Multiple Roots

Numerical Analysis · Axiom Academy

When roots repeat: challenges and solutions A root p has multiplicity m if f(x) = (x - p)^m · g(x) where g(p) ≠ 0. The function crosses the x-axis The function touches the x-axis Blue: (x-2)² - simple roots at x=1,3 | Orange: (x-2)² - double root at x=2 The Problem: Newton's Method Slows Down! For a root of multiplicity m, Newton's method convergence degrades from quadratic to linear ! At a double root, both f(x) and f'(x) approach 0, causing: Newton's method relies on the tangent line. At a multiple root: f'(p) = 0 (tangent is horizontal!) The linear approximation becomes poor The convergence rate drops to: Compare Newton's method on a simple vs double root: For the simple root, errors decrease like 10⁻², 10⁻⁴, 10⁻⁸... (doubling digits) For the double root, errors decrease like 10⁻¹, 10⁻², 10⁻³... (one digit at a time) Solution: Modified Newton's Method If we know the multiplicity m, we can restore quadratic convergence! This multiplies the correction by m, compensating for the reduced slope. Alternative: Use u(x) = f(x)/f'(x) If we don't know m, apply Newton to u(x) = f(x)/f'(x) instead! The function u(x) has only simple roots where f has roots of any multiplicity. Summary: Handling Multiple Roots For even-multiplicity roots (m = 2, 4, ...), the function doesn't change sign . Bisection requires a sign change, so it cannot find these roots! If Newton converges slowly (linearly), suspect a multiple root Check if f'(p) ≈ 0 at the apparent root

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