Accelerating the Speed of Convergence for High-Order Methods to Solve Equations
Abstract
:1. Introduction
- (P1)
- The local convergence analysis requires usually high-order derivatives, inverses of derivatives, or divided differences not on the methods. As demonstrated in the local analysis of convergence (LAC) in [11], the convergence order necessitates derivatives up to the sixth order, respectively, which are absent from the technique. These restrictions limit their use in a scenario, where . An inspiring and basic illustration is described by the function on , where is defined asNext, it is determined that the first three derivatives are
- (P2)
- There is no prior knowledge about the integer j, such that for each .
- (P3)
- There is no set containing only as a solution of (1).
- (P4)
- The results hold only on .
- (P5)
- The more important semi local results are not studied in [11].
2. Local Analysis of Convergence
- (H1)
- There exists CONDF , such that admits SMPS, which is denoted by . Define the interval .
- (H2)
- There exists CONDF , such that for , which is denoted byEquation admits SMPS in the interval , which is denoted by .Let and define the interval .
- (H3)
- There exists CONDF , such that equation admits SMPS in the interval , which is denoted by .Define the functions , , and by
- (H4)
- Equation admits SMPS in the interval , which is denoted by .Define that constant asIn Theorem 1, this constant is proven to be a radius of convergence for the method (2).It is worth noting that according to condition , function is CONDF and is SMPS for equation . By the definition of and the fact that the function is nondecreasing on , is the largest positive number in , such that . Therefore, all numbers will be such thatIf in particular, , this makes , and function is well defined. Hence, the convergence is assured, provided the rest of conditions – are satisfied. The scalar functions and N are connected to the operators on method (2).
- (H5)
- There exists and a solution for equation , such that and for each , we getDefine the domain .
- (H6)
- For each , we get
- (H7)
- .
- (i)
- Selections for L can be either or . Note that according to the condition , if , then solution is simple. If , condition does not necessarily imply that is a simple solution. Hence, method (2) can be used to find a solution of multiplicity that is greater than one. Consequently, the popular choice is not necessarily the most appropriate. The choice of has also been used, where is an auxiliary point.
- (ii)
- Conditions – are standard in the study of the convergence of iterative methods, if and functions and N are constant functions [3,9,10,19]. In this case, conditions and reduce to the usual center-Lipschitz and Lipschitz conditions. If one uses generalized continuity to replace the Lipschitz conditions, the results are more general, since they include Hölder and other continuity conditions (see the first two examples in Section 7).
- (iii)
- If one of the two versions of function is smaller than the other, then we use the smaller one in our calculations. However, if they cross, say for and for , then we choose
3. Semi-Local Analysis of Convergence
- (B1)
- There exists CONDF , such that equation admits SMPS, which is defined by .Define set .
- (B2)
- There exists CONDF and some .Define sequence for , some , and each and bySequence is proven to be majoring in Theorem 1. However, let us first develop a convergence condition for it.
- (B3)
- There exists , such that for each and , and .It follows by this condition and (24) that sequence is nondecreasing, bounded from above by , and is such that it is convergent to some .
- (B4)
- There exists and , such that and for each , we haveNotice that if , thenIt follows that
- (B5)
- Define set . For each ,
- (B6)
- .
- (1)
- Constant ρ can be switched with in condition .
- (2)
- Under conditions –, take and in Proposition 2.
4. Specializations and Numerical Experiments
5. Selection of the Majorant Functions for (27)
6. Selection of the Majorant Sequence for (27)
7. Numerical Problems
- Device Name: HP
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- Processor: Intel(R) Core(TM) i7-4790 CPU @ 3.60GHz 3.60 GHz
- System type: 64-bit operating system, x64-based processor
- Edition: Windows 10 Enterprise
- Version: 22H2
- OS Build: 19045.2006
7.1. Examples for LAC
j | s | |||||
---|---|---|---|---|---|---|
3 | 0.581977 | 0.38269 | 0.202479 | 0.13506 | - | 0.13506 |
4 | 0.581977 | 0.38269 | 0.202479 | 0.13506 | 0.11881 | 0.11881 |
j | s | |||||
---|---|---|---|---|---|---|
3 | 0.041667 | 0.020833 | 0.00949453 | 0.000568145 | - | 0.000568145 |
4 | 0.041667 | 0.020833 | 0.00949453 | 0.00055814 | 0.00186203 | 0.000568145 |
7.2. Examples for SLAC
Methods | n | CPU Timing | ||||
---|---|---|---|---|---|---|
Method (32) | 4 | 6.3492 | 2439.14 | |||
Method (33) | 3 | 8.2208 | 1620.13 |
Methods (2) | n | CPU Timing | ||||
---|---|---|---|---|---|---|
Method (32) | 4 | 6.0246 | 100.695 | |||
Method (33) | 3 | 8.3385 | 41.824 |
Methods | n | CPU Timing | ||||
---|---|---|---|---|---|---|
Method (32) | 4 | 6.0605 | 1617.33 | |||
Method (33) | 3 | 9.2706 | 976.077 |
Methods | n | CPU Timing | ||||
---|---|---|---|---|---|---|
Method (32) | 4 | 6.0298 | 203.797 | |||
Method (33) | 3 | 9.1334 | 103.344 |
8. Conclusions
- In conclusion, this study emphasizes the nature of convergence analysis in iterative techniques, particularly in the absence of explicit convergence guarantees.
- We rectify all these problems by using conditions only on the first derivation, which is on the method. This is how we extend its applicability. This includes Newmark, FEM, and Crant–Nicolson methods, which are used to solve complex problems.
- We also utilize extended continuity assumptions and present semi-local analysis, which further advances our knowledge of convergence tendencies in iterative approaches. By providing useful examples for real-world applications, we verified the semi-local convergence.
- With similar advantages, the method created in this study can also be applied to other iterative techniques that do not require the inverses of linear operators.
- The convergence conditions in [11] and in the present article are only sufficient. Our future work will also include necessary conditions.
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
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Behl, R.; Argyros, I.K.; Alharbi, S. Accelerating the Speed of Convergence for High-Order Methods to Solve Equations. Mathematics 2024, 12, 2785. https://doi.org/10.3390/math12172785
Behl R, Argyros IK, Alharbi S. Accelerating the Speed of Convergence for High-Order Methods to Solve Equations. Mathematics. 2024; 12(17):2785. https://doi.org/10.3390/math12172785
Chicago/Turabian StyleBehl, Ramandeep, Ioannis K. Argyros, and Sattam Alharbi. 2024. "Accelerating the Speed of Convergence for High-Order Methods to Solve Equations" Mathematics 12, no. 17: 2785. https://doi.org/10.3390/math12172785