NAG Library Function Document

1Purpose

nag_real_jacobian_elliptic (s21cac) evaluates the Jacobian elliptic functions sn, cn and dn.

2Specification

 #include #include
 void nag_real_jacobian_elliptic (double u, double m, double *sn, double *cn, double *dn, NagError *fail)

3Description

nag_real_jacobian_elliptic (s21cac) evaluates the Jacobian elliptic functions of argument $u$ and argument $m$,
 $snu∣m = sin⁡ϕ, cnu∣m = cos⁡ϕ, dnu∣m = 1-msin2⁡ϕ,$
where $\varphi$, called the amplitude of $u$, is defined by the integral
 $u=∫0ϕdθ 1-msin2⁡θ .$
The elliptic functions are sometimes written simply as $\mathrm{sn}u$, $\mathrm{cn}u$ and $\mathrm{dn}u$, avoiding explicit reference to the argument $m$.
Another nine elliptic functions may be computed via the formulae
 $cd⁡u = cn⁡u/dn⁡u sd⁡u = sn⁡u/dn⁡u nd⁡u = 1/dn⁡u dc⁡u = dn⁡u/cn⁡u nc⁡u = 1/cn⁡u sc⁡u = sn⁡u/cn⁡u ns⁡u = 1/sn⁡u ds⁡u = dn⁡u/sn⁡u cs⁡u = cn⁡u/sn⁡u$
(see Abramowitz and Stegun (1972)).
nag_real_jacobian_elliptic (s21cac) is based on a procedure given by Bulirsch (1960), and uses the process of the arithmetic-geometric mean (16.9 in Abramowitz and Stegun (1972)). Constraints are placed on the values of $u$ and $m$ in order to avoid the possibility of machine overflow.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Bulirsch R (1960) Numerical calculation of elliptic integrals and elliptic functions Numer. Math. 7 76–90

5Arguments

1:    $\mathbf{u}$doubleInput
2:    $\mathbf{m}$doubleInput
On entry: the argument $u$ and the argument $m$ of the functions, respectively.
Constraints:
• $\mathrm{abs}\left({\mathbf{u}}\right)\le \sqrt{\lambda }$, where $\lambda =1/{\mathbf{nag_real_safe_small_number}}$;
• if $\mathrm{abs}\left({\mathbf{u}}\right)<1/\sqrt{\lambda }$, $\mathrm{abs}\left({\mathbf{m}}\right)\le \sqrt{\lambda }$.
3:    $\mathbf{sn}$double *Output
4:    $\mathbf{cn}$double *Output
5:    $\mathbf{dn}$double *Output
On exit: the values of the functions $\mathrm{sn}u$, $\mathrm{cn}u$ and $\mathrm{dn}u$, respectively.
6:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_2
On entry, $\left|{\mathbf{m}}\right|$ is too large when used in conjunction with the supplied argument u: $\left|{\mathbf{m}}\right|=〈\mathit{\text{value}}〉$ it must be less than $〈\mathit{\text{value}}〉$.
On entry, $\left|{\mathbf{u}}\right|$ is too large: $\left|{\mathbf{u}}\right|=〈\mathit{\text{value}}〉$ it must be less than $〈\mathit{\text{value}}〉$.

7Accuracy

In principle the function is capable of achieving full relative precision in the computed values. However, the accuracy obtainable in practice depends on the accuracy of the standard elementary functions such as SIN and COS.

8Parallelism and Performance

nag_real_jacobian_elliptic (s21cac) is not threaded in any implementation.

None.

10Example

This example reads values of the argument $u$ and argument $m$ from a file, evaluates the function and prints the results.

10.1Program Text

Program Text (s21cace.c)

10.2Program Data

Program Data (s21cace.d)

10.3Program Results

Program Results (s21cace.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017