Limit Multiplication Conditions for Existence of Real Roots of Continuous Functions and Implications for Numerical Computation

Abstract

In engineering and applied science, conditions for existence of real roots of a function have useful implications. Solution of many problems such as optimization problems, stability analyses i.e. are based on existence and finding roots of characteristic or objective functions. This theoretical study presents limit conditions for existence of at least one real root of a continuous and differentiable function. These conditions are an elaboration of intermediate value theorem and Rolle’s theorem on the bases of limit theorem. The proposed conditions can be useful to numerical check or ensure the existence of real root solution of very complex engineering and science problems without solving the complicated equations. Computer based design and analysis tools may benefit from these conditions in solution of complicated engineering problems. 

Keywords

• Real roots,continuous functions,limit conditions,numerical methods

Sürekli Fonksiyonların Gerçek Köklerinin Olması İçin Limit Çarpma Koşulları ve Sayısal Hesaplama İçin Uygulaması

Abstract

Mühendislik ve uygulamalı bilimlerde, bir fonksiyonun gerçek köklerinin var olma koşullarının faydalı uygulamaları olur. Optimizasyon problemleri, kararlılık analizleri gibi bir çok problemin çözümü karakteristik veya objektif fonksiyonların köklerinin varlığına veya bulunabilmesine dayanır. Bu teorik çalışma, sürekli ve türevlenebilir bir fonksiyonun en azından bir gerçek kökünün varolabilmesi için sınır koşullarını sunmaktadır. Bu koşullar,  ara değer teoremi ve Rolle teoreminin limit koşullarda incelemesine dayanır. Önerilen koşullar, karmaşık mühendislik  denklemlerini çözmeden  gerçek köklerin varlığının nümerik olarak kontrol edilmesi veya garanti edilebilmesi için faydalı olabilir. Bilgisayar tabanlı tasarım ve analiz araçları karmaşık mühendislik problemlerinin çözümünde bu koşullardan yararlanabilir.

Keywords

• Reel Kökler,Sürekli Fonksiyonlar,Limit koşullar,Numerik Yöntemler

References

1. [1] C. Hewitt, "Real Roots of Univariate Polynomials with Real Coefficients",Lecture Note in Department of Mathematics, North Carolina State University, pp.1-17, 2012.
2. [2] S.B. Russ, "A Translation of Bolzano's Paper on the Intermediate Value Theorem", Historia Mathematics, Vol.7, pp.156-185, 1980.
3. [3] J. Stewart, "Calculus: Concepts and Contexts", Thomson Brooks/Cole, Belmont, CA, 3rd edition, 2006.
4. [4] N.B. Conkwright, "Introduction to the Theory of Equations", Ginn and Company, Boston, MA, 1957.
5. [5] Morris Marden, "The Search for a Rolle's Theorem in the Complex Domain", The American Mathematical Monthly, Vol. 92, No. 9, pp. 643-650, 1985.
6. [6] S.C. Chapra, R.P. Canale, "Numerical Methods for Engineers", 7th Edition McGraw-Hill Education, New York, 2015.
7. [7] E.J. Barbeau, "Polynomials": Edited by P.R. Halmoz in Problem Books in Mathematics, Springer-Verlag, New York, 1989.
8. [8] G.E. Collins, R. Loos, "Real Zeros of Polynomials", In: Buchberger B., Collins G.E., Loos R., Albrecht R. (eds) Computer Algebra. Computing Supplementa, vol 4. Springer, Vienna, 1983.

9. [9] S.M. Rump, "Ten methods to bound multiple roots of polynomials", Journal of Computational and Applied Mathematics 156, pp.403–432, 2003.
10. [10] E.M. Yambao, M. Carlota, B. Decena, "On sufficient condition for the existence of imaginary roots of a cubic polynomial equation", Acta Manilana Vol.60, pp. 15–18, 2012.
11. [11] C. Carstensen, M. Petkovic, "On iteration methods without derivatives for the simultaneous determination of polynomial zeros", Journal of Computational and Applied Mathematics Vol. 45, pp. 251–66, 1993.
12. [12] M.S. Petkovic, "A highly efficient root-solver of very fast convergence", Applied Mathematics and Computation Vol.205, pp.298–302, 2008.
13. [13] M.S. Petkovic, "The self-validated method for polynomial zeros of high efficiency", Journal of Computational and Applied Mathematics Vol.233(4), pp.1175–86, 2009.
14. [14] B. Mourrain, N.G. Pavlidis, D.K. Tasoulis, M.N. Vrahatis, "Determining the Number of Real Roots of Polynomials through Neural Networks", Computers and Mathematics with Applications, pp.1-10, 2006.
15. [15] S.J. Perantonis, N. Ampazis, S. Varoufakis and G. Antoniou, "Constrained learning in neural networks: Application to stable factorization of 2-D polynomials", Neural Processing Letters Vol.7, pp.5-14, 1998.