Research Article
BibTex RIS Cite

Newton Tabanlı Kök Bulma Yöntemleri İçin Simülatör Tasarımı

Year 2015, Volume: 19 Issue: 3, 327 - 337, 11.12.2015

Abstract

Mühendislik problemlerinin birçoğunda denklemlerin köklerinin hesaplanması gerekmektedir. Bunun için birçok yöntemler geliştirilmiştir. Ancak, özellikle gerçek zamanlı uygulamalarda köklerin en az işlemle, en kısa sürede, yüksek hassasiyetle bulunması istenen başlıca özelliktir. Gerçekleştirilen çalışmada, Newton tabanlı 42 yöntemi barındıran grafiksel arayüz programı geliştirilmiştir. Kullanıcı dostu ve eğitim amaçlı da kullanılabilecek simülatörde tanımlanan/girilen denklemlerin, belirtilen aralıkta ve istenen hassasiyette kökleri hesaplanabilmekte; köke yakınsama adımları (iterasyonları) hem sayısal hem de grafiksel (animasyonlu veya animasyonsuz) olarak görülebilmekte, yöntemlerle ilgili konu anlatımları sunulmaktadır. Ayrıca yöntemlerin performans analizleri (iterasyon sayısı, bulunan kök, hesaplama süresi)  de karşılaştırmalı olarak yapılabilmektedir. Böylece simülatör ile kullanıcılar farklı yöntemlerle kök bulma işlemlerini karşılaştırmalı olarak gerçekleştirebilmekte; öğrenciler bu alandaki yöntemleri görsel olarak öğrenip uygulayabilmekte; tasarımcılar sistemleri için performans açısından en uygun yöntemi kolaylıkla, etkin ve verimli bir şekilde seçebilmektedirler.

