nag_elliptic_integral_rj (s21bdc) : NAG Library, Mark 24

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{array}{lcl} 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{array}

For

n

sufficiently large,

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

and the function may be approximated by a fifth order power series

\begin{array}{l} 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{array}

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^{- \frac{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).

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.

On entry,

r = 0.0

.
Constraint:

r \neq 0.0

On entry,

x = ⟨value⟩

y = ⟨value⟩

and

z = ⟨value⟩

.
Constraint: at most one of x, y and z is

0.0

.
The function is undefined.

On entry,

U = ⟨value⟩

r = ⟨value⟩

x = ⟨value⟩

y = ⟨value⟩

and

z = ⟨value⟩

.
Constraint:

|r| \leq U

and

x \leq U

and

y \leq U

and

z \leq U

On entry,

L = ⟨value⟩

r = ⟨value⟩

x = ⟨value⟩

y = ⟨value⟩

and

z = ⟨value⟩

.
Constraint:

|r| \geq L

and at most one of x, y and z is less than

L

On entry,

x = ⟨value⟩

y = ⟨value⟩

and

z = ⟨value⟩

.
Constraint:

x \geq 0.0

and

y \geq 0.0

and

z \geq 0.0

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.

You should consult the s Chapter Introduction which shows the relationship of this function to the classical definitions of the elliptic integrals.

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).

This example 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 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 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 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_elliptic_integral_rj (s21bdc)