NAG Library Function Document

nag_general_elliptic_integral_f (s21dac)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_general_elliptic_integral_f (s21dac) returns the value of the general elliptic integral of the second kind Fz,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 Fz,k,a,b given by
Fz,k,a,b=0za+bζ2 1+ζ21+ζ21+k2ζ2 dζ,  
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 = 0 z d ζ 1 + ζ 2 1 + k 2 ζ 2 ,  
or F1z,k (the elliptic integral of the first kind) and
Fz,k,1,k2=0z1+k2ζ2 1+ζ21+ζ2 dζ,  
or F2z,k (the elliptic integral of the second kind). Note that the values of F1z,k and F2z,k are equal to tan-1z 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:
  • 0.0z.reλ;
  • absz.imλ, where λ6=1/nag_real_safe_small_number.
2:     akp doubleInput
On entry: the argument k of the function.
Constraint: absakp 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 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_general_elliptic_integral_f (s21dac) is not threaded in any implementation.

9
Further Comments

None.

10
Example

This example evaluates the elliptic integral of the first kind F1z,k given by
F1z,k=0zdζ 1+ζ21+k2ζ2 ,  
where z=1.2+3.7i and k=0.5, and prints the results.

10.1
Program Text

Program Text (s21dace.c)

10.2
Program Data

Program Data (s21dace.d)

10.3
Program Results

Program Results (s21dace.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017