References

  • «List of numerical analysis software,» 11 Mart 2015. [Çevrimiçi]. Available: http://en.wikipedia.org/wiki/List_of_numerical_analysis_software.
  • «Comparison of numerical analysis software,» 11 Mart 2015. [Çevrimiçi]. Available: http://en.wikipedia.org/wiki/Comparison_of_numerical_analysis_software.
  • «Keisan online calculator,» 11 Mart 2015. [Çevrimiçi]. Available: http://keisan.casio.com/menu/system/000000000980.
  • «Numerical analysis tools,» 11 Mart 2015. [Çevrimiçi]. Available: https://play.google.com/store/apps/details?id=com.ay0w.rootstati0n.natools.
  • «Practical Numerical Methods with Python,» 11 Mart 2015. [Çevrimiçi]. Available: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about.
  • «Mathematical Visualization Toolkit,» 11 MArt 2015. [Çevrimiçi]. Available: http://amath.colorado.edu/java/.
  • A. B. Hassan, M. S. Abolarin ve O. H. Jimoh, «The Application of Visual Basic Programming Language to Simulate Numerical Iterations,» Leonardo Journal of Sciences, no. 9, pp. 125-136, 2006.
  • P. Wlodkowski, «Teaching Numerical Methods in Engineering with Mathcad,» American Mathematical Society for Engineering Education, no. 2006-1549, 2006.
  • S. Yüncü ve C. Aslan, «Nümerik Yöntemlerde Hata Analizi ve Bir Nümerik Çözüm Paketinin Hazırlanması,» Gazi Üniv. Müh. Mim. Fak. Der., cilt 17, no. 2, pp. 87-102, 2002.
  • J. Carroll, «The Role of Computer Software in Numerical Analysis Teaching,» ACM SIGNUM Newsletter, cilt 27, no. 2, pp. 2-31, 1992.
  • C. Balsa, L. Alves, M. J. Pereira, P. J. Rodrigues ve R. P. Lopes, «Graphical Simulation of Numerical Algorithms - An Aproach based on Code Instrumentation and Java Technologies,» %1 içinde CSEDU, Porto, 2012.
  • R. L. Burden ve J. D. Faires, Numerical Analysis, Canada: Brooks/Cole Cengage Learning, 2011.
  • A. Gilat ve V. Subramaniam, Numerical Methods for Engineers and Scientists, USA: Wiley, 2014.
  • F. Vatansever, İleri Programlama Uygulamaları, Ankara: Seçkin Yayıncılık, 2006.
  • J. M. Ortega ve W. G. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, New York: Academic Press, 1970.
  • J. F. Traub, Iterative Methods for the Solution of Equations, New Jersey: Prentice-Hall, 1964.
  • S. D. Conte ve C. de Boor, Elementary Numerical Analysis, An Algorithmic Approach, McGraw-Hill, 1980.
  • I. K. Argyros, «A note on the Halley method in Banach spaces,» Appl. Math. Comput., cilt 58, pp. 215-224, 1993.
  • J. M. Gutiérrez ve M. A. Hernández, «An acceleration of Newton’s method: Super–Halley method,» Appl. Math. Comput., cilt 117, pp. 223-239, 2001.
  • A. B. Kasturiarachi, «Leap-frogging Newton’s method,» Int. J. Math. Education. Sci. Technol., cilt 33, no. 4, pp. 521-527, 2002.
  • J. R. Sharma, «A composite third order Newton–Steffensen method for solving nonlinear equations,» Appl. Math. Comput., cilt 169, pp. 242-246, 2005.
  • H. Ren, Q. Wu ve W. Bi, «A class of two-step Steffensen type methods with fourth-order convergence,» Appl. Math. Comput., cilt 209, pp. 206-210, 2009.
  • S. Weerakoon ve T. G. I. Fernando, «A variant of Newton’s method with accelerated third-order convergence,» Appl. Math. Lett., cilt 13, pp. 87-93, 2000.
  • M. Frontini ve E. Sormani, «Some variant of Newton’s method with third-order convergence,» Appl. Math. Comput., cilt 140, pp. 419-426, 2003.
  • H. H. H. Homeier, «A modified Newton method for root finding with cubic convergence,» J. Comput. Appl. Math., cilt 157, pp. 227-230, 2003.
  • A. Y. Özban, «Some new variants of Newton’s method,» Appl. Math. Lett., cilt 17, pp. 677-682, 2004.
  • H. H. H. Homeier, «On Newton-type methods with cubic convergence,» J. Comput. Appl. Math., cilt 176, pp. 425-432, 2005.
