Convergence of High-Order Derivative-Free Algorithms for the Iterative Solution of Systems of Not Necessarily Differentiable Equations
Abstract
:1. Introduction
- (C1)
- The convergence analysis is usually only local and where j is a natural number.
- (C2)
- The sufficient convergence hypotheses involve where order of convergence.
- (C3)
- No a priori and computational error distances are available.
- (C4)
- The isolation of the solution is not discussed.
- (C5)
- The semi-local convergence, which is considered more interesting and challenging than the local convergence, is not discussed.
- (C1)’
- The analysis is developed in Banach space.
- (C2)’
- The sufficient convergence hypotheses involve only the operators on the algorithm (see Algorithm 1), i.e., the divided differences. This is in contrast with the motivational work in [10] using hypotheses on high-order derivatives in the algorithm to show the convergence of the algorithm.
- (C3)’
- (C4)’
- The isolation of the solution is specified.and
- (C5)’
- The semi-local convergence analysis of the algorithm is studied.
Algorithm 1 |
Step 1: Given solve for |
Step 2: Set |
Step 3: Solve for |
Step 4: Solve for |
Step 5: Solve for |
Step 6: Ste |
Step 7: Solve for |
Step 8: Solve for |
Step 9: Solve for |
Step 10: Solve for |
Step 11: Solve for |
Step 12: Set |
Step 13: If STOP. Otherwise, repeat the process with |
2. Local Analysis
- (H1)
- Nondecreasing functions and continuous (NFC) exist, so that the equation
- (H2)
- NFC and exist, such that the equation admits MPS denoted by respectively, provided are provided asDefine parameter
- (H3)
- L is an invertible operator on E, such that for eachDefine the region with
- (H4)
- for and and are provided by the last two substeps of the algorithm.It is shown that exist (see Proof of Theorem 1).and
- (H5)
- where is the closure of
- (i)
- We can certainly choose in Proposition 1.
- (ii)
- Possible choice for the uncluttered functions can be obtained as follows:Thus, we can defineSimilarly, we setand
- (iii)
- A possible choice for L in local convergence studies may be , or , or any other linear operator satisfying the conditions (H1)–(H5) (see also the Example 1 in the Section 4).
3. Semi-Local Analysis
- (E1)
- NFC exists, such that equation
- (E2)
- exists, such that for eachIt follows that and exists, such that
- (E3)
- An invertible linear operator of L and exists, such that for eachNotice that by condition (E1)Thus, the linear operator is invertible and we can take
- (E4)
- Letfor eachand
- (E5)
- (i)
- The limit point can be replaced by in (E5) (provided in the condition (E1)).
- (ii)
- Suppose that all conditions (E1)–(E5) hold in Proposition 1. Then, set and
- (iii)
- Functions can be specified as in the local case by the following estimates:Hence, we can defineSimilarly, we chooseand
- (iv)
- A possible choice for L may be , provided that the operator is invertible or Other choices are possible, as long as conditions (E1)–(E4) are validated.
4. Numerical Examples
Algorithm 2 |
Step1: Given Solve for |
Step 2: Set |
Step 3: Solve for |
Step 4: Solve for |
Step 5: Solve for |
Step 6: Set |
Step 7: Solve for |
Step 8: Solve for |
Step 9: Solve for |
Step 10: Solve for |
Step 11: Set |
Step 12: If STOP. Otherwise, repeat the process with |
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Iteration | Solution | Time | |
---|---|---|---|
s | |||
1 | 0.52465745776846004734532218993115 | 0.862114977251371 | 9.196941 |
2 | 1.0666417888794666022247900197049 | 0.003827358620836 | 19.437201 |
3 | 1.0682235441972490182834127193622 | 1.110223024625157 × | 28.369249 |
4 | 1.0682235441972490182834711142631 | 1.110223024625157 × | 37.619249 |
Iteration | Solution | Time | |
---|---|---|---|
s | |||
1 | 0.20391080591998655968666298576863 | 1.081148328339054 × | 0.002347 |
2 | 0.20388835470224017654139458954887 | 1.110223024625157 × | 0.002501 |
3 | 0.20388835470224017654139458954887 | 1.110223024625157 × | 0.002839 |
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Regmi, S.; Argyros, I.K.; George, S. Convergence of High-Order Derivative-Free Algorithms for the Iterative Solution of Systems of Not Necessarily Differentiable Equations. Mathematics 2024, 12, 723. https://doi.org/10.3390/math12050723
Regmi S, Argyros IK, George S. Convergence of High-Order Derivative-Free Algorithms for the Iterative Solution of Systems of Not Necessarily Differentiable Equations. Mathematics. 2024; 12(5):723. https://doi.org/10.3390/math12050723
Chicago/Turabian StyleRegmi, Samundra, Ioannis K. Argyros, and Santhosh George. 2024. "Convergence of High-Order Derivative-Free Algorithms for the Iterative Solution of Systems of Not Necessarily Differentiable Equations" Mathematics 12, no. 5: 723. https://doi.org/10.3390/math12050723
APA StyleRegmi, S., Argyros, I. K., & George, S. (2024). Convergence of High-Order Derivative-Free Algorithms for the Iterative Solution of Systems of Not Necessarily Differentiable Equations. Mathematics, 12(5), 723. https://doi.org/10.3390/math12050723