nag_elliptic_integral_rd (s21bcc) : NAG Library, Mark 24

3 Description

nag_elliptic_integral_rd (s21bcc) calculates an approximate value for the integral

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

where

x

y \geq 0

, at most one of

x

and

y

is zero, and

z > 0

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 \\ μ_{n} & = & (x_{n} + y_{n} + 3 z_{n}) / 5 \\ 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{array}

For

n

sufficiently large,

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

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

\begin{array}{l} R_{D} (x, y, z) = & 3 \sum_{m = 0}^{n - 1} \frac{4^{- m}}{(z_{m} + λ_{n}) \sqrt{z_{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} + 3 Z_{n}^{m}) / 2 m .

The truncation error in this expansion is bounded by

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

and the recursive 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_{D} (x, x, x) = x^{- 3 / 2}

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

5 Arguments

1: x – doubleInput

2: y – doubleInput

3: z – doubleInput

On entry: the arguments

x

y

and

z

of the function.

Constraint:

x

y \geq 0.0

z > 0.0

and only one of x and y may be zero.

4: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

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.

NE_REAL_ARG_EQ

On entry, x and y are both

0.0

.
Constraint: at most one of x and y is

0.0

.
The function is undefined.

NE_REAL_ARG_GE

On entry,

U = ⟨value⟩

x = ⟨value⟩

y = ⟨value⟩

and

z = ⟨value⟩

.
Constraint:

x < U

and

y < U

and

z < U

.
There is a danger of setting underflow and the function returns zero.

NE_REAL_ARG_LE

On entry,

z = ⟨value⟩

.
Constraint:

z > 0.0

.
The function is undefined.

NE_REAL_ARG_LT

On entry,

L = ⟨value⟩

x = ⟨value⟩

y = ⟨value⟩

and

z = ⟨value⟩

.
Constraint:

z \geq L

and (

x \geq L

y \geq L

).
The function is undefined.

On entry,

x = ⟨value⟩

and

y = ⟨value⟩

.
Constraint:

x \geq 0.0

and

y \geq 0.0

.
The function is undefined.

7 Accuracy

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.

NAG Library Function Document

nag_elliptic_integral_rd (s21bcc)

+− 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_rd (s21bcc)

+− 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_rd (s21bcc)