Enhancing Equation Solving: Extending the Applicability of Steffensen-Type Methods
Abstract
:1. Introduction
- (L1)
- The local convergence analysis carried out in the case when , where k is a natural number.
- (L2)
- There are no computable error bounds on the distances . Therefore, we do not know a priori how many iterations must be carried out to reach a certain pre-decided error tolerance.
- (L3)
- There is no uniqueness of the solution results.
- (L4)
- The existence is assumed of derivatives that are not present in the method. As an example for method (2), consider and function defined by
- (L5)
- The choice of the initial point is a “shot in the dark”, since no computable radius of convergence is provided.
- (L6)
- (L1)′
- The convergence analysis is carried out in the setting of a Banach space.
- (L2)′
- A priori computable upper error bounds on the distances are provided. Hence, we know in advance the number of iterations to be carried out in order to achieve a desired error tolerance.
- (L3)′
- A neighborhood is specified that contains only one solution.
- (L4)′
- The convergence is established using only the operators in method (2).
- (L5)′
- The radius of convergence is determined. Hence, if we choose an initial point from the ball with this radius, the convergence is assured.
- (L6)′
- The semi-local convergence is developed by utilizing majorizing sequences.
2. Convergence Analysis I: Local
- (H1)
- There exist functions of which are increasing as well as continuous (IC) such that the equation admits a smallest positive solution (sps), denoted by . Let .
- (H2)
- There exists an IC function such that for the equation has a sps , where
- (H3)
- There exists an IC function such that the equation admits a sps . The set and the function are developed later. Define the parameters byThe functions and are associated with the data in method (2) as follows:
- (H4)
- There exists an operator P such that and for eachLet .
- (H5)
- (H6)
- for each
- (H7)
- , where .
- (H8)
- There exists a function IC such that for each
- (H9)
- There exists such that .
- (H7)′
- , where
3. Convergence II: Semi-Local
- (C1)
- and for some .
- (C2)
- There exists a linear operator P such that and, for each ,It follows by the first condition that there exists such that
- (C3)
- for each .
- (C4)
- There exists IC function such that
- (C5)
- (C6)
- The equation has a smallest positive solution and there exists such that .Let .
- (C7)
- , where .
- (i)
- The limit point can be replaced by in condition .
- (ii)
- It is clear that, under all conditions –, one can choose and in Proposition 1.
- (iii)
- Notice also that, as in the local case,provided thatfor some IC function .
4. Special Cases and Applications
- (1)
- , if the operator G is differentiable in the local convergence case.
- (2)
- , if the operator is not necessarily differentiable or if the operator is differentiable in the semi-local convergence case.
5. Numerical Applications
- (i)
- and
- (ii)
- ,
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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s0 | s1 | s2 | s | |||
---|---|---|---|---|---|---|
0.18365 | 0.19369 | 0.13771 | 0.13771 | |||
Cases | CPU Timing | |||||
Solver (3) | 4 | 4.0000 | 0.0237102 | |||
Solver (4) | 5 | 4.0000 | 0.0362553 | |||
Solver (5) | 4 | 4.0000 | 0.0251659 |
s | |||
---|---|---|---|
0.078031 | 0.070361 | 0.052654 | 0.052654 |
Cases | CPU Timing | |||||
---|---|---|---|---|---|---|
Solver (3) | 4 | 4.0001 | 8395.78 | |||
Solver (4) | 4 | 4.0002 | 13475.8 | |||
Solver (5) | 4 | 4.0002 | 29148.4 |
j | tj | wj |
---|---|---|
1 | ||
2 | ||
3 | ||
4 | ||
5 | ||
6 | ||
7 | ||
8 | ||
9 | ||
10 |
Cases | CPU Timing | |||||
---|---|---|---|---|---|---|
Solver (3) | 4 | 3.9999 | 0.893317 | |||
Solver (4) | 4 | 4.0000 | 0.796277 | |||
Solver (5) | 4 | 3.9999 | 0.760152 |
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Behl, R.; Argyros, I.K.; Alansari, M. Enhancing Equation Solving: Extending the Applicability of Steffensen-Type Methods. Mathematics 2023, 11, 4551. https://doi.org/10.3390/math11214551
Behl R, Argyros IK, Alansari M. Enhancing Equation Solving: Extending the Applicability of Steffensen-Type Methods. Mathematics. 2023; 11(21):4551. https://doi.org/10.3390/math11214551
Chicago/Turabian StyleBehl, Ramandeep, Ioannis K. Argyros, and Monairah Alansari. 2023. "Enhancing Equation Solving: Extending the Applicability of Steffensen-Type Methods" Mathematics 11, no. 21: 4551. https://doi.org/10.3390/math11214551
APA StyleBehl, R., Argyros, I. K., & Alansari, M. (2023). Enhancing Equation Solving: Extending the Applicability of Steffensen-Type Methods. Mathematics, 11(21), 4551. https://doi.org/10.3390/math11214551