s21bdc (ellipint_symm_3) : NAG Library CL Interface, Mark 27

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

The function may be called by the names: s21bdc, nag_specfun_ellipint_symm_3 or 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.

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 7 in the Introduction to the NAG Library CL Interface).

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.

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.

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.

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.

s21bdc is not threaded in any implementation.

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 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 CL Interface
s21bdc (ellipint_symm_3)

▸▿ 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 Interfaces21bdc (ellipint_​symm_​3)

▸▿ 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
s21bdc (ellipint_symm_3)