Derivative Free Conjugate Gradient Method via Broyden's Update for solving symmetric systems of nonlinear equations
The applications of mathematics in many areas of computing, scientific and engineering research mostly give rise to a systems of nonlinear equations. Various iterative methods have been developed to solve such equations, this includes Newton method, Quasi-Newton's etc. Over the years, there has...
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| Vydáno v: | Journal of physics. Conference series Ročník 1366; číslo 1; s. 12099 - 12105 |
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| Jazyk: | angličtina |
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01.11.2019
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| ISSN: | 1742-6588, 1742-6596 |
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| Abstract | The applications of mathematics in many areas of computing, scientific and engineering research mostly give rise to a systems of nonlinear equations. Various iterative methods have been developed to solve such equations, this includes Newton method, Quasi-Newton's etc. Over the years, there has been significant theoretical study on quasi-Newton methods for solving such systems, but unfortunately the methods suffers setback. To overcome such problems, a Derivative free Method for Solving Symmetric Systems of Nonlinear Equations Using Broyden's Update is presented. The modification is achieved by simply approximating the inverse Hessian matrix Qk+1−1 to QkI with (δk and I represents acceleration parameter and an identity matrix respectively) without computing any derivative. The method uses the symmetric structure of the system sufficiently and the generalized classical Broyden's update method for unconstrained optimization problems. The squared norm merit function is used, both the direction and the line search technique are derivative-free, this attractive feature of the proposed method makes it to have a very low storage requirement thereby solving large scale problems successfully. In an effort to solve nonlinear problems of the form F(x) = 0, 0, x ∈ Rn different initial starting points were used on a set of benchmark test problems, the output is based on number of iterations and CPU time. A comparison between the proposed method and the classical methods were made and found that the proposed method is efficient, robust and outperformed the existing method. |
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| AbstractList | The applications of mathematics in many areas of computing, scientific and engineering research mostly give rise to a systems of nonlinear equations. Various iterative methods have been developed to solve such equations, this includes Newton method, Quasi-Newton’s etc. Over the years, there has been significant theoretical study on quasi-Newton methods for solving such systems, but unfortunately the methods suffers setback. To overcome such problems, a Derivative free Method for Solving Symmetric Systems of Nonlinear Equations Using Broyden’s Update is presented. The modification is achieved by simply approximating the inverse Hessian matrix
Q
k
+
1
−
1
to
Q
k
I
with (
δ
k
and
I
represents acceleration parameter and an identity matrix respectively) without computing any derivative. The method uses the symmetric structure of the system sufficiently and the generalized classical Broyden’s update method for unconstrained optimization problems. The squared norm merit function is used, both the direction and the line search technique are derivative-free, this attractive feature of the proposed method makes it to have a very low storage requirement thereby solving large scale problems successfully. In an effort to solve nonlinear problems of the form
F
(
x
) = 0, 0,
x
∈
R
n
different initial starting points were used on a set of benchmark test problems, the output is based on number of iterations and CPU time. A comparison between the proposed method and the classical methods were made and found that the proposed method is efficient, robust and outperformed the existing method. The applications of mathematics in many areas of computing, scientific and engineering research mostly give rise to a systems of nonlinear equations. Various iterative methods have been developed to solve such equations, this includes Newton method, Quasi-Newton’s etc. Over the years, there has been significant theoretical study on quasi-Newton methods for solving such systems, but unfortunately the methods suffers setback. To overcome such problems, a Derivative free Method for Solving Symmetric Systems of Nonlinear Equations Using Broyden’s Update is presented. The modification is achieved by simply approximating the inverse Hessian matrix \(Q_{k + 1}^{ - 1}\) to \(Q_k^I\) with (δk and I represents acceleration parameter and an identity matrix respectively) without computing any derivative. The method uses the symmetric structure of the system sufficiently and the generalized classical Broyden’s update method for unconstrained optimization problems. The squared norm merit function is used, both the direction and the line search technique are derivative-free, this attractive feature of the proposed method makes it to have a very low storage requirement thereby solving large scale problems successfully. In an effort to solve nonlinear problems of the form F(x) = 0, 0, x ∈ Rn different initial starting points were used on a set of benchmark test problems, the output is based on number of iterations and CPU time. A comparison between the proposed method and the classical methods were made and found that the proposed method is efficient, robust and outperformed the existing method. The applications of mathematics in many areas of computing, scientific and engineering research mostly give rise to a systems of nonlinear equations. Various iterative methods have been developed to solve such equations, this includes