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 by repeatedly halving an interval \([a, b]\) where \(f\) changes sign, guaranteeing that each half still contains a root. At each step, the…

Fixed Point Iteration

Computational math

Linear convergence

Slow convergence

Fixed point iteration is a way to solve an equation after rewriting it in the form

Newton’s Method

Computational math

Fast convergence

Quadratic convergence

Newton’s method is one of the most common methods for solving equations of the form Newton’s method finds roots of \(f(x) = 0\) by repeatedly linearizing the function…

Secant Method

Computational math

Superlinear convergence

The secant method finds roots of \(f(x) = 0\) the way Newton’s method does — by repeatedly intersecting a line with the \(x\)-axis — but it replaces the tangent line with a…