  • K. Jisheng, L. Yitian ve W. Xiuhua, «Third-order modification of Newton’s method,» J. Comput. Appl. Math., cilt 205, pp. 1-5, 2007.
  • O. Y. Ababneh, «New Newton’s method with third-order convergence for solving nonlinear equations,» Int. Scholarly and Scientific Research & Innovation, cilt 6, pp. 1269-1271, 2012.
  • T. Lukić ve N. M. Ralević, «Geometric mean Newton’s method for simple and multiple roots,» Appl. Math. Lett., cilt 21, pp. 30-36, 2008.
  • G. Nedzhibov, «On a few iterative methods for solving nonlinear equations,» %1 içinde Application of Mathematics in Engineering and Economics’28, in: Proceedings of the XXVIII Summer School Sozopol’ 2002, Sofia, 2002.
  • V. I. Hasanov, I. G. Ivanov ve G. Nedzhibov, «A new modification of Newton's method,» Appl. Math. Eng., cilt 27, pp. 278-286, 2002.
  • A. Cordero ve J. R. Torregrosa, «Variants of Newton’s method using fifth-order quadrature formulas,» Appl. Math. Comput., cilt 190, pp. 686-698, 2007.
  • P. Jain, «Steffensen type methods for solving non-linear equations,» Appl. Math. Comput., cilt 194, pp. 527-533, 2007.
  • V. Pták ve F. A. Potra, Nondiscrete Induction and Iterative Processes, Pitman, Boston: Chapman & Hall / CRC Research Notes in Mathematics Series, vol. 103, 1984.
  • J. Kou, Y. Li ve X. Wang, «A modification of Newton method with third-order convergence,» Appl. Math. Comput., cilt 181, pp. 1106-1111, 2006.
  • J. Kou ve Y. Li, «Modified Chebyshev’s method free from second derivative for non-linear equations,» Applied Mathematics and Computation, cilt 187, p. 1027–1032, 2007.
  • G. Ardelean, «A new third-order newton-type iterative method for solving nonlinear equations,» Appl. Math. Comput., cilt 219, p. 9856–9864, 2013.
  • A. M. Ostrowski, Solutions of Equations and System of Equations, New York: Academic Press, 1966.
  • I. K. Argyros, D. Chen ve Q. Qian, «The Jarratt method in Banach space setting,» J. Comput. Appl. Math., cilt 51, pp. 103-106, 1994.
  • K. Jisheng, L. Yitian ve W. Xiuhua, «A composite fourth-order iterative method for solving non-linear equations,» Appl. Math. Comput., cilt 184, pp. 471-475, 2007.
  • X. Y. Wu, «A new continuation Newton-like method and its deformation,» Appl. Math. Comput., cilt 112, pp. 75-78, 2000.
  • P. Wang, «A third-order family of Newton-like iteration methods for solving nonlinear equations,» J. Numer. Math. Stoch., cilt 3, pp. 13-19, 2011.
  • J. Jayakumarand ve M. Kalyanasundaram, «Modified Newton's method using harmonic mean for solving nonlinear equations,» IOSR J. Math., cilt 7, pp. 93-97, 2013.
  • T. J. McDougall ve S. J. Wotherspoon, «A simple modification of Newton’s method to achieve convergence of order 1+√2,» Appl. Math. Lett., cilt 29, pp. 20-25, 2014.
  • A. K. Maheshwari, «A fourth order iterative method for solving nonlinear equations,» Appl. Math. Comput., cilt 211, pp. 383-391, 2009.
  • M. Dehghan ve M. Hajarian, «Fourth-order variants of Newton’s method without second derivatives for solving non-linear equations,» Engineering Computations: Int.J. for Computer-Aided Engineering and Software, cilt 29, no. 4, pp. 356-365, 2012.
  • R. F. King, «A family of fourth order methods for nonlinear equations,» SIAM J. Numer. Anal., cilt 10, pp. 876-879, 1973.
  • J. Kou, «The improvements of modified Newton’s method,» Appl. Math. Comput., cilt 189, pp. 602-609, 2007.
  • J. Kou, Y. Li ve X. Wang, «Some modifications of Newton’s method with fifth-order convergence,» J. Comput. Appl. Math., cilt 209, pp. 146-152, 2007.
  • M. K. Singh ve S. R. Singh, «Six-order modification of Newton’s method for solving nonlinear equations,» International Journal of Computational Cognition, cilt 9, pp. 66-71, 2011.
  • Mathworks, MATLAB, www.mathworks.com, 2007.

