nag_elliptic_integral_rj (s21bdc) : NAG Library, Mark 23

nag_elliptic_integral_rj (s21bdc) returns a value of the symmetrised elliptic integral of the third kind.

nag_elliptic_integral_rj (s21bdc) calculates an approximation to the integral

R_{J} (x, y, z, ρ) = \frac{3}{2} \int_{0}^{\infty} \frac{d t}{(t + ρ) \sqrt{(t + x) (t + y) (t + z)}}

where

x

y

z \geq 0

ρ \neq 0

and at most one of

x

y

and

z

is zero.

ρ < 0

, the result computed is the Cauchy principal value of the integral.

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

\begin{aligned} x_{0} & = x, y_{0} = y, z_{0} = z, ρ_{0} = ρ \\ μ_{n} & = (x_{n} + y_{n} + z_{n} + 2 ρ_{n}) / 5 \\ X_{n} & = 1 - x_{n} / μ_{n} \\ Y_{n} & = 1 - y_{n} / μ_{n} \\ Z_{n} & = 1 - z_{n} / μ_{n} \\ P_{n} & = 1 - ρ_{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 \\ ρ_{n + 1} & = (ρ_{n} + λ_{n}) / 4 \\ α_{n} & = {[ρ_{n} (\sqrt{x_{n}} + \sqrt{y_{n}} + \sqrt{z_{n}}) + \sqrt{x_{n} y_{n} z_{n}}]}^{2} \\ β_{n} & = ρ_{n} {(ρ_{n} + λ_{n})}^{2} . \end{aligned}

For

n

sufficiently large,

ε_{n} = \max (|X_{n}|, |Y_{n}|, |Z_{n}|, |P_{n}|) \sim 1 / 4^{n}

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

\begin{aligned} R_{J} (x, y, z, ρ) & = 3 \sum_{m = 0}^{n - 1} 4^{- m} R_{C} (α_{m}, β_{m}) \\ + \frac{4^{- n}}{\sqrt{μ_{n}^{3}}} (1 + \frac{3}{7} S_{n}^{(2)} + \frac{1}{3} S_{n}^{(3)} + \frac{3}{22} {(S_{n}^{(2)})}^{2} + \frac{3}{11} S_{n}^{(4)} + \frac{3}{13} S_{n}^{(2)} S_{n}^{(3)} + \frac{3}{13} S_{n}^{(5)}), \end{aligned}

where

S_{n}^{(m)} = (X_{n}^{m} + Y_{n}^{m} + Z_{n}^{m} + 2 P_{n}^{m}) / 2 m

The truncation error in this expansion is bounded by

3 ε_{n}^{6} / \sqrt{{(1 - ε_{n})}^{3}}

and the recursion process is terminated when this quantity is negligible compared with the machine precision. The function may fail either because it has been called with arguments outside the domain of definition or with arguments so extreme that there is an unavoidable danger of setting underflow or overflow.

Note:

R_{J} (x, x, x, x) = x^{- 3 / 2},

so there exists a region of extreme arguments for which the function value is not representable. .

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

On entry: the arguments

x

y

z

and

ρ

of the function.

Constraint: x, y,

z \geq 0.0

r \neq 0.0

and at most one of x, y and z may be zero.

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

On entry,

〈parameters〉

must not be equal to 0.0:

〈parameters〉 = 〈value〉

.
At least two of x, y and z are zero and the function is undefined.

On entry, r must not be equal to 0.0:

r = 〈value〉

.
The function is undefined.

On entry,

|r|

must not be greater than

〈value〉

|r| = 〈value〉

On entry,

x = 〈value〉

.
Constraint:

x \leq 〈value〉

On entry,

y = 〈value〉

.
Constraint:

y \leq 〈value〉

On entry,

z = 〈value〉

.
Constraint:

z \leq 〈value〉

On entry, either r is too close to zero, or any two of x, y and z are too close to zero; there is a danger of setting overflow.

On entry,

〈parameters〉

must not be less than

〈value〉

〈parameters〉 = 〈value〉

On entry, x must not be less than 0.0:

x = 〈value〉

On entry, y must not be less than 0.0:

y = 〈value〉

On entry, z must not be less than 0.0:

z = 〈value〉

.
The function is undefined.

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 the argument r is equal to any of the other arguments, the function reduces to the integral

R_{D}

, computed by nag_elliptic_integral_rd (s21bcc).

Symmetrised elliptic integrals returned by functions nag_elliptic_integral_rj (s21bdc), nag_elliptic_integral_rc (s21bac), nag_elliptic_integral_rf (s21bbc) and nag_elliptic_integral_rd (s21bcc) 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.

NAG Library Function Document

nag_elliptic_integral_rj (s21bdc)

+− 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_rj (s21bdc)

+− 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 Document

nag_elliptic_integral_rj (s21bdc)