s21da: FL CL CPP AD

NAG CL Interface
s21dac (ellipint_general_2)

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NAG Library Manual, Mark 28.3

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s21da: 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

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>

Complex

s21dac (Complex z, double akp, double a, double b, NagError *fail)

The function may be called by the names: s21dac, nag_specfun_ellipint_general_2 or nag_general_elliptic_integral_f.

3 Description

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

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: $z$ – Complex Input

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: $akp$ – double Input

On entry: the argument

k^{'}

of the function.

Constraint:

abs (akp) \leq λ

3: $a$ – double Input

On entry: the argument

a

of the function.

4: $b$ – double Input

On entry: the argument

b

of the function.

5: $fail$ – 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_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.
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: 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

s21dac is not threaded in any implementation.

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.

s21da: FL CL CPP AD

NAG CL Interfaces21dac (ellipint_​general_​2)

▸▿ 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
s21dac (ellipint_general_2)