본문 바로가기
KERD
논문KCI 등재

EXPANDING THE APPLICABILITY OF SECANT METHOD WITH APPLICATIONS

Ioannis K. Argyros|A. Alberto Magre˜n´an
대한수학회 (The Korean Mathematical Society)|2015.05|대한수학회보 (Bulletin of the KMS)|제52권 제3호|pp. 865-880

2

조회

0

다운로드

0

인용

0

참고문헌

키워드

Banach spacemajorizing sequencedivided differencesecant methodFr´echet–derivative
주제분류수학

초록

We present new sufficient convergence criteria for the convergence of the secant-method to a locally unique solution of a nonlinear equation in a Banach space. Our idea uses Lipschitz and center–Lipschitz instead of just Lipschitz conditions in the convergence analysis. The new convergence criteria can always be weaker than the corresponding ones in earlier studies. Numerical examples are also provided in this study to solve equations in cases not possible before.

Abstract

We present new sufficient convergence criteria for the convergence of the secant-method to a locally unique solution of a nonlinear equation in a Banach space. Our idea uses Lipschitz and center–Lipschitz instead of just Lipschitz conditions in the convergence analysis. The new convergence criteria can always be weaker than the corresponding ones in earlier studies. Numerical examples are also provided in this study to solve equations in cases not possible before.

참고문헌 (28)

  1. S. Amat, "On the local convergence of secant-type methods", Int. J. Comput. Math, 81(9), pp.1153~1161 (2004)
  2. S. Amat, "Reducing chaos and bifurcations in Newtontype methods", Abstr. Appl. Anal, 2013, pp.10~ (2013)
  3. S. Amat, "Adaptive approximation of nonlinear operators", Numer. Funct. Anal. Optim, 25(5-6), pp.397~405 (2004)
  4. S. Amat, "On a two step relaxed Newton-type methods", Appl. Math. Comput, 219(24), pp.11341~11347 (2013)
  5. I. K. Argyros, "A unifying local–semilocal convergence analysis and applications for two–point Newton–like methods in Banach space", J. Math. Anal. Appl, 298(2), pp.374~397 (2004)
  6. I. K. Argyros, "New sufficient convergence conditions for the Secant method", Chechoslovak Math. J, 55(130)(1), pp.175~187 (2005)
  7. I. K. Argyros, "Convergence and Applications of Newton–Type Iterations" (2008), Springer–Verlag Publ
  8. I. K. Argyros, "Weaker conditions for the convergence of Newton’s method", J. Complexity, 28(3), pp.364~387 (2012)
  9. I. K. Argyros, "A semilocal convergence for a uniparametric family of efficient secant-like methods", J. Function Spaces, 2014, pp.10~ (2014)[DOI]
  10. W. E. Bosarge, "A multipoint method of third order", J. Optimization Theory Appl, 4, pp.156~166 (1969)
  11. J. E. Dennis, "Toward a unified convergence theory for Newton–like methods", Nonlinear Functional Anal. and Appl.(Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin), pp.425~472 (1971), Academic Press
  12. M. Garc´ıa-Olivo, "El m´etodo de Chebyshev para el c´alculo de las ra´ıces de ecuaciones no lineales", PhD, pp.2013
  13. J. M. Guti´errez, "On the semilocal convergence of Newton-Kantorovich method under center-Lipschitz conditions", Appl. Math. Comput, 221, pp.79~88 (2013)
  14. M. A. Hern´andez, "Secant–like methods for solving nonlinear integral equations of the Hammerstein type", J. Comput. Appl. Math, 115, pp.245~254 (2000)
  15. M. A. Hern´andez, "Solving a special case of conservative problems by Secant–like method", Appl. Math. Comput, 169(2), pp.926~942 (2005)
  16. L. V. Kantorovich, "Functional Analysis" (1982), Pergamon Press
  17. P. Laasonen, "Ein ¨uberquadratisch konvergenter iterativer algorithmus", Ann. Acad. Sci Fenn. Ser I, 450, pp.1~10 (1969)
  18. A. A. Magre˜n´an, "Estudio de la din´amica del m´etodo de Newton amortiguado", PhD, pp.2013
  19. J. M. Ortega, "Iterative Solution of Nonlinear Equations in Several Variables" (1970), Academic Press
  20. A. M. Ostrowski, "Solution of equations in Euclidian and Banach Spaces" (1972), Academic Press
  21. F. A. Potra, "An error analysis for the secant method", Numer. Math, 38(3), pp.427~445 (1982)
  22. F. A. Potra, "Sharp error bounds for a class of Newton–like methods", Libertas Math, 5, pp.71~84 (1985)
  23. F. A. Potra, "Nondiscrete Induction and Iterative Processes" (1984), Pitman
  24. J. W. Schmidt, "Untere Fehlerschranken f¨ur Regula–Falsi Verhafren", Period. Math. Hungar, 9(3), pp.241~247 (1978)
  25. A. S. Sergeev, "The method of Chords", Sibirsk, Mat. ˇ Z, 11, pp.282~289 (1961)
  26. S. Ulm, "A majorant principle and the method of secants", Izv. Akad. Nauk Eston. SSR, Ser. Fiz.-Mat, 13, pp.217~227 (1964)
  27. T. Yamamoto, "A convergence theorem for Newton–like methods in Banach spaces", Numer. Math, 51(5), pp.545~557 (1987)
  28. M. A. Wolfe, "Extended iterative methods for the solution of operator equations", Numer. Math, 31(2), pp.153~174 (1978)