Design the Simulator for Root-finding based on Newton's Methods

Year 2015, Volume: 19 Issue: 3, 327 - 337, 11.12.2015

Abstract

In most of the engineering problems, the calculation of the roots of the equations is required. Many methods have been developed for this purpose. However, especially to obtain the roots in real-time applications with a minimum operation in minimum time, and having high accuracy are main properties. In the performed study, a graphical user interface program that contains 42 Newton-based methods is developed. In the user friendly simulator which can be used also for educational purposes, the roots of the defined/entered equations can be calculated within the specified range with a desired precision; the convergence steps (iterations) to the root can be seen as both numerical and graphical (animated or non-animated), the descriptions of subjects related to the methods are presented. Also, the performance analyzes (the iteration number, the obtained root, the computation time) of the methods can be performed comparatively. Thus, the users can perform the root-finding operations with different methods comparatively by the simulator; the students can learn and apply the methods visually in this field; the designers can choose the most appropriate method in terms of performance easily, effectively and efficiently for their systems.

References

  • «List of numerical analysis software,» 11 Mart 2015. [Çevrimiçi]. Available: http://en.wikipedia.org/wiki/List_of_numerical_analysis_software.
  • «Comparison of numerical analysis software,» 11 Mart 2015. [Çevrimiçi]. Available: http://en.wikipedia.org/wiki/Comparison_of_numerical_analysis_software.
  • «Keisan online calculator,» 11 Mart 2015. [Çevrimiçi]. Available: http://keisan.casio.com/menu/system/000000000980.
  • «Numerical analysis tools,» 11 Mart 2015. [Çevrimiçi]. Available: https://play.google.com/store/apps/details?id=com.ay0w.rootstati0n.natools.
  • «Practical Numerical Methods with Python,» 11 Mart 2015. [Çevrimiçi]. Available: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about.
  • «Mathematical Visualization Toolkit,» 11 MArt 2015. [Çevrimiçi]. Available: http://amath.colorado.edu/java/.
  • A. B. Hassan, M. S. Abolarin ve O. H. Jimoh, «The Application of Visual Basic Programming Language to Simulate Numerical Iterations,» Leonardo Journal of Sciences, no. 9, pp. 125-136, 2006.
  • P. Wlodkowski, «Teaching Numerical Methods in Engineering with Mathcad,» American Mathematical Society for Engineering Education, no. 2006-1549, 2006.
  • S. Yüncü ve C. Aslan, «Nümerik Yöntemlerde Hata Analizi ve Bir Nümerik Çözüm Paketinin Hazırlanması,» Gazi Üniv. Müh. Mim. Fak. Der., cilt 17, no. 2, pp. 87-102, 2002.
  • J. Carroll, «The Role of Computer Software in Numerical Analysis Teaching,» ACM SIGNUM Newsletter, cilt 27, no. 2, pp. 2-31, 1992.
  • C. Balsa, L. Alves, M. J. Pereira, P. J. Rodrigues ve R. P. Lopes, «Graphical Simulation of Numerical Algorithms - An Aproach based on Code Instrumentation and Java Technologies,» %1 içinde CSEDU, Porto, 2012.
  • R. L. Burden ve J. D. Faires, Numerical Analysis, Canada: Brooks/Cole Cengage Learning, 2011.
  • A. Gilat ve V. Subramaniam, Numerical Methods for Engineers and Scientists, USA: Wiley, 2014.
  • F. Vatansever, İleri Programlama Uygulamaları, Ankara: Seçkin Yayıncılık, 2006.
  • J. M. Ortega ve W. G. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, New York: Academic Press, 1970.
  • J. F. Traub, Iterative Methods for the Solution of Equations, New Jersey: Prentice-Hall, 1964.
  • S. D. Conte ve C. de Boor, Elementary Numerical Analysis, An Algorithmic Approach, McGraw-Hill, 1980.
  • I. K. Argyros, «A note on the Halley method in Banach spaces,» Appl. Math. Comput., cilt 58, pp. 215-224, 1993.
  • J. M. Gutiérrez ve M. A. Hernández, «An acceleration of Newton’s method: Super–Halley method,» Appl. Math. Comput., cilt 117, pp. 223-239, 2001.
  • A. B. Kasturiarachi, «Leap-frogging Newton’s method,» Int. J. Math. Education. Sci. Technol., cilt 33, no. 4, pp. 521-527, 2002.
  • J. R. Sharma, «A composite third order Newton–Steffensen method for solving nonlinear equations,» Appl. Math. Comput., cilt 169, pp. 242-246, 2005.
  • H. Ren, Q. Wu ve W. Bi, «A class of two-step Steffensen type methods with fourth-order convergence,» Appl. Math. Comput., cilt 209, pp. 206-210, 2009.
