📂Numerical Analysis

Rate of Convergence in Numerical Analysis

Definition¹

If for a sequence $\left\{ x_{n} \right\}$ converging to $\alpha$ there exists a $c \ge 0$ satisfying $$ | \alpha - x_{n+1} | \le c | \alpha - x_{n} | ^{p} $$ for an order $p \ge 1$, then $\left\{ x_{n} \right\}$ is said to converge to $\alpha$ with order $p$ at rate of convergence $c$.

Explanation

In particular, when $p=1$ together with the condition $c < 1$, it is called Linear Convergence. Similarly, when $p=2$ it is called Quadratic Convergence, and when $p=3$ it is called Cubic Convergence.

While pure analysis only cares about whether something converges, in numerical analysis the speed of convergence also matters.