hypergeo (version 1.2-14)

shanks: Evaluation of the hypergeometric function using Shanks's method

Description

Evaluation of the hypergeometric function using Shanks transformation of successive sums

Usage

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

Arguments

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

Author

Robin K. S. Hankin

Details

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.

References

• Shanks, D. (1955). “Non-linear transformation of divergent and slowly convergent sequences”, Journal of Mathematics and Physics 34:1-42

See Also

buhring

Examples

hypergeo_shanks(1/2,1/3,pi,z= 0.1+0.2i)