Root Finding

Root finding involves determining the values \(x\) where \(f(x) = 0\), critical for solving equations numerically.

Newton’s Method

Iteratively refine guesses using \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\), requiring the function’s derivative.

Examples

For \(f(x) = x^2 - 2\), starting at \(x_0 = 1\), Newton’s method converges to \(\sqrt{2}\).

Other Methods

Bisection (bracketing) and Secant (derivative-free) are alternatives, each with trade-offs in speed and stability.