Müller's Method

Numerical Analysis · Axiom Academy

Quadratic interpolation for finding complex roots The secant method uses two points to draw a line . What if we use three points to fit a parabola ? This is Muller's method - it uses quadratic interpolation to find roots! Unlike Newton and secant, Muller's method can find complex roots even when starting with real numbers! This makes it especially useful for polynomials. Given three points x₀, x₁, x₂, we fit a parabola through (x₀, f(x₀)), (x₁, f(x₁)), (x₂, f(x₂)), then find where the parabola crosses zero. The parabola is written in Newton form centered at x₂: Setting P(x) = 0 and solving the quadratic gives us x₃, our next approximation. After some algebra, the next iterate is: Why the ± in the quadratic formula? We choose the sign that makes the denominator larger in magnitude . This picks the root of the parabola closest to x₂. If b > 0, use + sign. If b < 0, use - sign. When b² - 4ac < 0, we get complex roots! Muller's method handles this naturally by using complex arithmetic throughout. Watch Muller's method find roots of f(x) = x³ - x - 2. Muller's method can find complex roots of real polynomials! Consider f(x) = x² + 1, which has no real roots. Newton's method would fail, but Muller's method finds: Polynomial root-finding: Find all roots, real and complex Control systems: Eigenvalue computation Signal processing: Filter design Summary: When to Use Muller's Method Finding complex roots of real polynomials When you want faster convergence than secant

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