  • S. Weerakoon ve T. G. I. Fernando, «A variant of Newton’s method with accelerated third-order convergence,» Appl. Math. Lett., cilt 13, pp. 87-93, 2000.
  • M. Frontini ve E. Sormani, «Some variant of Newton’s method with third-order convergence,» Appl. Math. Comput., cilt 140, pp. 419-426, 2003.
  • H. H. H. Homeier, «A modified Newton method for root finding with cubic convergence,» J. Comput. Appl. Math., cilt 157, pp. 227-230, 2003.
  • A. Y. Özban, «Some new variants of Newton’s method,» Appl. Math. Lett., cilt 17, pp. 677-682, 2004.
  • H. H. H. Homeier, «On Newton-type methods with cubic convergence,» J. Comput. Appl. Math., cilt 176, pp. 425-432, 2005.
  • K. Jisheng, L. Yitian ve W. Xiuhua, «Third-order modification of Newton’s method,» J. Comput. Appl. Math., cilt 205, pp. 1-5, 2007.
  • O. Y. Ababneh, «New Newton’s method with third-order convergence for solving nonlinear equations,» Int. Scholarly and Scientific Research & Innovation, cilt 6, pp. 1269-1271, 2012.
  • T. Lukić ve N. M. Ralević, «Geometric mean Newton’s method for simple and multiple roots,» Appl. Math. Lett., cilt 21, pp. 30-36, 2008.
  • G. Nedzhibov, «On a few iterative methods for solving nonlinear equations,» %1 içinde Application of Mathematics in Engineering and Economics’28, in: Proceedings of the XXVIII Summer School Sozopol’ 2002, Sofia, 2002.
  • V. I. Hasanov, I. G. Ivanov ve G. Nedzhibov, «A new modification of Newton's method,» Appl. Math. Eng., cilt 27, pp. 278-286, 2002.
  • A. Cordero ve J. R. Torregrosa, «Variants of Newton’s method using fifth-order quadrature formulas,» Appl. Math. Comput., cilt 190, pp. 686-698, 2007.
  • P. Jain, «Steffensen type methods for solving non-linear equations,» Appl. Math. Comput., cilt 194, pp. 527-533, 2007.
  • V. Pták ve F. A. Potra, Nondiscrete Induction and Iterative Processes, Pitman, Boston: Chapman & Hall / CRC Research Notes in Mathematics Series, vol. 103, 1984.
  • J. Kou, Y. Li ve X. Wang, «A modification of Newton method with third-order convergence,» Appl. Math. Comput., cilt 181, pp. 1106-1111, 2006.
  • J. Kou ve Y. Li, «Modified Chebyshev’s method free from second derivative for non-linear equations,» Applied Mathematics and Computation, cilt 187, p. 1027–1032, 2007.
  • G. Ardelean, «A new third-order newton-type iterative method for solving nonlinear equations,» Appl. Math. Comput., cilt 219, p. 9856–9864, 2013.
  • A. M. Ostrowski, Solutions of Equations and System of Equations, New York: Academic Press, 1966.
  • I. K. Argyros, D. Chen ve Q. Qian, «The Jarratt method in Banach space setting,» J. Comput. Appl. Math., cilt 51, pp. 103-106, 1994.
  • K. Jisheng, L. Yitian ve W. Xiuhua, «A composite fourth-order iterative method for solving non-linear equations,» Appl. Math. Comput., cilt 184, pp. 471-475, 2007.
  • X. Y. Wu, «A new continuation Newton-like method and its deformation,» Appl. Math. Comput., cilt 112, pp. 75-78, 2000.
  • P. Wang, «A third-order family of Newton-like iteration methods for solving nonlinear equations,» J. Numer. Math. Stoch., cilt 3, pp. 13-19, 2011.
  • J. Jayakumarand ve M. Kalyanasundaram, «Modified Newton's method using harmonic mean for solving nonlinear equations,» IOSR J. Math., cilt 7, pp. 93-97, 2013.
  • T. J. McDougall ve S. J. Wotherspoon, «A simple modification of Newton’s method to achieve convergence of order 1+√2,» Appl. Math. Lett., cilt 29, pp. 20-25, 2014.
  • A. K. Maheshwari, «A fourth order iterative method for solving nonlinear equations,» Appl. Math. Comput., cilt 211, pp. 383-391, 2009.
  • M. Dehghan ve M. Hajarian, «Fourth-order variants of Newton’s method without second derivatives for solving non-linear equations,» Engineering Computations: Int.J. for Computer-Aided Engineering and Software, cilt 29, no. 4, pp. 356-365, 2012.
  • R. F. King, «A family of fourth order methods for nonlinear equations,» SIAM J. Numer. Anal., cilt 10, pp. 876-879, 1973.
  • J. Kou, «The improvements of modified Newton’s method,» Appl. Math. Comput., cilt 189, pp. 602-609, 2007.
  • J. Kou, Y. Li ve X. Wang, «Some modifications of Newton’s method with fifth-order convergence,» J. Comput. Appl. Math., cilt 209, pp. 146-152, 2007.
  • M. K. Singh ve S. R. Singh, «Six-order modification of Newton’s method for solving nonlinear equations,» International Journal of Computational Cognition, cilt 9, pp. 66-71, 2011.
  • Mathworks, MATLAB, www.mathworks.com, 2007.
There are 52 citations in total.

