NAG Library Routine Document

S21CBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

S21CBF evaluates the Jacobian elliptic functions

sn z

cn z

and

dn z

for a complex argument

z

2 Specification

SUBROUTINE S21CBF (

Z, AK2, SN, CN, DN, IFAIL)

INTEGER	IFAIL
REAL (KIND=nag_wp)	AK2
COMPLEX (KIND=nag_wp)	Z, SN, CN, DN

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 parameter (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 parameter

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 8.

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 Parameters

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 – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. 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,	$AK2 < 0.0$ ,
or	$AK2 > 1.0$ ,
or	$abs (Re (Z)) > \sqrt{λ}$ ,
or	$abs (Im (Z)) > \sqrt{λ}$ , where $λ = 1 / X02AMF$ .

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 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).

9 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.

NAG Library Routine DocumentS21CBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

S21CBF