Newton’s Method

Definition

Newton’s method is an iterative root-finding algorithm for solving $f(x)=0$. It requires an analytical derivative $f^{\prime }(x)$ and is among the fastest methods when applicable.

Intuition

At each step, replace the curve $f$ with its tangent line at the current guess and use where that line crosses zero as the next guess. Because the tangent is a good local approximation, each iterate is typically much closer to the root than the previous one — convergence is quadratic near the root.

Formal Description

At each iterate $x_n$, approximate $f$ by its tangent line:

$$y-f(x_n)=f^{\prime }(x_n)(x-x_n).$$

Setting $y=0$ gives the next iterate:

$$x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}.$$

An initial guess $x_0$ close to the root $r$ is required.

Example — estimating $\sqrt{2}$: Solve $f(x)=x^2-2=0$ with $f^{\prime }(x)=2x$:

$$x_{n+1}=x_n-\frac{x_n^2-2}{2x_n}=\frac{x_n^2+2}{2x_n}.$$

Starting from $x_0=1$, this converges rapidly to $\sqrt{2}\approx 1.41421$.

Applications

Used in optimisation (solving $\nabla f=0$), in numerical solvers for ODEs and PDEs, and in computational geometry. Variants (quasi-Newton methods) approximate $f^{\prime }$ when it is expensive to compute exactly.

Trade-offs

Convergence is not guaranteed globally: a poor initial guess can cause divergence, cycling, or convergence to the wrong root. The method requires $f^{\prime }(x_n)\ne 0$ at each step and knowledge of $f^{\prime }$ analytically or numerically. Near a root with multiplicity $>1$, convergence degrades from quadratic to linear.

Links

• Limits and Continuity — convergence relies on continuity of $f$ and $f^{\prime }$