The localization method of extremum point for unimodal function
Keywords:
extremum, unimodal function, one-dimensional search, piecewise linear approximation, weighted average operation, characteristic values, efficiency indexAbstract
The combination of numerical methods such as Regula falsi method and secant method for direct search of extremum of unimodal function on the given interval is considered. The proposed combination does not require any prior analysis of character of the functions to begin its search for an extremum. The unique method with a minimum of memory depth in the search area is implemented. It is universal and independent of the class of minimized function. Accepted a posteriori approach allows to find the extremum of non-differentiable functions, including algorithmically defined functions. The method is quite general. It provides a guaranteed convergence to the extreme point due to the use ща the weighted average method for realizing solutions. If the minimized function in a given interval is not unimodal, the suggested method is always provides obtaining at least a relative minimum. The stated method can be easily extended to the multidimensional case.The massive computational experiments on smooth and non-smooth functions are carried out. The application of the proposed method to the convex-concave functions with a first-order gap, to functions with a asymmetrical character in vicinity of solution, as well as empirically given functions of complex geometry. It is shown that the efficiency index of combination methods exceeds index of the individual methods with the same initial conditions.
References
Vasil’ev, F. P. (1980) Chislennyie metodyi resheniya ekstremalnyih zadach [Numerical methods for solving extreme problems]. – Moscow: Nauka. – 520 p.
Aoki, M. (1977) Vvedenie v metodyi optimizatsii [Introduction To Optimization Techniques]. – Moscow: Nauka. – 344 p.
Shor, N. Z. (1979) Metodyi minimizatsii nedifferentsiruemyih funktsiy i ih prilozheniya [Methods of minimizing non-differentiable functions and their applications]. – Kiev: Naukova dumka. – 200 p.
Yasmin, N. (2012) Some derivative free iterative methods for solving non-linear equations // Academic Research Intern.– Vol. 2, No. 1. – P. 75-82.
Soleymani, F. (2002) New derivative-free quasi-secant algorithm for solving non-linear equations // World Academy Sciences, Eng. and Technology. – Vol. 31. – P. 719-721.
Vorontsova, E.A. (2012) Byistroshodyaschiysya algoritm lineynogo poiska v nedifferentsiruemoy optimizatsii [Fast convergent algorithm for linear search in the non-differentiable optimization]. // Modelirovanie sistem. –№2 (32). – P. 39-48.
Traub, J.F. (1985) Iteratsionnyie metodyi resheniya uravneniy [Iterative methods for the solution of equations].– Moscow: Mir. – 264 p.
Ganshin, G.S. (1973) K teorii iteratsionnyih protsessov [On the theory of iterative processes].// Vyichisl. i prikl. matematika. – Kiev: Izd-vo Kiev. universiteta. –№ 19. – P. 143-147.
Meleshko, V.I. (1973). Gradientnyie metodyi optimizatsii s pamyatyu [Gradient optimization methods with memory]. // Izv. AN SSSR. Tehn. kibernetika. –– Vol. 1, №1. – P. 38-51.
Ostrovskiy, A.M. (1963) Reshenie uravneniy i sistem uravneniy [Solution of equations and systems of equations]. – Moscow: Izd-vo inostr. lit. – 219 p.
Box, M.J., Davies D., Swann W.H. (1969) Non-linear optimization techniques.– Edinburgh: Oliver&Boyd. – 60 p.
Powell, M.J.D. (1958) An iterative method for finding stationary values of a function of several variаbles / M.J.D. Powell // Comp. J. – Vol. 5, No 2. – P. 147-151.
Melent’ev, P.V. (1937) Neskolko novyih metodov i priemov priblizhennyih vyichisleniy [Several new methods and techniques of approximate calculations]. – Leningrad-Moscow: Gl. red. tehn. teoret. lit. – 148 p.
Chen, J., Li W. (2006) An exponential regula falsi method for solving nonlinear equations // Numerical Algoritms. – Vol. 41, No 4. – P. 327-338.
Soleymani, F. (2011) Computing simple roots by a sixth order iterative method // Int. J. Pure and Appl. Maths.– Vol. 69, No 1. – P. 41-48.
Virchenko, N.A., Lyashko I.I., Shvetsov K.I. (1979) Grafiki funktsiy. Spravochnik [Graphs of functions. Handbook]. – Kiev: Naukova dumka. – 320 p.
Thukral, R. (2012) New family higher order Steffensen-type methods for solving nonlinear equations // J. Modern Methods in Numerical Maths. – Vol. 3, No 1. – P. 1-10.
Soleymani, F., Sharifi M. (2009) A new derivative-free quasi-Secant algorithm for solving non-linear equations // Intern. J. Math. Comp., Phys. Electr. and Computer Eng. – Vol. 3., No 7. – P. 554-556.
Downloads
Published
Issue
Section
License
Copyright (c) 2016 Г. А. Шелудько, С. В. Угримов
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
All authors agree with the following conditions:
- The authors reserve the right to claim authorship of their work and transfer to the journal the right of first publication of the work under the license agreement (the agreement).
- Authors have a right to conclude independently additional agreement on non-exclusive spreading the work in the form in which it was published by the jpurnal (for example, to place the work in institution repository or to publish as a part of a monograph), providing a link to the first publication of the work in this journal.
- Journal policy allows authors to place the manuscript in the Internet (for example, in the institution repository or on a personal web sites) both before its submission to the editorial board and during its editorial processing, as this ensures the productive scientific discussion and impact positively on the efficiency and dynamics of citation of published work (see The Effect of Open Access).
