Особливості застосування чисельних методів в нелінійних задачах

Abstract

The article presents a comparative characteristic of classical numerical methods for calculating nonlinear equations with one variable, namely the half-division method, the chord method, and Newton's method, as well as their practical implementation. The choice of method for calculating the roots of a nonlinear equation depends on the specific conditions of the problem and the accuracy requirements. The common feature of calculating nonlinear equations using these methods is that in the first step of the calculations, a check is performed for the definiteness of the function on the segment. In this case, the segment has only one simple root and the condition for the function to be definite is satisfied. It has been proven that the method of half division is advisable to use for rough determination of the roots of a nonlinear equation and, provided that the accuracy is increased, the amount of computational work increases significantly. What is special about using the chord method is that to calculate a nonlinear equation using this method, it is necessary to determine the fixed end of the function curve and to use the appropriate mathematical apparatus to calculate the root. In addition, the function must be twice differentiated. Newton's method is effective under strict constraints on the nature of the function, namely, the function must be twice differentiated, the first derivative of the function is not zero, and the first and second derivatives of the function retain constant signs. Previously, the roots of the nonlinear equation were determined using the Maple V R4 computer algebra system, and to separate the roots, numerical methods for solving nonlinear equations were used, which allow determining one root on a given segment. For software implementations of calculating a nonlinear equation using the half-division, chord, and Newton methods, the Visual Studio Code programming environment was used. Solving a single nonlinear equation by the division-by-half method, the chord method, and Newton's method helped to compare the speed of convergence. The use of computer algebra software environments and systems in the calculation of nonlinear problems has increased the efficiency of the practical application of numerical methods.