NewtonRaphsonmenetelmän

The Newton-Raphson method, also known as Newton's method, is a powerful numerical technique for finding successively better approximations to the roots (or zeroes) of a real-valued function. It is an iterative process that begins with an initial guess for the root. The core idea is to use the tangent line to the function at the current guess to estimate the location of the root.

The formula for the Newton-Raphson method is derived from Taylor series expansion. If $x_n$ is the current approximation of a root of the function $f(x)$, then the next approximation, $x_{n+1}$, is given by the formula: $x_{n+1} = x_n - f(x_n) / f'(x_n)$, where $f'(x_n)$ is the derivative of the function evaluated at $x_n$. The method converges quadratically if the initial guess is sufficiently close to the actual root and the derivative is non-zero at the root.

However, the Newton-Raphson method has limitations. It may fail to converge if the initial guess is poor, if the derivative is zero or close to zero near the root, or if the function has multiple roots. Despite these drawbacks, it remains a widely used and efficient method for root-finding in various scientific and engineering applications due to its rapid convergence when applicable.