nag_elliptic_integral_rf (s21bbc) : NAG Library, Mark 23

nag_elliptic_integral_rf (s21bbc) returns a value of the symmetrised elliptic integral of the first kind.

nag_elliptic_integral_rf (s21bbc) calculates an approximation to the integral

R_{F} (x, y, z) = \frac{1}{2} \int_{0}^{\infty} \frac{d t}{\sqrt{(t + x) (t + y) (t + z)}}

where

x

,

y

,

z \geq 0

and at most one is zero.

The basic algorithm, which is due to Carlson (1979) and Carlson (1988), is to reduce the arguments recursively towards their mean by the rule:

x_{0} = \min (x, y, z), z_{0} = \max (x, y, z), y_{0} =

remaining third intermediate value argument. (This ordering, which is possible because of the symmetry of the function, is done for technical reasons related to the avoidance of overflow and underflow.)

\begin{aligned} μ_{n} & = (x_{n} + y_{n} + z_{n}) / 3 \\ X_{n} & = 1 - x_{n} / μ_{n} \\ Y_{n} & = 1 - y_{n} / μ_{n} \\ Z_{n} & = 1 - z_{n} / μ_{n} \\ λ_{n} & = \sqrt{x_{n} y_{n}} + \sqrt{y_{n} z_{n}} + \sqrt{z_{n} x_{n}} \\ x_{n + 1} & = (x_{n} + λ_{n}) / 4 \\ y_{n + 1} & = (y_{n} + λ_{n}) / 4 \\ z_{n + 1} & = (z_{n} + λ_{n}) / 4 \end{aligned}

ε_{n} = \max (|X_{n}|, |Y_{n}|, |Z_{n}|)

and the function may be approximated adequately by a 5th-order power series:

R_{F} (x, y, z) = \frac{1}{\sqrt{μ_{n}}} (1 - \frac{E_{2}}{10} + \frac{E_{2}^{2}}{24} - \frac{3 E_{2} E_{3}}{44} + \frac{E_{3}}{14})

where

E_{2} = X_{n} Y_{n} + Y_{n} Z_{n} + Z_{n} X_{n}, E_{3} = X_{n} Y_{n} Z_{n}

.

The truncation error involved in using this approximation is bounded by

ε_{n}^{6} / 4 (1 - ε_{n})

and the recursive process is stopped when this truncation error is negligible compared with the machine precision.

Within the domain of definition, the function value is itself representable for all representable values of its arguments. However, for values of the arguments near the extremes the above algorithm must be modified so as to avoid causing underflows or overflows in intermediate steps. In extreme regions arguments are pre-scaled away from the extremes and compensating scaling of the result is done before returning to the calling program.

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

Carlson B C (1979) Computing elliptic integrals by duplication Numerische Mathematik 33 1–16

Carlson B C (1988) A table of elliptic integrals of the third kind Math. Comput. 51 267–280

In principle the function is capable of producing full machine precision. However round-off errors in internal arithmetic will result in slight loss of accuracy. This loss should never be excessive as the algorithm does not involve any significant amplification of round-off error. It is reasonable to assume that the result is accurate to within a small multiple of the machine precision.

If two arguments are equal, the function reduces to the elementary integral

R_{C}

, computed by nag_elliptic_integral_rc (s21bac).

Symmetrised elliptic integrals returned by functions nag_elliptic_integral_rf (s21bbc), nag_elliptic_integral_rc (s21bac), nag_elliptic_integral_rd (s21bcc) and nag_elliptic_integral_rj (s21bdc) can be related to the more traditional canonical forms (see Abramowitz and Stegun (1972)), as described in the s Chapter Introduction.

This example program simply generates a small set of nonextreme arguments which are used with the function to produce the table of low accuracy results.

None.

NAG Library Function Document

nag_elliptic_integral_rf (s21bbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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 Function Documentnag_elliptic_integral_rf (s21bbc)