ruutmeetod

Ruutmeetod, also known as the quadratic method or root-finding method, is a numerical technique used to approximate the roots of a polynomial equation. It is a specific application of Newton's method, adapted for the context of finding roots. The core idea of Ruutmeetod is to iteratively refine an initial guess of a root by using the value of the polynomial and its derivative at the current guess.

The formula for the Ruutmeetod iteration is given by:

x_(n+1) = x_n - f(x_n) / f'(x_n)

where x_n is the current approximation of the root, x_(n+1) is the next approximation, f(x) is the polynomial function, and f'(x) is its derivative.

The method starts with an initial guess, x_0, which is believed to be close to a root. This guess is then plugged into the formula to calculate a new, hopefully better, approximation. This process is repeated until the difference between successive approximations is smaller than a predefined tolerance, or until a maximum number of iterations is reached.

Ruutmeetod is known for its rapid convergence when the initial guess is sufficiently close to a root and the derivative is non-zero. However, it can fail to converge or converge to an unintended root if the initial guess is poor, if the derivative is zero or close to zero at an approximation, or if the function has multiple roots close to each other. The effectiveness of Ruutmeetod depends heavily on the nature of the polynomial and the chosen starting point.