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Polynomial Approximation of the Sine Function Using Genetic Algorithms and Fractal-Based Coefficients
Introduction
Polynomial approximation is a fundamental concept in numerical analysis and computational mathematics. The goal is to approximate complex functions, like trigonometric or exponential functions, using polynomials. Polynomials are preferred because they are simple, continuous, and differentiable functions that are easy to compute.
The Weierstrass approximation theorem assures us that any continuous function on a closed interval can be approximated by polynomials to any desired degree of accuracy. However, the challenge lies in finding the optimal polynomial that minimizes the error of approximation, especially for functions with intricate behaviour like the sine function.
In this article, we explore my hybrid method that combines fractal-based initialization, genetic algorithms (GAs), and gradient-based optimization to generate a polynomial that closely approximates the sine function. We will break down the mathematics, walk through the code, discuss real-world applications, and compare this method to existing approaches.
