Sequence transformation

In mathematics, a sequence transformation is a transformation that is used to numerically calculate the limit value of a slowly convergent sequence or series , or the antilimes of a divergent series .

For a given sequence

{\ displaystyle S = \ {s_ {n} \} _ {n \ in N_ {0}}}

is the transformed sequence

{\ displaystyle T (S) = S '= \ {s' _ {n} \} _ {n \ in N_ {0}}}

.

The elements of the transformed sequence are normally calculated as a function of a finite number of elements of the original sequence. So there is an illustration of the form ${\ displaystyle s' _ {n}}$ ${\ displaystyle F}$

{\ displaystyle F: (s_ {n}, s_ {n + 1}, \ dots, s_ {n + k}) \ to s' _ {n}}

with a finite . In the simplest case, these and the are real or complex numbers . In general, these are elements of a vector space or an algebra . ${\ displaystyle k}$ ${\ displaystyle s_ {n}}$ ${\ displaystyle s' _ {n}}$

The transformed sequence is said to converge faster than the original sequence if

{\ displaystyle \ lim _ {n \ to \ infty} {\ frac {s' _ {n} -s} {s_ {n} -s}} = 0}

where the (anti-) limes of is. If the original sequence converges slowly, this is called convergence acceleration . ${\ displaystyle s}$ ${\ displaystyle S}$

If the mapping is linear in each argument, i. i.e., if ${\ displaystyle F}$

{\ displaystyle s' _ {n} = \ sum _ {m = 0} ^ {k} c_ {m} s_ {n + m}}

for constants

{\ displaystyle c_ {0}, \ dots, c_ {k}}

holds, then the sequence transformation is called a linear sequence transformation , otherwise a non-linear sequence transformation . ${\ displaystyle T}$

A sequence transformation can be used to accelerate the convergence of a convergent series or as a summation method for a divergent series : For a series

{\ displaystyle R = \ sum _ {i = 0} ^ {\ infty} a_ {i}}

just look at the result

{\ displaystyle S = \ {s_ {n} \} _ {n = 0} ^ {\ infty}}

the partial sums

{\ displaystyle s_ {n} = \ sum _ {i = 0} ^ {n} a_ {i}}

and applies a suitable sequence transformation to this.

Important examples of nonlinear sequence transformations are Padé approximants for power series and Levin-like sequence transformations .

Particularly non-linear sequence transformations often result in highly efficient extrapolation methods .

literature

C. Brezinski and M. Redivo Zaglia: Extrapolation Methods. Theory and Practice. North Holland, 1991.
GA Baker, Jr. and P. Graves-Morris: Padé Approximants. Cambridge UP 1996.