Editing Jacobi elliptic functions (section)

==Fast computation==
The theta function ratios provide an efficient way of computing the Jacobi elliptic functions. There is an alternative method, based on the [[arithmetic-geometric mean]] and [[Landen's transformation]]s:<ref name="sala"/>

Initialize
:<math>a_0=1,\, b_0=\sqrt{1-m}</math>
where <math>0<m<1</math>.
Define
:<math>a_n=\frac{a_{n-1}+b_{n-1}}{2},\, b_n=\sqrt{a_{n-1}b_{n-1}},\, c_n=\frac{a_{n-1}-b_{n-1}}{2}</math>
where <math>n\ge 1</math>.
Then define
:<math>\varphi_N=2^N a_N u</math>
for <math>u\in\mathbb{R}</math> and a fixed <math>N\in\mathbb{N}</math>. If
:<math>\varphi_{n-1}=\frac{1}{2}\left(\varphi_n+\arcsin \left(\frac{c_n}{a_n}\sin \varphi_n\right)\right)</math>
for <math>n\ge 1</math>, then
:<math>\operatorname{am}(u,m)=\varphi_0,\quad \operatorname{zn}(u,m)=\sum_{n=1}^N c_n\sin\varphi_n</math>
as <math>N\to\infty</math>. This is notable for its rapid convergence. It is then trivial to compute all Jacobi elliptic functions from the Jacobi amplitude <math>\operatorname{am}</math> on the real line.<ref group="note">For the <math>\operatorname{dn}</math> function,
<math>\operatorname{dn}(u,m)=\frac{\operatorname{cn}(u,m)}{\operatorname{sn}(K(m)-u,m)}</math> can be used.</ref>

In conjunction with the addition theorems for elliptic functions (which hold for complex numbers in general) and the Jacobi transformations, the method of computation described above can be used to compute all Jacobi elliptic functions in the whole complex plane.

Another method of fast computation of the Jacobi elliptic functions via the arithmetic–geometric mean, avoiding the computation of the Jacobi amplitude, is due to Herbert E. Salzer:<ref>{{cite journal |last=Salzer |first=Herbert E. |date=July 1962 |title=Quick calculation of Jacobian elliptic functions|journal=Communications of the ACM|volume=5|issue=7|pages=399|doi=10.1145/368273.368573 |s2cid=44953400 |doi-access=free }}</ref>

Let
:<math>0\le m\le 1,\,0\le u\le K(m),\, a_0=1,\, b_0=\sqrt{1-m},</math>
:<math>a_{n+1}=\frac{a_n+b_n}{2},\, b_{n+1}=\sqrt{a_n b_n},\,c_{n+1}=\frac{a_n-b_n}{2}.</math>
Set
:<math>\begin{align}y_N&=\frac{a_N}{\sin (a_Nu)}\\
y_{N-1}&=y_N+\frac{a_Nc_N}{y_N}\\
y_{N-2}&=y_{N-1}+\frac{a_{N-1}c_{N-1}}{y_{N-1}}\\
\vdots&=\vdots\\
y_0&=y_1+\frac{m}{4y_1}.\end{align}</math>
Then
:<math>\begin{align}\operatorname{sn}(u,m)&=\frac{1}{y_0}\\
\operatorname{cn}(u,m)&=\sqrt{1-\frac{1}{y_0^2}}\\
\operatorname{dn}(u,m)&=\sqrt{1-\frac{m}{y_0^2}}\end{align}</math>
as <math>N\to\infty</math>.

Yet, another method for a rapidly converging fast computation of the Jacobi elliptic sine function found in the literature is shown below.<ref>{{Cite journal |last=Smith |first=John I. |date=May 5, 1971 |title=The Even- and Odd-Mode Capacitance Parameters for Coupled Lines in Suspended Substrate |url=https://ieeexplore.ieee.org/document/1127543 |journal=IEEE Transactions on Microwave Theory and Techniques |volume=MTT-19 |issue=5 |pages=430 |doi=10.1109/TMTT.1971.1127543 |bibcode=1971ITMTT..19..424S |via=IEEE Xplore}}</ref>

Let:

:<math>\begin{align}
&a_0 = u    &b_0 = \frac{1-\sqrt{1-m}}{1+\sqrt{1-m}} \\
&a_1 = \frac{a_0}{1+b_0} &b_1 = \frac{1-\sqrt{1-b^2_0 }}{1+\sqrt{1-b^2_0}}\\
&\vdots = \vdots &\vdots = \vdots \\
&a_n = \frac{a_{n-1}}{1+b_{n-1}}  &b_n = \frac{1-\sqrt{1-b^2_{n-1}}}{1+\sqrt{1-b^2_{n-1}}}\\
\end{align}</math>

Then set:

:<math>\begin{align}
y_{n+1} &= \sin(a_n)   \\
y_{n} &= \frac{y_{n+1}(1+b_n)}{1+y^2_{n+1}b_n} \\
\vdots &= \vdots\\
y_0 &= \frac{y_1(1+b_0)}{1+y^2_1b_0} \\
\end{align}</math>

Then:

:<math>\operatorname{sn}(u,m) = y_0 \text{ as }n \rightarrow\infty</math>.