Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	ak2
Complex (Kind=nag_wp), Intent (In)	::	z
Complex (Kind=nag_wp), Intent (Out)	::	sn, cn, dn

C Header Interface

#include <nag.h>

void	s21cbf_ (const Complex z, const double ak2, Complex sn, Complex cn, Complex dn, Integer ifail)

The routine may be called by the names s21cbf or nagf_specfun_jacellip_complex.

3 Description

s21cbf evaluates the Jacobian elliptic functions

sn (z ∣ k)

cn (z ∣ k)

and

dn (z ∣ k)

given by

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

where

z

is a complex argument,

k

is a real argument (the modulus) with

k^{2} \leq 1

and

ϕ

(the amplitude of

z

) is defined by the integral

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

The above definitions can be extended for values of

k^{2} > 1

(see Salzer (1962)) by means of the formulae

\begin{array}{lcl} sn (z ∣ k) & = & k_{1} sn (k z ∣ k_{1}) \\ cn (z ∣ k) & = & dn (k z ∣ k_{1}) \\ dn (z ∣ k) & = & cn (k z ∣ k_{1}), \end{array}

where

k_{1} = 1 / k

Special values include

\begin{array}{lcl} sn (z ∣ 0) & = & \sin z \\ cn (z ∣ 0) & = & \cos z \\ dn (z ∣ 0) & = & 1 \\ sn (z ∣ 1) & = & \tanh z \\ cn (z ∣ 1) & = & sech z \\ dn (z ∣ 1) & = & sech z . \end{array}

These functions are often simply written as

sn z

cn z

and

dn z

, thereby avoiding explicit reference to the argument

k

. They can also be expressed in terms of Jacobian theta functions (see s21ccf).

Another nine elliptic functions may be computed via the formulae

\begin{array}{lcl} cd z & = & cn z / dn z \\ sd z & = & sn z / dn z \\ nd z & = & 1 / dn z \\ dc z & = & dn z / cn z \\ nc z & = & 1 / cn z \\ sc z & = & sn z / cn z \\ ns z & = & 1 / sn z \\ ds z & = & dn z / sn z \\ cs z & = & cn z / sn z \end{array}

(see Abramowitz and Stegun (1972)).

The values of

sn z

cn z

and

dn z

are obtained by calls to s21caf. Further details can be found in Section 9.

4 References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Salzer H E (1962) Quick calculation of Jacobian elliptic functions Comm. ACM 5 399

5 Arguments

1: $z$ – Complex (Kind=nag_wp) Input

On entry: the argument

z

of the functions.

Constraints:

$abs (Re (z)) \leq = \sqrt{λ}$ ;
$abs (Im (z)) \leq \sqrt{λ}$ , where $λ = 1 / x02amf$ .

2: $ak2$ – Real (Kind=nag_wp) Input

On entry: the value of

k^{2}

Constraint:

0.0 \leq ak2 \leq 1.0

3: $sn$ – Complex (Kind=nag_wp) Output

4: $cn$ – Complex (Kind=nag_wp) Output

5: $dn$ – Complex (Kind=nag_wp) Output

On exit: the values of the functions

sn z

cn z

and

dn z

, 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, $| Im (z) |$ is too large: $| Im (z) | = ⟨ value ⟩$ . It must be less than $⟨ value ⟩$ .

On entry, $| Re (z) |$ is too large: $| Re (z) | = ⟨ value ⟩$ . It must be less than $⟨ value ⟩$ .

On entry, $ak2 = ⟨ value ⟩$ .
Constraint: $ak2 \leq 1.0$ .

On entry, $ak2 = ⟨ value ⟩$ .
Constraint: $ak2 \geq 0.0$ .

$ifail = - 99$: 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

s21cbf is not threaded in any implementation.

9 Further Comments

The values of

sn z

cn z

and

dn z

are computed via the formulae

\begin{array}{lcccl} sn z & = & \frac{sn (u, k) dn (v, k^{'})}{1 - {dn}^{2} (u, k) {sn}^{2} (v, k^{'})} & + & i \frac{cn (u, k) dn (u, k) sn (v, k^{'}) cn (v, k^{'})}{1 - {dn}^{2} (u, k) {sn}^{2} (v, k^{'})} \\ cn z & = & \frac{cn (u, k) cn (v, k^{'})}{1 - {dn}^{2} (u, k) {sn}^{2} (v, k^{'})} & - & i \frac{sn (u, k) dn (u, k) sn (v, k^{'}) dn (v, k^{'})}{1 - {dn}^{2} (u, k) {sn}^{2} (v, k^{'})} \\ dn z & = & \frac{dn (u, k) cn (v, k^{'}) dn (v, k^{'})}{1 - {dn}^{2} (u, k) {sn}^{2} (v, k^{'})} & - & i \frac{k^{2} sn (u, k) cn (u, k) sn (v, k^{'})}{1 - {dn}^{2} (u, k) {sn}^{2} (v, k^{'})}, \end{array}

where

z = u + i v

and

k^{'} = \sqrt{1 - k^{2}}

(the complementary modulus).

10 Example

This example evaluates

sn z

cn z

and

dn z

z = - 2.0 + 3.0 i

when

k = 0.5

, and prints the results.

s21cb: FL CL CPP AD

NAG FL Interfaces21cbf (jacellip_​complex)

▸▿ 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
s21cbf (jacellip_complex)