Kiihdytysmenetelmä

Kiihdytysmenetelmä, meaning acceleration method in Finnish, refers to a general approach in various fields, particularly in mathematics and physics, to speed up the convergence of sequences or series. Instead of directly computing terms of a sequence or partial sums of a series, an acceleration method transforms the original sequence into a new one that converges to the same limit but at a much faster rate. This can significantly reduce the computational effort required to obtain a desired level of accuracy.

The core idea behind most acceleration methods is to exploit the structure of the convergence. If a sequence converges slowly, it often implies that the error term decreases in a predictable way, for example, like $1/n$ or $1/n^2$. Acceleration methods aim to create a new sequence where this error term decreases much more rapidly, such as $1/n^2$ or even exponentially. This is typically achieved through linear combinations of existing terms of the sequence.

One of the most well-known and widely used acceleration methods is Aitken's delta-squared process. This method is particularly effective for sequences that converge linearly. Other notable acceleration methods include Richardson extrapolation, Shanks transformations, and various methods based on Padé approximants. The choice of acceleration method often depends on the specific properties of the sequence or series being accelerated.

Acceleration methods find applications in diverse areas, including numerical analysis for solving equations, eigenvalue problems, and integral equations. In physics, they can be used to speed up calculations in quantum mechanics, statistical mechanics, and computational fluid dynamics. The ability to achieve faster convergence makes these methods invaluable for simulations and computations where computational resources are a limiting factor.