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In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequence space).Sequence transformations include linear mappings such as discrete convolution with another sequence and resummation of a sequence and nonlinear mappings, more generally.
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 combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function.
These transformations typically involve integral formulas applied to a sequence generating function (see integral transformations) or weighted sums over the higher-order derivatives of these functions (see derivative transformations). Given a sequence, {} =, the ordinary generating function (OGF) of the sequence, denoted (), and the exponential ...
Aitken's delta-squared process is an acceleration of convergence method and a particular case of a nonlinear sequence transformation. A sequence X = ( x n ) {\textstyle X=(x_{n})} that converges to a limiting value ℓ {\textstyle \ell } is said to converge linearly , or more technically Q-linearly, if there is some number μ ∈ ( 0 , 1 ...
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 .
Legendre transformation; Möbius transformation; Perspective transform (computer graphics) Sequence transform; Watershed transform (digital image processing) Wavelet transform (orthonormal) Y-Δ transform (electrical circuits)
In numerical analysis, the Shanks transformation is a non-linear series acceleration method to increase the rate of convergence of a sequence. This method is named after Daniel Shanks, who rediscovered this sequence transformation in 1955. It was first derived and published by R. Schmidt in 1941.