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§22.20 Methods of Computation

Contents

§22.20(i) Via Theta Functions

A powerful way of computing the twelve Jacobian elliptic functions for real or complex values of both the argument z and the modulus k is to use the definitions in terms of theta functions given in §22.2, obtaining the theta functions via methods described in §20.14.

§22.20(ii) Arithmetic-Geometric Mean

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Notes:
See Walker (1996, pp. 141–143).
Keywords:
Jacobian elliptic functions, arithmetic-geometric mean
Referenced by:
‣ §23.22(ii)
Permalink:
http://dlmf.nist.gov/22.20.ii

Given real or complex numbers a_{0},b_{0}, with b_{0}/a_{0} not real and negative, define

22.20.1
a_{n}=\tfrac{1}{2}\left(a_{{n-1}}+b_{{n-1}}\right),
b_{n}=\left(a_{{n-1}}b_{{n-1}}\right)^{{1/2}},
c_{n}=\tfrac{1}{2}\left(a_{{n-1}}-b_{{n-1}}\right),

for n\geq 1, where the square root is chosen so that \mathop{\mathrm{ph}\/}\nolimits b_{n}=\tfrac{1}{2}(\mathop{\mathrm{ph}\/}\nolimits a_{{n-1}}+\mathop{\mathrm{ph}\/}\nolimits b_{{n-1}}), where \mathop{\mathrm{ph}\/}\nolimits a_{{n-1}} and \mathop{\mathrm{ph}\/}\nolimits b_{{n-1}} are chosen so that their difference is numerically less than \pi. Then as n\to\infty sequences \{ a_{n}\}, \{ b_{n}\} converge to a common limit M=M(a_{0},b_{0}), the arithmetic-geometric mean of a_{0},b_{0}. And since

22.20.2\max\left(\left|a_{n}-M\right|,\left|b_{n}-M\right|,\left|c_{n}\right|\right)\leq\text{(const.)}\times 2^{{-2^{n}}},

convergence is very rapid.

For x real and k\in(0,1), use (22.20.1) with a_{0}=1, b_{0}=k^{{\prime}}\in(0,1), c_{0}=k, and continue until c_{N} is zero to the required accuracy. Next, compute \phi _{N},\phi _{{N-1}},\dots,\phi _{0}, where

22.20.3\phi _{N}=2^{N}a_{N}x,
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Symbols:
x: real, a_{n}: numbers and \phi _{N}: approximations
Referenced by:
‣ §22.20(ii)
Permalink:
http://dlmf.nist.gov/22.20.E3
Encodings:
TeX, pMML, png
22.20.4\phi _{{n-1}}=\frac{1}{2}\left(\phi _{n}+\mathop{\mathrm{arcsin}\/}\nolimits\!\left(\frac{c_{n}}{a_{n}}\mathop{\sin\/}\nolimits\phi _{n}\right)\right),

and the inverse sine has its principal value (§4.23(ii)). Then

22.20.5
\mathop{\mathrm{sn}\/}\nolimits\left(x,k\right)=\mathop{\sin\/}\nolimits\phi _{0},
\mathop{\mathrm{cn}\/}\nolimits\left(x,k\right)=\mathop{\cos\/}\nolimits\phi _{0},
\mathop{\mathrm{dn}\/}\nolimits\left(x,k\right)=\frac{\mathop{\cos\/}\nolimits\phi _{0}}{\mathop{\cos\/}\nolimits\!\left(\phi _{1}-\phi _{0}\right)},

and the subsidiary functions can be found using (22.2.10).

See also Wachspress (2000).

Example

To compute \mathop{\mathrm{sn}\/}\nolimits, \mathop{\mathrm{cn}\/}\nolimits, \mathop{\mathrm{dn}\/}\nolimits to 10D when x=0.8, k=0.65.

Four iterations of (22.20.1) lead to c_{4}=\Sci{6.5}{-12}. From (22.20.3) and (22.20.4) we obtain \phi _{1}=1.40213\; 91827 and \phi _{0}=0.76850\; 92170. Then from (22.20.5), \mathop{\mathrm{sn}\/}\nolimits\left(0.8,0.65\right)=0.69506\; 42165, \mathop{\mathrm{cn}\/}\nolimits\left(0.8,0.65\right)=0.71894\; 76580, \mathop{\mathrm{dn}\/}\nolimits\left(0.8,0.65\right)=0.89212\; 34349.

§22.20(iii) Landen Transformations

By application of the transformations given in §§22.7(i) and 22.7(ii), k or k^{{\prime}} can always be made sufficently small to enable the approximations given in §22.10(ii) to be applied. The rate of convergence is similar to that for the arithmetic-geometric mean.

Example

To compute \mathop{\mathrm{dn}\/}\nolimits\left(x,k\right) to 6D for x=0.2, k^{2}=0.19, k^{{\prime}}=0.9.

From (22.7.1), k_{1}=\tfrac{1}{19} and x/(1+k_{1})=0.19. From the first two terms in (22.10.6) we find \mathop{\mathrm{dn}\/}\nolimits\left(0.19,\tfrac{1}{19}\right)=0.999951. Then by using (22.7.4) we have \mathop{\mathrm{dn}\/}\nolimits\left(0.2,\sqrt{0.19}\right)=0.996253.

If needed, the corresponding values of \mathop{\mathrm{sn}\/}\nolimits and \mathop{\mathrm{cn}\/}\nolimits can be found subsequently by applying (22.10.4) and (22.7.2), followed by (22.10.5) and (22.7.3).

§22.20(iv) Lattice Calculations

If either \tau or q=e^{{i\pi\tau}} is given, then we use k={\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(0,q\right)/{\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right), k^{{\prime}}={\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(0,q\right)/{\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right), K=\frac{1}{2}\pi{\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right), and K^{{\prime}}=-i\tau K, obtaining the values of the theta functions as in §20.14.

If k,k^{{\prime}} are given with k^{2}+{k^{{\prime}}}^{2}=1 and \imagpart{k^{{\prime}}}/\imagpart{k}<0, then K,K^{{\prime}} can be found from

22.20.6
K=\frac{\pi}{2M(1,k^{{\prime}})},
K^{{\prime}}=\frac{\pi}{2M(1,k)},

using the arithmetic-geometric mean.

Example 1

If k=k^{{\prime}}=1/\sqrt{2}, then three iterations of (22.20.1) give M=0.84721\; 30848, and from (22.20.6) K=\pi/(2M)=1.85407\; 46773 —in agreement with the value of \left(\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{4}\right)\right)^{2}/\left(4\sqrt{\pi}\right); compare (23.17.3) and (23.22.2).

Example 2

If k^{{\prime}}=1-i, then four iterations of (22.20.1) give K=1.23969\; 74481+i0.56499\; 30988.

§22.20(v) Inverse Functions

See Wachspress (2000).

§22.20(vi) Related Functions

\mathop{\mathrm{am}\/}\nolimits\left(x,k\right) can be computed from its definition (22.16.1) or from its Fourier series (22.16.9). Alternatively, Sala (1989) shows how to apply the arithmetic-geometric mean to compute \mathop{\mathrm{am}\/}\nolimits\left(x,k\right).

Jacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14. Jacobi’s zeta function can then be found by use of (22.16.32).

§22.20(vii) Further References

For additional information on methods of computation for the Jacobi and related functions, see the introductory sections in the following books: Lawden (1989), Curtis (1964b), Milne-Thomson (1950), and Spenceley and Spenceley (1947).

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