Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms
Abstract
:1. Introduction
2. Development of Method
3. Main Result
Some Special Cases
4. Numerical Results
- (i)
- The number of iterations taken by the algorithms and the satisfaction of the stopping criterion .
- (ii)
- The first three iterations’ error .
- (iii)
- The calculated convergence order (CCO).
- (iv)
- The total time (in seconds) consumed by the algorithms.
Problem | Root | Multiplicity | Initial Guess | Results’ Table |
---|---|---|---|---|
2: Isentropic supersonic flow problem [53] | ||||
1.8411… | 3 | 1.5 & 1.7 | Table 2 | |
3: Planck law of radiation problem [54] | ||||
4.9651… | 4 | 2.5 & 5.5 | Table 3 | |
4: Kepler’s problem [49] | ||||
0.8093… | 5 | 1 & 1.4 | Table 4 | |
5: Complex root problem | ||||
i | 6 | 1.1 i & 1.3 i | Table 5 |
Methods | CCO | CPU-Time | ||||
---|---|---|---|---|---|---|
LLC | 4 | 4 | 2.1224 | |||
LCN | 4 | 4 | 2.4660 | |||
SS | 4 | 4 | 2.4653 | |||
ZCS | 4 | 4 | 2.4492 | |||
SBL | 4 | 4 | 2.7765 | |||
NM1 | 4 | 4 | 1.8522 | |||
NM2 | 4 | 4 | 1.8878 | |||
NM3 | 4 | 4 | 1.8867 | |||
NM4 | 4 | 4 | 1.9042 | |||
LLC | 4 | 4 | 2.6054 | |||
LCN | 4 | 4 | 2.8556 | |||
SS | 4 | 4 | 2.9173 | |||
ZCS | 4 | 4 | 2.8868 | |||
SBL | 4 | 4 | 3.2764 | |||
NM1 | 4 | 4 | 2.4214 | |||
NM2 | 4 | 4 | 2.5285 | |||
NM3 | 4 | 4 | 2.5432 | |||
NM4 | 4 | 4 | 2.4967 |
Methods | CCO | CPU-Time | ||||
---|---|---|---|---|---|---|
LLC | 5 | 4 | 0.8893 | |||
LCN | 5 | 4 | 1.2483 | |||
SS | 5 | 4 | 1.2791 | |||
ZCS | 5 | 4 | 1.2172 | |||
SBL | 5 | 4 | 1.4986 | |||
NM1 | 5 | 4 | 0.5946 | |||
NM2 | 5 | 4 | 0.6385 | |||
NM3 | 5 | 4 | 0.6843 | |||
NM4 | 5 | 4 | 0.6682 | |||
LLC | 4 | 4 | 0.6545 | |||
LCN | 4 | 4 | 1.0772 | |||
SS | 4 | 4 | 1.0301 | |||
ZCS | 4 | 4 | 1.0142 | |||
SBL | 4 | 4 | 1.2646 | |||
NM1 | 3 | 0 | 4 | 0.4534 | ||
NM2 | 3 | 0 | 4 | 0.5173 | ||
NM3 | 3 | 0 | 4 | 0.5238 | ||
NM4 | 3 | 0 | 4 | 0.4923 |
Methods | CCO | CPU-Time | ||||
---|---|---|---|---|---|---|
LLC | 4 | 4 | 0.8422 | |||
LCN | 4 | 4 | 0.9675 | |||
SS | 4 | 4 | 0.9833 | |||
ZCS | 4 | 4 | 0.9215 | |||
SBL | 4 | 4 | 1.1863 | |||
NM1 | 4 | 4 | 0.5865 | |||
NM2 | 4 | 4 | 0.6246 | |||
NM3 | 4 | 4 | 0.5934 | |||
NM4 | 4 | 4 | 0.6440 | |||
LLC | 4 | 4 | 0.8114 | |||
LCN | 4 | 4 | 1.0320 | |||
SS | 4 | 4 | 1.0146 | |||
ZCS | 4 | 4 | 0.9998 | |||
SBL | 4 | 4 | 1.1393 | |||
NM1 | 4 | 4 | 0.6155 | |||
NM2 | 4 | 4 | 0.6442 | |||
NM3 | 4 | 4 | 0.6561 | |||
NM4 | 4 | 4 | 0.6240 |
Methods | CCO | CPU-Time | ||||
---|---|---|---|---|---|---|
LLC | 4 | 4 | 1.7604 | |||
LCN | 4 | 4 | 2.4491 | |||
SS | 4 | 4 | 2.4653 | |||
ZCS | 4 | 4 | 2.5426 | |||
SBL | 4 | 4 | 3.1042 | |||
NM1 | 4 | 4 | 0.5311 | |||
NM2 | 4 | 4 | 0.5380 | |||
NM3 | 4 | 4 | 0.5935 | |||
NM4 | 4 | 4 | 0.5468 | |||
LLC | 4 | 4 | 1.6234 | |||
LCN | 4 | 4 | 2.5432 | |||
SS | 4 | 4 | 2.6216 | |||
ZCS | 4 | 4 | 2.6217 | |||
SBL | 4 | 4 | 3.1824 | |||
NM1 | 4 | 4 | 0.5162 | |||
NM2 | 4 | 4 | 0.6243 | |||
NM3 | 4 | 4 | 0.5922 | |||
NM4 | 4 | 4 | 0.5610 |
Methods | CCO | CPU-Time | ||||
---|---|---|---|---|---|---|
LLC | 6 | 4 | 0.0787 | |||
LCN | 6 | 4 | 0.0786 | |||
SS | 6 | 4 | 0.1095 | |||
ZCS | 6 | 4 | 0.0786 | |||
SBL | 6 | 4 | 0.0938 | |||
NM1 | 6 | 4 | 0.0753 | |||
NM2 | 6 | 4 | 0.0821 | |||
NM3 | 6 | 4 | 0.0778 | |||
NM4 | 6 | 4 | 0.0782 | |||
LLC | 6 | 4 | 0.0982 | |||
LCN | 6 | 4 | 0.1102 | |||
SS | 6 | 4 | 0.1096 | |||
ZCS | 6 | 4 | 0.1145 | |||
SBL | 6 | 4 | 0.0944 | |||
NM1 | 6 | 4 | 0.0682 | |||
NM2 | 6 | 4 | 0.0924 | |||
NM3 | 6 | 4 | 0.0947 | |||
NM4 | 6 | 4 | 0.0936 |
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Kumar, S.; Sharma, J.R.; Bhagwan, J.; Jäntschi, L. Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms. Symmetry 2023, 15, 1249. https://doi.org/10.3390/sym15061249
Kumar S, Sharma JR, Bhagwan J, Jäntschi L. Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms. Symmetry. 2023; 15(6):1249. https://doi.org/10.3390/sym15061249
Chicago/Turabian StyleKumar, Sunil, Janak Raj Sharma, Jai Bhagwan, and Lorentz Jäntschi. 2023. "Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms" Symmetry 15, no. 6: 1249. https://doi.org/10.3390/sym15061249
APA StyleKumar, S., Sharma, J. R., Bhagwan, J., & Jäntschi, L. (2023). Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms. Symmetry, 15(6), 1249. https://doi.org/10.3390/sym15061249