Newton–Raphson Method

Summary

Newton–Raphson linearizes $f$ at the current guess and takes the root of that tangent line. For simple roots with a good start, convergence is locally quadratic.

Prerequisites

• Differentiable $f$ with computable $f^{\prime }$
• Optional: Secant Method as a derivative-free cousin

Problem Type

Solve $f(x)=0$ when $f^{\prime }$ is available and a reasonable initial guess is known.

Method Definition

Geometrically, $x_{n+1}$ is where the tangent at $(x_n,f(x_n))$ hits the $x$ -axis.^[1]

Assumptions / Requirements

• $f\in C^2$ near the root (for the classical local theory)
• Simple root: $f(r)=0$ and $f^{\prime }(r)\ne 0$
• Initial $x_0$ sufficiently close to $r$
• $f^{\prime }(x_n)\ne 0$ at every iterate

Algorithm

1. Choose $x_0$ , tolerance $\epsilon$ , max iterations.
2. Compute $x_{n+1}=x_n-f(x_n)/f^{\prime }(x_n)$ .
3. Stop if $|x_{n+1}-x_n|<\epsilon$ or $|f(x_{n+1})|<\epsilon$ .
4. Fail if $f^{\prime }(x_n)=0$ or the budget is exhausted.

Convergence

For a simple root and $x_0$ close enough, the error satisfies $e_{n+1}\approx C{e}_n^2$ (quadratic). Multiple roots reduce the order to linear unless the method is modified.

Error / Accuracy

Use step size and residual together. Quadratic convergence means digits roughly double each successful iteration near the root.

Worked Example

$f(x)=x^3-2x-5$ , $f^{\prime }(x)=3x^2-2$ , start $x_0=2$ .

Residual at $x_3$ is already near machine noise for double precision in a few more steps.

Common Failure Modes

• $f^{\prime }(x_n)=0$ or tiny (step blows up)
• Poor initial guess → divergence or attraction to a different root
• Multiple roots → slow linear convergence and unstable division
• Cycles for some nonlinear $f$

Connections

• Secant Method
• Bisection Method for a safe global phase before Newton polish
• Root Finding

References