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The binomial transform and the Stirling transform are two linear transformations of a more general type. An example of a nonlinear sequence transformation is Aitken's delta-squared process, used to improve the rate of convergence of a slowly convergent sequence. An extended form of this is the Shanks transformation. The Möbius transform is ...
Examples of such nonlinear sequence transformations are Padé approximants, the Shanks transformation, and Levin-type sequence transformations. Especially nonlinear sequence transformations often provide powerful numerical methods for the summation of divergent series or asymptotic series that arise for instance in perturbation theory , and ...
In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926. [1] It is most useful for accelerating the convergence of a sequence that is converging linearly.
Legendre transformation; Möbius transformation; Perspective transform (computer graphics) Sequence transform; Watershed transform (digital image processing) Wavelet transform (orthonormal) Y-Δ transform (electrical circuits)
The most important cases are sequences of real or complex numbers, and these spaces, together with linear subspaces, are known as sequence spaces. Operators on these spaces are known as sequence transformations. Bounded linear operators over a Banach space form a Banach algebra in respect to the standard
Partial summation of a sequence is an example of a linear sequence transformation, and it is also known as the prefix sum in computer science. The inverse transformation for recovering a sequence from its partial sums is the finite difference , another linear sequence transformation.
In linear algebra, linear transformations can be represented by matrices.If is a linear transformation mapping to and is a column vector with entries, then there exists an matrix , called the transformation matrix of , [1] such that: = Note that has rows and columns, whereas the transformation is from to .
An example of Richardson extrapolation method in two dimensions. In numerical analysis , Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value A ∗ = lim h → 0 A ( h ) {\displaystyle A^{\ast }=\lim _{h\to 0}A(h)} .
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