Evaluation of the hypergeometric function using Shanks's method
Evaluation of the hypergeometric function using Shanks transformation of successive sums
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
Shanks, D. (1955). “Non-linear transformation of divergent and slowly convergent sequences”, Journal of Mathematics and Physics 34:1-42
hypergeo_shanks(1/2,1/3,pi,z= 0.1+0.2i)
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