Convergence Rates

Numerical Analysis · Axiom Academy

Understanding linear, superlinear, and quadratic convergence 1. What is Order of Convergence? If eₙ = |xₙ - p| is the error at iteration n, a method has order p if: The larger p is, the faster the error shrinks. Let's see what this means in practice. With linear convergence , each iteration multiplies the error by a constant factor c < 1. The error decreases steadily, but not spectacularly. 3. Quadratic Convergence (p = 2) With quadratic convergence , the error is squared each iteration. The number of correct digits roughly doubles with each step! The order of convergence has big practical implications: High order = fewer iterations to reach desired accuracy Cost per iteration matters — Newton needs derivatives! Reliability vs speed trade-off — bisection is slow but safe

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