Newton method, Quasi-Newton's etc. Over the years, there has been significant theoretical study on quasi-Newton methods for solving such systems, but unfortunately the methods suffers setback. To overcome such problems, a Derivative free Method for Solving Symmetric Systems of Nonlinear Equations Using Broyden's Update is presented. The modification is achieved by simply approximating the inverse Hessian matrix Qk+1−1 to QkI with (δk and I represents acceleration parameter and an identity matrix respectively) without computing any derivative. The method uses the symmetric structure of the system sufficiently and the generalized classical Broyden's update method for unconstrained optimization problems. The squared norm merit function is used, both the direction and the line search technique are derivative-free, this attractive feature of the proposed method makes it to have a very low storage requirement thereby solving large scale problems successfully. In an effort to solve nonlinear problems of the form F(x) = 0, 0, x ∈ Rn different initial starting points were used on a set of benchmark test problems, the output is based on number of iterations and CPU time. A comparison between the proposed method and the classical methods were made and found that the proposed method is efficient, robust and outperformed the existing method. |
| Author | Mohamad, Fatma Susilawati Waziri, M.Y. Dauda, M.K. Mamat, Mustafa Magaji, Abubakar S. |
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| Cites_doi | 10.1007/s101070100263 10.1016/j.cam.2012.10.002 10.1007/s10957-014-0547-1 10.17654/MS100111787 10.1017/S0334270000005208 10.1090/S0025-5718-1965-0198670-6 10.1063/1.4966066 10.1080/10556780008805782 10.1016/j.amc.2015.03.048 |
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| DOI | 10.1088/1742-6596/1366/1/012099 |
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| References | Li (JPCS_1366_1_012099bib1) 2000; 13 Waziri (JPCS_1366_1_012099bib5) 2013; 10 Dolan (JPCS_1366_1_012099bib2) 2002; 91 Dauda (JPCS_1366_1_012099bib6) 2016; 1782 Zhou (JPCS_1366_1_012099bib11) 2015; 164 Andrei (JPCS_1366_1_012099bib4) 2013; 241 Waziri (JPCS_1366_1_012099bib12) 2012 Griewank (JPCS_1366_1_012099bib8) 1986; 28 Broyden (JPCS_1366_1_012099bib3) 1965; 19 Liu (JPCS_1366_1_012099bib10) 2015; 4 Gaohang (JPCS_1366_1_012099bib13) 2010; 6 Nocedal (JPCS_1366_1_012099bib7) 2006 Mustafa (JPCS_1366_1_012099bib9) 2016; 100 |
| References_xml | – volume: 91 start-page: 201 year: 2002 ident: JPCS_1366_1_012099bib2 article-title: Benchmarking Optimization Software with Performance Profiles publication-title: Math. Program. Ser. A doi: 10.1007/s101070100263 – volume: 241 start-page: 19 year: 2013 ident: JPCS_1366_1_012099bib4 article-title: A simple three-term conjugate algorithm for unconstrained Optimization publication-title: Journal of Computational and Applied Mathematics doi: 10.1016/j.cam.2012.10.002 – year: 2006 ident: JPCS_1366_1_012099bib7 – volume: 164 start-page: 277 year: 2015 ident: JPCS_1366_1_012099bib11 article-title: Convergence properties of an iterative method for solving symmetric non-linear equations publication-title: Journal of Optimization Theory and Applications doi: 10.1007/s10957-014-0547-1 – start-page: 6 year: 2012 ident: JPCS_1366_1_012099bib12 article-title: A two-step matrix-free secant method for solving large-scale systems of nonlinear equations – volume: 10 start-page: 86 year: 2013 ident: JPCS_1366_1_012099bib5 article-title: Two-Step Derivative-Free Diagonally Newton’s Method for Large-Scale Nonlinear Equations publication-title: World Applied Sciences Journal 21 (Special Issue of Applied Math) – volume: 100 start-page: 1787 year: 2016 ident: JPCS_1366_1_012099bib9 article-title: Inexact CG-Method via SR1 Update for Solving Systems of Nonlinear Equations publication-title: Far East Journal of Mathematical Sciences (FJMS) doi: 10.17654/MS100111787 – volume: 28 start-page: 75 year: 1986 ident: JPCS_1366_1_012099bib8 article-title: The ‘global’ convergence of Broyden-like methods with suitable line search publication-title: Journal of Australia Mathematical Society Ser. B doi: 10.1017/S0334270000005208 – volume: 19 start-page: 577 year: 1965 ident: JPCS_1366_1_012099bib3 article-title: Class of Methods for Solving Nonlinear Simultaneous Equations publication-title: Mathematics of Computation doi: 10.1090/S0025-5718-1965-0198670-6 – volume: 6 start-page: 149 year: 2010 ident: JPCS_1366_1_012099bib13 article-title: Derivative-Free for Solving large-scale Nonlinear Systems of Equations publication-title: Journal of Management Optimization – volume: 1782 year: 2016 ident: JPCS_1366_1_012099bib6 article-title: Improved Quasi-Newton Method Via PSB Update for Solving Systems of Nonlinear equations publication-title: AIP Conference Proceedings (ICOQSIA 2016) doi: 10.1063/1.4966066 – volume: 13 start-page: 181 year: 2000 ident: JPCS_1366_1_012099bib1 article-title: Derivative-Free Line Search and Global Convergence of Broyden-Like Methods for Nonlinear Equations publication-title: Optimization Methods and Software doi: 10.1080/10556780008805782 – volume: 4 start-page: 341 year: 2015 ident: JPCS_1366_1_012099bib10 article-title: Spectral DY Type Projection Method for Nonlinear Monotone Systems of Equations publication-title: Journal of Computation of Mathematics doi: 10.1016/j.amc.2015.03.048 |
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| SubjectTerms | Applications of mathematics Computation Conjugate gradient method Engineering research Hessian matrices Iterative methods Mathematical analysis Methods Nonlinear equations Nonlinear systems Optimization Physics Quasi Newton methods Robustness (mathematics) |
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| Title | Derivative Free Conjugate Gradient Method via Broyden's Update for solving symmetric systems of nonlinear equations |
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