More Numerically Accurate Algorithm for Stiff Matrix Exponential
Abstract
:1. Introduction and Related Work
2. The L-EXPM Algorithm
2.1. Algorithm
Algorithm 1 L-EXPM |
|
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 () and an amplitude () and generate matrices with eigenvalues that are in the range ().
- Matrices for which the eigenvalues are approaching 0: we generate matrices with eigenvalues that satisfy the following formula: .
- Matrices with large diameters: we generate matrices with eigenvalues that satisfy the formula, where a and b are picked randomly such that .
- Matrices that have a large condition number: we generate matrices with eigenvalues that satisfy the formula , such that .
- 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 , where 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
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Matrix Size | Algorithm | Type 1 | Type 2 | Type 3 | Type 4 | Type 5 | Type 6 | Type 7 |
---|---|---|---|---|---|---|---|---|
Naive | ||||||||
Pade | ||||||||
Newton | ||||||||
Lagrange | ||||||||
L-EXPM | ||||||||
Naive | ||||||||
Pade | ||||||||
Newton | ||||||||
Lagrange | ||||||||
L-EXPM | ||||||||
Naive | ||||||||
Pade | ||||||||
Newton | ||||||||
Lagrange | ||||||||
L-EXPM | ||||||||
Naive | Err | Err | Err | Err | ||||
Pade | Err | Err | ||||||
Newton | Err | Err | Err | Err | ||||
Lagrange | Err | Err | Err | Err | ||||
L-EXPM | Err | Err |
Algorithm | Metric | 3 × 3 | 10 × 10 | 100 × 100 | 1000 × 1000 |
---|---|---|---|---|---|
Pade | Error | ||||
Time | |||||
L-EXPM | Error | ||||
Time |
Index | Component | Description | Worst Case Complexity | Average Relative Numerical Error | Is Leaf Node |
---|---|---|---|---|---|
1 | Check form | The matrix is diagonal, trigonal, Jordan, or full | 0 | No | |
2 | Is symmetric | The matrix is symmetric or not | 0 | No | |
3 | Large diameter | The diameter of the matrix is larger than some threshold or not | 0 | No | |
4 | Large algebraic multiplicity | There are eigenvalues with algebraic multiplicity that are larger than some threshold or not | 0 | No | |
5 | Large condition number | The condition number of the matrix is larger than some threshold x or not | 0 | No | |
6 | Single eigenvalue | Does the matrix have a single eigenvalue | 0 | No | |
7 | Complex eigenvalues | Eigenvalues are complex with a big imaginary part | 0 | No | |
8 | Close eigenvalues | Eigenvalues such that | 0 | No | |
9 | Diagonal | Diagonal matrix exponential | 0.13 | Yes | |
10 | Jordan | Jordan matrix exponential | 21.21 | Yes | |
11 | Eigenvector algorithms | TRED2 [40] and TQL2 [8] | 18.89 | Yes | |
12 | Ill condition algorithms | IMPSUB [8] | 47.05 | Yes | |
13 | L-EXPM | The proposed L-EXPM algorithm | 8.45 | Yes | |
14 | Eigenvalue-based approximation | Lagrange algorithm [8] | 54.98 | Yes | |
15 | Different eigenvalue approximation | Newton algorithm [8] | 52.73 | Yes | |
16 | Power series approximation | Pade approximation algorithm [8] | 11.09 | Yes | |
17 | Naive | Naive algorithm | 78.20 | Yes |
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Lazebnik, T.; Bunimovich-Mendrazitsky, S. More Numerically Accurate Algorithm for Stiff Matrix Exponential. Mathematics 2024, 12, 1151. https://doi.org/10.3390/math12081151
Lazebnik T, Bunimovich-Mendrazitsky 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 Bunimovich-Mendrazitsky. 2024. "More Numerically Accurate Algorithm for Stiff Matrix Exponential" Mathematics 12, no. 8: 1151. https://doi.org/10.3390/math12081151