NAG C Library Function Document

nag_real_jacobian_elliptic (s21cac)

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NAG C Library Manual, Mark 26

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

s Chapter Contents

s Chapter Introduction

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

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

2

Specification

#include <nag.h>

#include <nags.h>

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

3

Description

nag_real_jacobian_elliptic (s21cac) evaluates the Jacobian elliptic functions of argument

u

and argument

m

\begin{array}{lcl} sn (u ∣ m) & = & \sin ϕ, \\ cn (u ∣ m) & = & \cos ϕ, \\ dn (u ∣ m) & = & \sqrt{1 - m \sin^{2} ϕ}, \end{array}

where

ϕ

, called the amplitude of

u

, is defined by the integral

u = \int_{0}^{ϕ} \frac{d θ}{\sqrt{1 - m \sin^{2} θ}} .

The elliptic functions are sometimes written simply as

sn u

cn u

and

dn u

, avoiding explicit reference to the argument

m

Another nine elliptic functions may be computed via the formulae

\begin{array}{lcl} 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 \end{array}

(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.

4

References

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

5

Arguments

1: $u$ – doubleInput

2: $m$ – doubleInput

On entry: the argument

u

and the argument

m

of the functions, respectively.

Constraints:

$abs (u) \leq \sqrt{λ}$ , where $λ = 1 / nag_real_safe_small_number$ ;
if $abs (u) < 1 / \sqrt{λ}$ , $abs (m) \leq \sqrt{λ}$ .

3: $sn$ – double *Output

4: $cn$ – double *Output

5: $dn$ – double *Output

On exit: the values of the functions

sn u

cn u

and

dn u

, respectively.

6: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error 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.
NE_BAD_PARAM: On entry, argument $〈value〉$ 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, $|m|$ is too large when used in conjunction with the supplied argument u: $|m| = 〈value〉$ it must be less than $〈value〉$ .

On entry, $|u|$ is too large: $|u| = 〈value〉$ it must be less than $〈value〉$ .

7

Accuracy

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.

8

Parallelism and Performance

nag_real_jacobian_elliptic (s21cac) is not threaded in any implementation.

9

Further Comments

None.

10

Example

This example reads values of the argument

u

and argument

m

from a file, evaluates the function and prints the results.

NAG C Library Function Document

nag_real_jacobian_elliptic (s21cac)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results