Derivative free Davidon-Fletcher-Powell (DFP) for solving symmetric systems of nonlinear equations
Research from the work of engineers, economist, modelling, industry, computing, and scientist are mostly nonlinear equations in nature. Numerical solution to such systems is widely applied in those areas of mathematics. Over the years, there has been significant theoretical study to develop method...
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my-unisza-ir.17232020-11-22T03:20:14Z http://eprints.unisza.edu.my/1723/ Derivative free Davidon-Fletcher-Powell (DFP) for solving symmetric systems of nonlinear equations Mamat, M. Dauda, M.K. Mohamed, M.A. Waziri, M.Y. Mohamad, F.S. Abdullah, H. QA Mathematics QA75 Electronic computers. Computer science Research from the work of engineers, economist, modelling, industry, computing, and scientist are mostly nonlinear equations in nature. Numerical solution to such systems is widely applied in those areas of mathematics. Over the years, there has been significant theoretical study to develop methods for solving such systems, despite these efforts, unfortunately the methods developed do have deficiency. In a contribution to solve systems of the form F (x) = 0, x is an element of R-n, a derivative free method via the classical Davidon-Fletcher-Powell (DFP) update is presented. This is achieved by simply approximating the inverse Hessian matrix with Q(k+1)(-1) to theta I-k. The modified method satisfied the descent condition and possess local superlinear convergence properties. Interestingly, without computing any derivative, the proposed method never fail to converge throughout the numerical experiments. The output is based on number of iterations and CPU time, different initial starting points were used on a solve 40 benchmark test problems. With the aid of the squared norm merit function and derivative-free line search technique, the approach yield a method of solving symmetric systems of nonlinear equations that is capable of significantly reducing the CPU time and number of iteration, as compared to its counterparts. A comparison between the proposed method and classical DFP update were made and found that the proposed methodis the top performer and outperformed the existing method in almost all the cases. In terms of number of iterations, out of the 40 problems solved, the proposed method solved 38 successfully, (95%) while classical DFP solved 2 problems (i.e. 05%). In terms of CPU time, the proposed method solved 29 out of the 40 problems given, (i.e. 72.5%) successfully whereas classical DFP solves 11 (27.5%). The method is valid in terms of derivation, reliable in terms of number of iterations and accurate in terms of CPU time. Thus, suitable and achived the objective. 2018 Conference or Workshop Item NonPeerReviewed image en http://eprints.unisza.edu.my/1723/1/FH03-FIK-18-13683.jpg text en http://eprints.unisza.edu.my/1723/2/FH03-FIK-19-23937.pdf Mamat, M. and Dauda, M.K. and Mohamed, M.A. and Waziri, M.Y. and Mohamad, F.S. and Abdullah, H. (2018) Derivative free Davidon-Fletcher-Powell (DFP) for solving symmetric systems of nonlinear equations. In: International Conference on Operations Research 2017, 12 Oct 2017, Tangerang Selatan, Indonesia. |
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QA Mathematics QA75 Electronic computers. Computer science Mamat, M. Dauda, M.K. Mohamed, M.A. Waziri, M.Y. Mohamad, F.S. Abdullah, H. Derivative free Davidon-Fletcher-Powell (DFP) for solving symmetric systems of nonlinear equations |
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Research from the work of engineers, economist, modelling, industry, computing, and scientist are mostly nonlinear
equations in nature. Numerical solution to such systems is widely applied in those areas of mathematics. Over the
years, there has been significant theoretical study to develop methods for solving such systems, despite these efforts,
unfortunately the methods developed do have deficiency. In a contribution to solve systems of the form F (x) = 0, x is
an element of R-n, a derivative free method via the classical Davidon-Fletcher-Powell (DFP) update is presented. This
is achieved by simply approximating the inverse Hessian matrix with Q(k+1)(-1) to theta I-k. The modified method
satisfied the descent condition and possess local superlinear convergence properties. Interestingly, without
computing any derivative, the proposed method never fail to converge throughout the numerical experiments. The
output is based on number of iterations and CPU time, different initial starting points were used on a solve 40
benchmark test problems. With the aid of the squared norm merit function and derivative-free line search technique,
the approach yield a method of solving symmetric systems of nonlinear equations that is capable of significantly
reducing the CPU time and number of iteration, as compared to its counterparts. A comparison between the proposed
method and classical DFP update were made and found that the proposed methodis the top performer and
outperformed the existing method in almost all the cases. In terms of number of iterations, out of the 40 problems
solved, the proposed method solved 38 successfully, (95%) while classical DFP solved 2 problems (i.e. 05%). In terms
of CPU time, the proposed method solved 29 out of the 40 problems given, (i.e. 72.5%) successfully whereas classical
DFP solves 11 (27.5%). The method is valid in terms of derivation, reliable in terms of number of iterations and
accurate in terms of CPU time. Thus, suitable and achived the objective. |
format |
Conference or Workshop Item |
author |
Mamat, M. Dauda, M.K. Mohamed, M.A. Waziri, M.Y. Mohamad, F.S. Abdullah, H. |
author_facet |
Mamat, M. Dauda, M.K. Mohamed, M.A. Waziri, M.Y. Mohamad, F.S. Abdullah, H. |
author_sort |
Mamat, M. |
title |
Derivative free Davidon-Fletcher-Powell (DFP) for solving symmetric systems of nonlinear equations |
title_short |
Derivative free Davidon-Fletcher-Powell (DFP) for solving symmetric systems of nonlinear equations |
title_full |
Derivative free Davidon-Fletcher-Powell (DFP) for solving symmetric systems of nonlinear equations |
title_fullStr |
Derivative free Davidon-Fletcher-Powell (DFP) for solving symmetric systems of nonlinear equations |
title_full_unstemmed |
Derivative free Davidon-Fletcher-Powell (DFP) for solving symmetric systems of nonlinear equations |
title_sort |
derivative free davidon-fletcher-powell (dfp) for solving symmetric systems of nonlinear equations |
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2018 |
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http://eprints.unisza.edu.my/1723/1/FH03-FIK-18-13683.jpg http://eprints.unisza.edu.my/1723/2/FH03-FIK-19-23937.pdf http://eprints.unisza.edu.my/1723/ |
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