Rate of convergence

In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence ${\displaystyle (x_{n})}$ that converges to ${\displaystyle L}$ is said to have order of convergence ${\displaystyle q\geq 1}$ and rate of convergence ${\displaystyle \mu }$ if

${\displaystyle \lim _{n\rightarrow \infty }{\frac {\left|x_{n+1}-L\right|}{\left|x_{n}-L\right|^{q}}}=\mu .}$^[1]

The rate of convergence ${\displaystyle \mu }$ is also called the asymptotic error constant. Note that this terminology is not standardized and some authors will use rate where this article uses order (e.g., ^[2]).

In practice, the rate and order of convergence provide useful insights when using iterative methods for calculating numerical approximations. If the order of convergence is higher, then typically fewer iterations are necessary to yield a useful approximation. Strictly speaking, however, the asymptotic behavior of a sequence does not give conclusive information about any finite part of the sequence.

Similar concepts are used for discretization methods. The solution of the discretized problem converges to the solution of the continuous problem as the grid size goes to zero, and the speed of convergence is one of the factors of the efficiency of the method. However, the terminology, in this case, is different from the terminology for iterative methods.

Series acceleration is a collection of techniques for improving the rate of convergence of a series discretization. Such acceleration is commonly accomplished with sequence transformations.