![[newton-raphson.png|200]] The Newton-Raphson method (Newton's method) is an algorithm that finds roots of a [[Function|function]] by producing successively better approximations to the roots. - In certain circumstances the successive approximations may diverge or fall out of the domain of the function, in which case a better initial approximation should be chosen. > [!summary] Procedure > > Let $f$ be a (real) function, $x_0$ be an initial guess for the root of $f$, and $f^\prime$ be its [[Derivative|derivative]], if the initial guess is close enough, then > $ > x_{1} = x_0 - \frac{f(x_0)}{f^\prime(x_0)} > $ > is a better approximation of the root $x_0$. > - $x_1$ is the intersection of the x-axis and the [[Tangent Line|tangent line]] of the graph at $(x_0, f(x_0)$. > - The process can be repeated as follows $x_{n+1} = x_n - \frac{f(x_n)}{f^\prime(x_n)}$ until a sufficiently precise value is reached.