Interpolation сubic polynomials on golden section grids for optimization and solving nonlinear equations of a single variable

Abstract

Interpolation cubic polynomials constructed on golden section grids possess unique properties that form the basis of an algorithm for the approximate solution of nonlinear equations and the search for extremal points of continuous single-variable functions. Since the interval is reduced by the golden ratio at each iteration, and the golden section grid requires the computation of only one new point per step, the algorithm demonstrates a high rate of implementation.

The extrema and zeros of the cubic polynomial are determined analytically, which enables rapid approximation of both the extremum search problem and the solution of nonlinear equations for continuous functions defined on finite intervals. The coefficients of the cubic polynomial are linear functions of the golden ratio parameter, resulting in minimal computational error.

As the interval narrows, the accuracy of the cubic polynomial’s approximation to a continuous function increases; therefore, solving the problems of extremum search and nonlinear equation solving with the use of cubic interpolation does not require reducing the interval length  to machine precision. This allows the construction of rhombastic algorithms for continuous functions of complex nature (where  is a constant and  is the machine epsilon). Keywords: interpolation cubic polynomial, golden section grid, function optimization, solving nonlinear equations, convergence acceleration.