The localization method of extremum point for unimodal function

• Date: 2016
• Main Authors: Шелудько, Г. А., Угримов, С. В.
• Format: Article
• Language: Russian
• Published: Інститут енергетичних машин і систем ім. А. М. Підгорного Національної академії наук України 2016
• Subjects: extremum; unimodal function; one-dimensional search; piecewise linear approximation; weighted average operation; characteristic values; efficiency index
• Online Access: https://journals.uran.ua/jme/article/view/65241
• Journal Title: Journal of Mechanical Engineering

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.