shanks: Evaluation of the hypergeometric function using Shanks's...
In hypergeo: The Gauss Hypergeometric Function

Description Usage Arguments Details Note Author(s) References See Also Examples

Evaluation of the hypergeometric function using Shanks transformation of successive sums

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hypergeo_shanks(A,B,C,z,maxiter=20)
genhypergeo_shanks(U,L,z,maxiter=20)
shanks(Last,This,Next)

`A,B,C`	Parameters (real or complex)
`U,L`	Upper and lower vectors
`z`	Primary complex argument
`maxiter`	Maximum number of iterations
`Last,This,Next`	Three successive convergents

The Shanks transformation of successive partial sums is

S(n)=\frac{A_{n+1}A_{n-1}-A_n^2}{A_{n+1}-2A_n+A_{n-1}}

and if the A_n tend to a limit then the sequence S(n) often converges more rapidly than A_n. However, the denominator is susceptible to catastrophic rounding under fixed-precision arithmetic and it is difficult to know when to stop iterating.

The

Robin K. S. Hankin