NAG Library Function Document

nag_general_elliptic_integral_f (s21dac)

+− 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_general_elliptic_integral_f (s21dac) returns the value of the general elliptic integral of the second kind

F (z, k^{'}, a, b)

for a complex argument

z

2 Specification

#include <nag.h>

#include <nags.h>

Complex

nag_general_elliptic_integral_f (Complex z, double akp, double a, double b, NagError *fail)

3 Description

nag_general_elliptic_integral_f (s21dac) evaluates an approximation to the general elliptic integral of the second kind

F (z, k^{'}, a, b)

given by

F (z, k^{'}, a, b) = \int_{0}^{z} \frac{a + b ζ^{2}}{(1 + ζ^{2}) \sqrt{(1 + ζ^{2}) (1 + {k^{'}}^{2} ζ^{2})}} d ζ,

where

a

and

b

are real arguments,

z

is a complex argument whose real part is non-negative and

k^{'}

is a real argument (the complementary modulus). The evaluation of

F

is based on the Gauss transformation. Further details, in particular for the conformal mapping provided by

F

, can be found in Bulirsch (1960).

Special values include

F (z, k^{'}, 1, 1) = \int_{0}^{z} \frac{d ζ}{\sqrt{(1 + ζ^{2}) (1 + k^{'} 2 ζ^{2})}},

F_{1} (z, k^{'})

(the elliptic integral of the first kind) and

F (z, k^{'}, 1, {k^{'}}^{2}) = \int_{0}^{z} \frac{\sqrt{1 + {k^{'}}^{2} ζ^{2}}}{(1 + ζ^{2}) \sqrt{1 + ζ^{2}}} d ζ,

F_{2} (z, k^{'})

(the elliptic integral of the second kind). Note that the values of

F_{1} (z, k^{'})

and

F_{2} (z, k^{'})

are equal to

\tan^{- 1} (z)

in the trivial case

k^{'} = 1

nag_general_elliptic_integral_f (s21dac) is derived from an Algol 60 procedure given by Bulirsch (1960). Constraints are placed on the values of

z

and

k^{'}

in order to avoid the possibility of machine overflow.

4 References

Bulirsch R (1960) Numerical calculation of elliptic integrals and elliptic functions Numer. Math. 7 76–90

5 Arguments

1: z – ComplexInput

On entry: the argument

z

of the function.

Constraints:

$0.0 \leq z . re \leq λ$ ;
$abs (z . im) \leq λ$ , where $λ^{6} = 1 / nag_real_safe_small_number$ .

2: akp – doubleInput

On entry: the argument

k^{'}

of the function.

Constraint:

abs (akp) \leq λ

3: a – doubleInput

On entry: the argument

a

of the function.

4: b – doubleInput

On entry: the argument

b

of the function.

5: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_COMPLEX: On entry, $|z . im|$ is too large: $|z . im| = ⟨value⟩$ . It must not exceed $⟨value⟩$ .
On entry, $z . re < 0.0$ : $z . re = ⟨value⟩$ .
On entry, $z . re$ is too large: $z . re = ⟨value⟩$ . It must not exceed $⟨value⟩$ .
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.
NE_REAL: On entry, $|akp|$ is too large: $|akp| = ⟨value⟩$ . It must not exceed $⟨value⟩$ .
NE_S21_CONV: The iterative procedure used to evaluate the integral has failed to converge.

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 atan2 and log.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example evaluates the elliptic integral of the first kind

F_{1} (z, k^{'})

given by

F_{1} (z, k^{'}) = \int_{0}^{z} \frac{d ζ}{\sqrt{(1 + ζ^{2}) (1 + {k^{'}}^{2} ζ^{2})}},

where

z = 1.2 + 3.7 i

and

k^{'} = 0.5

, and prints the results.

NAG Library Function Documentnag_general_elliptic_integral_f (s21dac)

+− 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 Library Function Document

nag_general_elliptic_integral_f (s21dac)