22 Jacobian Elliptic Functions Computation 22.19 Physical Applications 22.21 Tables

§22.20 Methods of Computation

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Keywords:: Jacobian elliptic functions, computation
Permalink:: http://dlmf.nist.gov/22.20
See also:: Annotations for Ch.22

§22.20(i) Via Theta Functions
§22.20(ii) Arithmetic-Geometric Mean
§22.20(iii) Landen Transformations
§22.20(iv) Lattice Calculations
§22.20(v) Inverse Functions
§22.20(vi) Related Functions
§22.20(vii) Further References

§22.20(i) Via Theta Functions

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Permalink:: http://dlmf.nist.gov/22.20.i
See also:: Annotations for §22.20 and Ch.22

A powerful way of computing the twelve Jacobian elliptic functions for real or complex values of both the argument and the modulus 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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Keywords:: Jacobian elliptic functions, arithmetic-geometric mean
Notes:: See Walker (1996, pp. 141–143).
Referenced by:: item 1
Permalink:: http://dlmf.nist.gov/22.20.ii
See also:: Annotations for §22.20 and Ch.22

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

22.20.1

$a_{n}=\tfrac{1}{2}\left(a_{n-1}+b_{n-1}\right),$

$b_{n}=\left(a_{n-1}b_{n-1}\right)^{1/2},$

$c_{n}=\tfrac{1}{2}\left(a_{n-1}-b_{n-1}\right),$

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Symbols:: $a_{n}$ : numbers, $b_{n}$ : numbers, $c_{n}$ : numbers and : positive
Referenced by:: §22.20(ii), §22.20(iv), §22.20(iv), §22.20(ii)
Permalink:: http://dlmf.nist.gov/22.20.E1
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §22.20(ii), §22.20 and Ch.22

for $n\geq 1$ , where the square root is chosen so that $\operatorname{ph}b_{n}=\tfrac{1}{2}(\operatorname{ph}a_{n-1}+\operatorname{ph}% b_{n-1})$ , where $\operatorname{ph}a_{n-1}$ and $\operatorname{ph}b_{n-1}$ 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 $M=M\left(a_{0},b_{0}\right)$ , the arithmetic-geometric mean of $a_{0},b_{0}$ . And since

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

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Symbols:: $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $a_{n}$ : numbers, $b_{n}$ : numbers, $c_{n}$ : numbers and : positive
Permalink:: http://dlmf.nist.gov/22.20.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §22.20(ii), §22.20 and Ch.22

convergence is very rapid.

For 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:: : 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
See also:: Annotations for §22.20(ii), §22.20 and Ch.22

22.20.4 $\phi_{n-1}=\frac{1}{2}\left(\phi_{n}+\operatorname{arcsin}\left(\frac{c_{n}}{a% _{n}}\sin\phi_{n}\right)\right),$

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Symbols:: $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\sin\NVar{z}$ : sine function, $a_{n}$ : numbers, $c_{n}$ : numbers, : positive and $\phi_{N}$ : approximations
Referenced by:: §22.20(ii)
Permalink:: http://dlmf.nist.gov/22.20.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §22.20(ii), §22.20 and Ch.22

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

22.20.5

$\operatorname{sn}\left(x,k\right)=\sin\phi_{0},$

$\operatorname{cn}\left(x,k\right)=\cos\phi_{0},$

$\operatorname{dn}\left(x,k\right)=\frac{\cos\phi_{0}}{\cos\left(\phi_{1}-\phi_% {0}\right)},$

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Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, : real, : modulus and $\phi_{N}$ : approximations
Referenced by:: (22.20.5), §22.20(ii), Erratum (V1.0.10) for Equation (22.20.5), Erratum (V1.0.10) for Equation (22.20.5)
Permalink:: http://dlmf.nist.gov/22.20.E5
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
Note (effective with 1.0.10):: A note was added after (22.20.5) to deal with cases when computation of $\operatorname{dn}\left(x,k\right)$ with (22.20.5) becomes numerically unstable near .
Suggested 2014-10-20 by Hartmut Henkel
See also:: Annotations for §22.20(ii), §22.20 and Ch.22

and the subsidiary functions can be found using (22.2.10). This formula for $\operatorname{dn}$ becomes unstable near . If only the value of $\operatorname{dn}\left(x,k\right)$ at is required then the exact value is in the table 22.5.1. If both and are real then $\operatorname{dn}$ is strictly positive and $\operatorname{dn}\left(x,k\right)=\sqrt{1-k^{2}{\operatorname{sn}}^{2}\left(x,% k\right)}$ which follows from (22.6.1). If either or is complex then (22.2.6) gives the definition of $\operatorname{dn}\left(x,k\right)$ as a quotient of theta functions.