관련 논문AI추천

A UNIFIED CONVERGENCE ANALYSIS FOR SECANT-TYPE METHODSAngel Alberto Magrenan, Ioannis Konstantinos Argyros대한수학회 · 대한수학회지2014 · 제51권 제6호On the solution of nonlinear equations containing a non-differentiable termIoannis K. Argyros영남수학회 · East Asian Mathematical Journal2008 · 제24권 제3호Enlarging the ball of convergence of secant-like methods for non-differentiable operatorsHongmin Ren, IOANNISK.ARGYROS대한수학회 · 대한수학회지2018 · 제55권 제1호APPROXIMATING SOLUTIONS OF EQUATIONSBY COMBINING NEWTON-LIKE METHODSIoannis K. Argyros한국수학교육학회 · 이론수학과 교직수학2008 · 제15권 제1호LOCAL CONVERGENCE OF THE SECANT METHOD UPPER HOLDER CONTINUOUS DIVIDED DIFFERENCESIoannis K. Argyros영남수학회 · East Asian Mathematical Journal2008 · 제24권 제1호CONVERGENCE THEOREMS FOR NEWTON'S AND MODIFIED NEWTON'S METHODSIoannis K. Argyros한국수학교육학회 · 이론수학과 교직수학2009 · 제16권 제4호

이 간행물의 다른 논문 (28건)

THE FUNDAMENTAL THEOREM OF ALGEBRA AND THE DIVERGENCE OF RECIPROCALS OF PRIMES LOOKED AT THROUGH BERGMAN SPACESYunus E. Zeytuncupp. 699-705GENERALIZATION OF INEQUALITIES ANALOGOUS TO HERMITE–HADAMARD INEQUALITY VIA FRACTIONAL INTEGRALSKiran Nazeer, Muhammad Iqbal, Muhammad Iqbal Bhattipp. 707-716CLASSIFICATION ON ARITHMETIC FUNCTIONS AND CORRESPONDING FREE-MOMENT L-FUNCTIONS조일우pp. 717-734AN ESTIMATE OF HEMPEL DISTANCE FOR BRIDGE SPHERESAyako Idopp. 735-740THE n-TH TWISTED CHANGHEE POLYNOMIALS AND NUMBERS권종겸, 표성수, 박진우, 임석훈pp. 741-749
원문 보기