Details

Primary Language Turkish
Subjects Engineering
Journal Section Research Articles
Authors

Fahri Vatansever

Metin Hatun This is me

Publication Date December 11, 2015
Submission Date March 25, 2015
Acceptance Date May 12, 2015
Published in Issue Year 2015 Volume: 19 Issue: 3

Cite

APA Vatansever, F., & Hatun, M. (2015). Newton Tabanlı Kök Bulma Yöntemleri İçin Simülatör Tasarımı. Sakarya University Journal of Science, 19(3), 327-337.
AMA Vatansever F, Hatun M. Newton Tabanlı Kök Bulma Yöntemleri İçin Simülatör Tasarımı. SAUJS. December 2015;19(3):327-337.
Chicago Vatansever, Fahri, and Metin Hatun. “Newton Tabanlı Kök Bulma Yöntemleri İçin Simülatör Tasarımı”. Sakarya University Journal of Science 19, no. 3 (December 2015): 327-37.
EndNote Vatansever F, Hatun M (December 1, 2015) Newton Tabanlı Kök Bulma Yöntemleri İçin Simülatör Tasarımı. Sakarya University Journal of Science 19 3 327–337.
IEEE F. Vatansever and M. Hatun, “Newton Tabanlı Kök Bulma Yöntemleri İçin Simülatör Tasarımı”, SAUJS, vol. 19, no. 3, pp. 327–337, 2015.
ISNAD Vatansever, Fahri - Hatun, Metin. “Newton Tabanlı Kök Bulma Yöntemleri İçin Simülatör Tasarımı”. Sakarya University Journal of Science 19/3 (December 2015), 327-337.
JAMA Vatansever F, Hatun M. Newton Tabanlı Kök Bulma Yöntemleri İçin Simülatör Tasarımı. SAUJS. 2015;19:327–337.
MLA Vatansever, Fahri and Metin Hatun. “Newton Tabanlı Kök Bulma Yöntemleri İçin Simülatör Tasarımı”. Sakarya University Journal of Science, vol. 19, no. 3, 2015, pp. 327-3.
Vancouver Vatansever F, Hatun M. Newton Tabanlı Kök Bulma Yöntemleri İçin Simülatör Tasarımı. SAUJS. 2015;19(3):327-3.