100: Everything You Need to Know about Newton's Method of Approximating Polynomials (Pure and Applied Mathematics)
Newton's method, also known as the Newton-Raphson method, is an algorithm used to find the roots of a polynomial function. It was developed in the late 1600s. For a function f(x), finding the value of x for which f(x) = 0 is the goal of Newton's algorithm. Image Credit: GeoGebra Here's an algorithm you can use: Choose an initial guess for the root, let's call it x₀. Calculate the function value and its derivative at the current guess. These are denoted as f(x₀) and f'(x₀), respectively. Use the formula: x₁ = x₀ - f(x₀) / f'(x₀) to update the guess. This formula is derived from the tangent line approximation to the function at x₀, assuming it is differentiable. Repeat the last two sentences until the desired level of accuracy is achieved or until a specified number of iterations have been performed. You will know your answer is correct as the result of every iteration converges to the same value. What to know about the algorithm: Newton's algorithm typical...