s21ca: FL CL CPP AD

NAG FL Interface
s21caf (jacellip_real)

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NAG Library Manual, Mark 27.3

Interfaces: FL CL CPP AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

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

1 Purpose

s21caf evaluates the Jacobian elliptic functions sn, cn and dn.

2 Specification

Fortran Interface

Subroutine s21caf (

u, m, sn, cn, dn, ifail)

Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	u, m
Real (Kind=nag_wp), Intent (Out)	::	sn, cn, dn

C Header Interface

#include <nag.h>

void	s21caf_ (const double u, const double m, double sn, double cn, double dn, Integer ifail)

The routine may be called by the names s21caf or nagf_specfun_jacellip_real.

3 Description

s21caf 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)).

s21caf 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: $u$ – Real (Kind=nag_wp) Input

2: $m$ – Real (Kind=nag_wp) Input

On entry: the argument

u

and the argument

m

of the functions, respectively.

Constraints:

$abs (u) \leq \sqrt{λ}$ , where $λ = 1 / x02amf$ ;
if $abs (u) < 1 / \sqrt{λ}$ , $abs (m) \leq \sqrt{λ}$ .

3: $sn$ – Real (Kind=nag_wp) Output

4: $cn$ – Real (Kind=nag_wp) Output

5: $dn$ – Real (Kind=nag_wp) Output

On exit: the values of the functions

sn u

cn u

and

dn u

, respectively.

6: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $| u |$ is too large: $| u | = ⟨ value ⟩$ it must be less than $⟨ value ⟩$ .

$ifail = 2$: On entry, $| m |$ is too large when used in conjunction with the supplied argument u: $| m | = ⟨ value ⟩$ it must be less than $⟨ value ⟩$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

In principle the routine 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

s21caf 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 FL Interfaces21caf (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 FL Interface
s21caf (jacellip_real)