Numerical values of the Jacobi elliptic function sn: Wolfram Alpha vs. Maple vs. C++?

Question

I have a problem to check the validity of an algorithm I've implemented in C++ to compute the Jacobi elliptic function $\mathrm{sn}(u, k)$ (inspired and improved from Numerical Recipes 3rd edition). I would like to test my algorithm for $u=0.2$ and $k=0.5$. I obtain the following results:

C++ (double precision): $\mathrm{sn}(0.2, 0.5) = 0.198345391214472$
Maple: $\mathrm{sn}(0.2, 0.5) = 0.198345391214472$
Wolfram Alpha: $\mathrm{sn}(0.2, 0.5) = 0.198021742981970$

My result is consistent with the result of Maple. But my big problem is that Maple and Wolfram Alpha do not give the same value...

So what is the exact value of the 15 first decimals of $\mathrm{sn}(0.2, 0.5)$ ?

score 8 · Accepted Answer · edited Apr 13 '17 at 12:19

8

Alpha will give you the same $0.1983453912\cdots$ result with $\ \operatorname{sn}(0.2,0.5^2)$

Mathematica and the old and excellent reference Abramowitz&Stegun use the 'parameter' $m=k^2$ instead of the 'modulus' $k$ used by your code, Maple, MATLAB and many other online references like Dlmf and MathWorld which is used when you click at the right on 'Definition' in... Alpha !

ADDITION: A fine discussion about this problem as well as the notation allowing a clear distinction between the conventions in use was given by J. M. here.

edited Apr 13 '17 at 12:19

Community

1

answered Apr 28 '13 at 10:35

Raymond Manzoni

43,021
5
86
140

@J.M. : To be honest your link was rather hard to find (I added the elliptic-functions tag there) but pretty useful ! – Raymond Manzoni Apr 29 '13 at 12:11
Technically, the function the guy in the other question was asking about is an elliptic integral and not an elliptic function, but... :) It's one of the answers I'm proud of writing. – J. M. ain't a mathematician Apr 29 '13 at 12:13

Numerical values of the Jacobi elliptic function sn: Wolfram Alpha vs. Maple vs. C++?

1 Answers1

Linked