Rate of convergence (Ofer Abarbanel online library)

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, the concept of rate of convergence is of practical importance when working 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.

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.

Acceleration of convergence

Many methods exist to increase the rate of convergence of a given sequence, i.e. to transform a given sequence into one converging faster to the same limit. Such techniques are in general known as “series acceleration”. The goal of the transformed sequence is to reduce the computational cost of the calculation. One example of series acceleration is Aitken’s delta-squared process.

References

  1. ^{\displaystyle q} may be non-integer. For example, the secant method has, in the case of convergence to a regular, simple root, convergence order φ ≈ 1.618.

Ofer Abarbanel – Executive Profile

Ofer Abarbanel online library

Ofer Abarbanel online library

Ofer Abarbanel online library