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.
Formulation For a sequence { a m } m ∈ N {\displaystyle \left\{a_{m}\right\}_{m\in \mathbb {N} }} the series
A = ∑ m = 0 ∞ a m {\displaystyle A=\sum _{m=0}^{\infty }a_{m}\,}
is to be determined. First, the partial sum A n {\displaystyle A_{n}} is defined as:
A n = ∑ m = 0 n a m {\displaystyle A_{n}=\sum _{m=0}^{n}a_{m}\,}
and forms a new sequence { A n } n ∈ N {\displaystyle \left\{A_{n}\right\}_{n\in \mathbb {N} }} . Provided the series converges, A n {\displaystyle A_{n}} will also approach the limit A {\displaystyle A} as n → ∞ . {\displaystyle n\to \infty .}
The Shanks transformation S ( A n ) {\displaystyle S(A_{n})} of the sequence A n {\displaystyle A_{n}} is the new sequence defined by
S ( A n ) = A n + 1 A n − 1 − A n 2 A n + 1 − 2 A n + A n − 1 = A n + 1 − ( A n + 1 − A n ) 2 ( A n + 1 − A n ) − ( A n − A n − 1 ) {\displaystyle S(A_{n})={\frac {A_{n+1}\,A_{n-1}\,-\,A_{n}^{2}}{A_{n+1}-2A_{n}+A_{n-1}}}=A_{n+1}-{\frac {(A_{n+1}-A_{n})^{2}}{(A_{n+1}-A_{n})-(A_{n}-A_{n-1})}}}
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