s21ca: FL CL CPP AD

NAG CL Interface
s21cac (jacellip_real)

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NAG Library Manual, Mark 28.3

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s21ca: FL CL CPP AD

▸▿ 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

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CL Name Style:
Short (a00aac)
Long (impl_details)
Full (nag_info_impl_details)

1 Purpose

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

2 Specification

#include <nag.h>

void	s21cac (double u, double m, double sn, double cn, double dn, NagError fail)

The function may be called by the names: s21cac, nag_specfun_jacellip_real or nag_real_jacobian_elliptic.

3 Description

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)).

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: $u$ – double Input

2: $m$ – double Input

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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface 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

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.

s21ca: FL CL CPP AD

NAG CL Interfaces21cac (jacellip_​real)

▸▿ 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

NAG CL Interface
s21cac (jacellip_real)