Root Finding

Author

Dhruv Azad

Published

June 25, 2026

Root finding methods are numerical techniques for solving equations of the form \(f(x)=0\). These methods are central in computational mathematics because many problems cannot be solved exactly with algebra alone.

Bisection Method

Computational Math

Guaranteed Convergence

The bisection method finds a root of \(f(x)=0\) by repeatedly halving an interval \([a,b]\) where \(f\) changes sign, keeping whichever half still brackets the root.

Fixed Point Iteration

Computational Math

Linear Convergence

Slow Convergence

Fixed point iteration solves \(x=g(x)\) by repeatedly applying \(g\): \(x_{n+1}=g(x_n)\). If the sequence converges to a value \(\alpha\), then \(\alpha=g(\alpha)\) — a…

Newton’s Method

Computational Math

Fast Convergence

Quadratic Convergence

Newton’s method finds roots of \(f(x)=0\) by repeatedly linearizing: from \(x_n\), it follows the tangent line at \((x_n, f(x_n))\) to its \(x\)-intercept,

Secant Method

Computational Math

Superlinear Convergence

The secant method finds roots of \(f(x)=0\) like Newton’s method, but replaces the tangent line with the secant line through the two most recent approximations, so no…