Newton's Method Convergence

Numerical Analysis · Axiom Academy

LESSON Newton's Method Convergence Understanding when and why Newton's method converges quadratically If f has a simple root r (f(r) = 0, f'(r) ≠ 0), and f'' is continuous near r, then Newton's method converges quadratically for initial guesses sufficiently close to r. The error eₙ = xₙ - r satisfies a precise relationship: Newton's method only converges if you start "close enough" to the root. The set of starting points that converge is called the basin of attraction . For complex functions, these basins can have fractal boundaries! This is why Newton's method can be unpredictable. 4. Multiple Roots: Linear Convergence When r is a multiple root (f(r) = f'(r) = 0), convergence slows to linear !

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