Wynn's Epsilon Method

Wynn's epsilon method is a recursive algorithm for computing Shanks' transformation of a sequence. It is used for numerical sums and products by transforming their partial sums or partial products.

In particular, the method provides an efficient algorithm for implementing transformations of the form

where is the th partial sum of a sequence , with and nonzero transformation denominator. The transformation can yield convergence improvement (Hamming 1986, p. 205). Set , , and use

for , 1, ..., wherever the denominator is nonzero. For a positive integer , the values give Shanks' transformation of order , based on , , . The case gives Aitken's delta-squared process, with . Higher orders generally differ from repeated applications of Aitken's delta-squared process.

Wynn's epsilon method can be invoked in numerical summation and multiplication using Method -> "WynnEpsilon" in the Wolfram Language's NSum and NProduct commands. It is also utilized in the routine NLimit[expr, x -> x0] in the Wolfram Language package NumericalCalculus` .

Wynn's epsilon method is a member of a large family of similar so-called lozenge, or rhombus, transformations (Hamming 1986, p. 207).