Example

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See also:: Annotations for §22.20(ii), §22.20 and Ch.22

To compute $\operatorname{sn}$ , $\operatorname{cn}$ , $\operatorname{dn}$ to 10D when , .

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), $\operatorname{sn}\left(0.8,0.65\right)=0.69506\;42165$ , $\operatorname{cn}\left(0.8,0.65\right)=0.71894\;76580$ , $\operatorname{dn}\left(0.8,0.65\right)=0.89212\;34349$ .

§22.20(iii) Landen Transformations

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Keywords:: Jacobian elliptic functions, Landen transformations, ascending, descending
Permalink:: http://dlmf.nist.gov/22.20.iii
See also:: Annotations for §22.20 and Ch.22

By application of the transformations given in §§22.7(i) and 22.7(ii), 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

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See also:: Annotations for §22.20(iii), §22.20 and Ch.22

To compute $\operatorname{dn}\left(x,k\right)$ to 6D for , $k^{2}=0.19$ , $k^{\prime}=0.9$ .

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 $\operatorname{dn}\left(0.19,\tfrac{1}{19}\right)=0.999951$ . Then by using (22.7.4) we have $\operatorname{dn}\left(0.2,\sqrt{0.19}\right)=0.996253$ .

If needed, the corresponding values of $\operatorname{sn}$ and $\operatorname{cn}$ 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

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Keywords:: Jacobian elliptic functions, computation, lattice
Permalink:: http://dlmf.nist.gov/22.20.iv
See also:: Annotations for §22.20 and Ch.22

If either $\tau$ or $q=e^{i\pi\tau}$ is given, then we use $k={\theta_{2}}^{2}\left(0,q\right)/{\theta_{3}}^{2}\left(0,q\right)$ , $k^{\prime}={\theta_{4}}^{2}\left(0,q\right)/{\theta_{3}}^{2}\left(0,q\right)$ , $K=\frac{1}{2}\pi{\theta_{3}}^{2}\left(0,q\right)$ , 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 $\Im k^{\prime}/\Im k<0$ , then $K,K^{\prime}$ can be found from

22.20.6

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

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

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Symbols:: $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : modulus and $k^{\prime}$ : complementary modulus
Referenced by:: §22.20(iv), (23.22.1), §23.22(ii)
Permalink:: http://dlmf.nist.gov/22.20.E6
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §22.20(iv), §22.20 and Ch.22

using the arithmetic-geometric mean.

Example 1

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See also:: Annotations for §22.20(iv), §22.20 and Ch.22

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 $\left(\Gamma\left(\tfrac{1}{4}\right)\right)^{2}/\left(4\sqrt{\pi}\right)$ ; compare (23.17.3) and (23.22.2).

Example 2

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See also:: Annotations for §22.20(iv), §22.20 and Ch.22

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

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Keywords:: computation, inverse Jacobian elliptic functions
Permalink:: http://dlmf.nist.gov/22.20.v
See also:: Annotations for §22.20 and Ch.22

See Wachspress (2000).

§22.20(vi) Related Functions

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Keywords:: Jacobi’s epsilon function, Jacobi’s zeta function, amplitude ( $\operatorname{am}$ ) function, computation
Permalink:: http://dlmf.nist.gov/22.20.vi
See also:: Annotations for §22.20 and Ch.22

$\operatorname{am}\left(x,k\right)$ 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 $\operatorname{am}\left(x,k\right)$ .

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

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Keywords:: Jacobian elliptic functions, computation
Permalink:: http://dlmf.nist.gov/22.20.vii
See also:: Annotations for §22.20 and Ch.22

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).