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 convolution with another sequence, and resummation of a sequence and, more generally, are commonly used for series acceleration, that is, for improving the rate of convergence of a slowly convergent sequence or series. Sequence transformations are also commonly used to compute the antilimit of a divergent series numerically, and are used in conjunction with extrapolation methods.
Classical examples for sequence transformations include the binomial transform, Möbius transform, Stirling transform and others.
For a given sequence
the transformed sequence is
where the members of the transformed sequence are usually computed from some finite number of members of the original sequence, i.e.
for some which often depends on (cf. e.g. Binomial transform). In the simplest case, the and the are real or complex numbers. More generally, they may be elements of some vector space or algebra.
In the context of acceleration of convergence, the transformed sequence is said to converge faster than the original sequence if
where is the limit of , assumed to be convergent. In this case, convergence acceleration is obtained. If the original sequence is divergent, the sequence transformation acts as extrapolation method to the antilimit .
If the mapping is linear in each of its arguments, i.e., for
for some constants (which may depend on n), the sequence transformation is called a linear sequence transformation. Sequence transformations that are not linear are called nonlinear sequence transformations.
Simplest examples of (linear) sequence transformations include shifting all elements, (resp. = 0 if n + k < 0) for a fixed k, and scalar multiplication of the sequence.
A less trivial example would be the discrete convolution with a fixed sequence. A particularly basic form is the difference operator, which is convolution with the sequence and is a discrete analog of the derivative. The binomial transform is another linear transformation of a still 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 also a nonlinear transformation, only possible for integer sequences.
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In linear algebra, a circulant matrix is a square matrix in which each row vector is rotated one element to the right relative to the preceding row vector. It is a particular kind of Toeplitz matrix.
In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence that converges to is said to have order of convergence and rate of convergence if
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