Comparative numerical laboratory

Root-Finding Methods on the Same Problem — Chapter 2

Compare bisection, Newton, secant, and fixed-point iteration for \(f(x)=x^3-2\).

\[ f(x)=x^3-2, x_\ast=\sqrt[3]{2}. \]

Methods compared

Method	Convenience	Inconvenience / warning
Bisection	Guaranteed under a sign change; bracket is preserved.	Linear convergence; slow when high precision is required.
Newton	Quadratic local convergence for simple roots.	Needs derivative and a good initial value.
Secant	Derivative-free and superlinear.	No bracket preservation; can fail from poor starts.
Fixed point	Useful when a good contraction is available.	Convergence depends entirely on the chosen map.

left bracket \(a\) right bracket \(b\) initial value \(x_0\) secant second value \(x_1\) iterations \(n\)