Rate of convergence

In numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, this concept is of practical importance in dealing with a sequence of successive approximations for an iterative method, as then typically fewer iterations are needed to yield a useful approximation if the rate of convergence is higher. This may even make the difference between needing ten or a million iterations insignificant.

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.

Suppose that the sequence {x_k} converges to the number L.

We say that this sequence converges linearly to L, if there exists a number μ ∈ (0, 1) such that

The number μ is called the rate of convergence.

If the sequence converges, and

If the sequence converges sublinearly and additionally

then it is said the sequence {x_k} converges logarithmically to L.

The next definition is used to distinguish superlinear rates of convergence. We say that the sequence converges with order q to L for q>1 if

In particular, convergence with order

This is sometimes called Q-linear convergence, Q-quadratic convergence, etc., to distinguish it from the definition below. The Q stands for "quotient," because the definition uses the quotient between two successive terms.

A practical method to calculate the rate of convergence for a sequence is to calculate the following sequence which is converging to p.

The drawback of the above definitions is that these do not catch some sequences which still converge reasonably fast, but whose "speed" is variable, such as the sequence {b_k} below. Therefore, the definition of rate of convergence is sometimes extended as follows.

...
Wikipedia