {10}^{2}\pm 0.3\times {10}^{2}$  $8.5\times {10}^{5}\pm 1.1\times {10}^{5}$ 
Index  Component  Description  Worst Case Complexity $\left(\mathit{C}\right)$  Average Relative Numerical Error $\left(\mathit{E}\right)$  Is Leaf Node 

1  Check form  The matrix is diagonal, trigonal, Jordan, or full  $O\left({n}^{2}\right)$  0  No 
2  Is symmetric  The matrix is symmetric or not  $O\left({n}^{2}\right)$  0  No 
3  Large diameter  The diameter of the matrix is larger than some threshold or not  $O\left({n}^{3}\right)$  0  No 
4  Large algebraic multiplicity  There are eigenvalues with algebraic multiplicity that are larger than some threshold or not  $O\left({n}^{3}\right)$  0  No 
5  Large condition number  The condition number of the matrix is larger than some threshold x or not  $O\left({n}^{3}\right)$  0  No 
6  Single eigenvalue  Does the matrix have a single eigenvalue  $O\left({n}^{3}\right)$  0  No 
7  Complex eigenvalues  Eigenvalues are complex with a big imaginary part  $O\left({n}^{3}\right)$  0  No 
8  Close eigenvalues  Eigenvalues ${\lambda}_{i},{\lambda}_{j}$ such that ${\lambda}_{i}{\lambda}_{j}<const\wedge {\lambda}_{i}\ne {\lambda}_{j}$  $O\left({n}^{3}\right)$  0  No 
9  Diagonal  Diagonal matrix exponential  $O\left(n\right)$  0.13  Yes 
10  Jordan  Jordan matrix exponential  $O\left({n}^{4}\right)$  21.21  Yes 
11  Eigenvector algorithms  TRED2 [40] and TQL2 [8]  $O\left({n}^{4}\right)$  18.89  Yes 
12  Ill condition algorithms  IMPSUB [8]  $O\left({n}^{4}\right)$  47.05  Yes 
13  LEXPM  The proposed LEXPM algorithm  $O\left({n}^{4}\right)$  8.45  Yes 
14  Eigenvaluebased approximation  Lagrange algorithm [8]  $O\left({n}^{5}\right)$  54.98  Yes 
15  Different eigenvalue approximation  Newton algorithm [8]  $O\left({n}^{5}\right)$  52.73  Yes 
16  Power series approximation  Pade approximation algorithm [8]  $O\left({n}^{4}\right)$  11.09  Yes 
17  Naive  Naive algorithm  $O\left({n}^{4}\right)$  78.20  Yes 
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Lazebnik, T.; BunimovichMendrazitsky, S. More Numerically Accurate Algorithm for Stiff Matrix Exponential. Mathematics 2024, 12, 1151. https://doi.org/10.3390/math12081151
Lazebnik T, BunimovichMendrazitsky S. More Numerically Accurate Algorithm for Stiff Matrix Exponential. Mathematics. 2024; 12(8):1151. https://doi.org/10.3390/math12081151
Chicago/Turabian StyleLazebnik, Teddy, and Svetlana BunimovichMendrazitsky. 2024. "More Numerically Accurate Algorithm for Stiff Matrix Exponential" Mathematics 12, no. 8: 1151. https://doi.org/10.3390/math12081151