# Newton's Method for Approximating Roots Newton's method for approximating roots uses an iterative approach to generate a [[Sequence]] that converges on a function's root. The recursive definition of the sequence being: $x_1 = x_0 - \frac {f(x_0)} {f'(x_0)}$ Instead of determining the root of the actual function, Newton's method determines the root of the function's linear approximation at the guess value (sum of the first two elements of its [[Taylor Series]]). That root of the linear approximation then becomes the new guess, and the process is repeated until the difference between the last two elements is within a predefined precision. <img src="https://ds055uzetaobb.cloudfront.net/uploads/79q4c4Sf0F-c4q1p5.svg?width=800" style="background: white; padding: 10px; width:300px" /> ## The Initial Value Newton's method needs to be seeded with a 'guess' for the root value, that needs to be located between the two critical points closest to the root. For the purposes of the previous requirement the edge of the domain is counted as a critical point (even at infinity). Put differently: the seed needs to be in the general neighborhood of the root. If this requirement is not satisfied, Newton's method is not guaranteed to produce a converging sequence. ## Deriving the Recursive Formula Linear approximation at $x_0$ is per definition $L(x) = f(x_0) + f'(x)(x-x_0)$. We're looking for its root so: $ \begin{align} 0 = f(x_0) + f'(x)x - f'(x)x_0 \\ f'(x)x = f'(x)x_0-f(x_0) \\ x = x_0 - \frac{f(x_0)}{f'(x_0)} \end{align} $