2018_A Family of Iterative Methods Via Modified Rational Approximation Model For Solving Nonlinear Problems
| Format: | General Document |
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| collectionurl | https://intelek.unisza.edu.my/intelek/pages/search.php?search=!collection3 |
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| country | Malaysia |
| date | 2018-08-08 |
| format | General Document |
| id | 16161 |
| institution | UniSZA |
| originalfilename | A FAMILY OF ITERATIVE METHODS VIA MODIFIED RATIONAL APPROXIMATION MODEL FOR SOLVING NONLINEAR PROBLEMS (PHD_2018).pdf |
| person | Kamilu Uba Kamfa |
| recordtype | oai_dc |
| resourceurl | https://intelek.unisza.edu.my/intelek/pages/view.php?ref=16161 |
| sourcemedia | Server storage Scanned document |
| spelling | 16161 https://intelek.unisza.edu.my/intelek/pages/view.php?ref=16161 https://intelek.unisza.edu.my/intelek/pages/search.php?search=!collection3 General Document Malaysia Library Staff (Top Management) Library Staff (Management) Library Staff (Support) Terengganu Faculty of Informatics & Computing English application/pdf 1.5 Server storage Scanned document Universiti Sultan Zainal Abidin UniSZA Private Access UNIVERSITI SULTAN ZAINAL ABIDIN SAMBox 2.3.4; modified using iTextSharp™ 5.5.10 ©2000-2016 iText Group NV (AGPL-version) Copyright©PWB2025 2018-08-08 196 A FAMILY OF ITERATIVE METHODS VIA MODIFIED RATIONAL APPROXIMATION MODEL FOR SOLVING NONLINEAR PROBLEMS (PHD_2018).pdf Nonlinear equations—Numerical solutions Kamilu Uba Kamfa 2018_A Family of Iterative Methods Via Modified Rational Approximation Model For Solving Nonlinear Problems Nonlinear problems emerge in some areas of science, engineering, social science, economics and art when designing various kind of events. This type of problems are reduced to solve a system of nonlinear equations (SNE) or nonlinear unconstrained optimization problems(NUO) or least square problems. All the problems are equivalent under fairly reasonable hypothesis, but they are not treat with the same algorithm. The famous and well known method for solving SNE is the Newton method (NM). Whereas, the Conjugate Gradient (CG) method is largely considered for solving NUO. Both methods are easy to implement and converges rapidly. Nevertheless, an iteration of NM turns out to be computational expensive, this is due to the fact that, the method requires finding and storage the n n Jacobian matrix which involves the first derivatives of the system and solving the Newton equation. In addition the convergence may even be lost when the Jacobian is singular at the solution. These draw back is more noticeable when handling large scale SNE. This lead to the first idea of this thesis. The CG search direction with PRP parameter is very efficient in term of the numerical performance, but it fails in term of the global convergence for the general function under Wolfe line search technique. In addition the trust region and sufficient properties of the search direction could not be established for this method. This lead to the second idea of this thesis. This study proposed several derivative-free Newton-like methods via a modified rational approximation model and various three-term CG methods using a new conjugate parameter based on a Wolfe line search. In particular, a famous Jacobian approximation such as the Broyden update (BM), Broyden, Fletcher, Goldfarb and Shannon update (BFGS) and the inverse diagonal Newton-like update (IDN). And a new derivative free rational approximation model were used to developed each method for solving SNE. The advantage of this novel approach when applied to both small and large-scale system of a nonlinear problem has been in the reduction of the computational cost, memory requirement, CPU time as well as convergence properties. Extensive computational experiments have been carried out using various benchmark problems to demonstrate the impact of the proposed methods compared with other variant of NM and CG methods respectively. The comparison is based on the number of iteration, processing time and the residual norm at the stopping point F(xk) . The residual norm indicates how close the approximate solution is, to the exact solution. The results shows that all the new methods outperformance the classical NM, CG and some of their variant in term of storage, computational cost and CPU time. In addition, the convergence properties of all the proposed methods have been proved. Dissertations, Academic Iterative Numerical Methods Rational Approximation Techniques Solving Nonlinear Problems Thesis |
| spellingShingle | 2018_A Family of Iterative Methods Via Modified Rational Approximation Model For Solving Nonlinear Problems |
| state | Terengganu |
| subject | Nonlinear equations—Numerical solutions Dissertations, Academic |
| summary | Nonlinear problems emerge in some areas of science, engineering, social science, economics and art when designing various kind of events. This type of problems are reduced to solve a system of nonlinear equations (SNE) or nonlinear unconstrained optimization problems(NUO) or least square problems. All the problems are equivalent under fairly reasonable hypothesis, but they are not treat with the same algorithm. The famous and well known method for solving SNE is the Newton method (NM). Whereas, the Conjugate Gradient (CG) method is largely considered for solving NUO. Both methods are easy to implement and converges rapidly. Nevertheless, an iteration of NM turns out to be computational expensive, this is due to the fact that, the method requires finding and storage the n n Jacobian matrix which involves the first derivatives of the system and solving the Newton equation. In addition the convergence may even be lost when the Jacobian is singular at the solution. These draw back is more noticeable when handling large scale SNE. This lead to the first idea of this thesis. The CG search direction with PRP parameter is very efficient in term of the numerical performance, but it fails in term of the global convergence for the general function under Wolfe line search technique. In addition the trust region and sufficient properties of the search direction could not be established for this method. This lead to the second idea of this thesis. This study proposed several derivative-free Newton-like methods via a modified rational approximation model and various three-term CG methods using a new conjugate parameter based on a Wolfe line search. In particular, a famous Jacobian approximation such as the Broyden update (BM), Broyden, Fletcher, Goldfarb and Shannon update (BFGS) and the inverse diagonal Newton-like update (IDN). And a new derivative free rational approximation model were used to developed each method for solving SNE. The advantage of this novel approach when applied to both small and large-scale system of a nonlinear problem has been in the reduction of the computational cost, memory requirement, CPU time as well as convergence properties. Extensive computational experiments have been carried out using various benchmark problems to demonstrate the impact of the proposed methods compared with other variant of NM and CG methods respectively. The comparison is based on the number of iteration, processing time and the residual norm at the stopping point F(xk) . The residual norm indicates how close the approximate solution is, to the exact solution. The results shows that all the new methods outperformance the classical NM, CG and some of their variant in term of storage, computational cost and CPU time. In addition, the convergence properties of all the proposed methods have been proved. |
| title | 2018_A Family of Iterative Methods Via Modified Rational Approximation Model For Solving Nonlinear Problems |
| title_full | 2018_A Family of Iterative Methods Via Modified Rational Approximation Model For Solving Nonlinear Problems |
| title_fullStr | 2018_A Family of Iterative Methods Via Modified Rational Approximation Model For Solving Nonlinear Problems |
| title_full_unstemmed | 2018_A Family of Iterative Methods Via Modified Rational Approximation Model For Solving Nonlinear Problems |
| title_short | 2018_A Family of Iterative Methods Via Modified Rational Approximation Model For Solving Nonlinear Problems |
| title_sort | 2018_a family of iterative methods via modified rational approximation model for solving nonlinear problems |