s21bcf : NAG Library, Mark 26

s21bcf returns a value of the symmetrised elliptic integral of the second kind, via the function name.

Fortran Interface

Function s21bcf (

x, y, z, ifail)

Real (Kind=nag_wp)	::	s21bcf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x, y, z

C Header Interface

#include nagmk26.h

double

s21bcf_ ( const double *x, const double *y, const double *z, Integer *ifail)

s21bcf 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 routine 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.

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

and

z

of the function.

Constraint:

x

y \geq 0.0

z > 0.0

and only one of x and y may be zero.

On entry: ifail must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this argument, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

On entry, either x or y is negative, or both x and y are zero; the function is undefined.

On entry,

z \leq 0.0

; the function is undefined.

On entry, either z is too close to zero or both x and y are too close to zero: there is a danger of setting overflow. See also the Users' Note for your implementation.

On entry, at least one of x, y and z is too large: there is a danger of setting underflow. On soft failure the routine returns zero. See also the Users' Note for your implementation.

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

In principle the routine 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.

s21bcf 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.

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

NAG Library Routine Document

s21bcf (ellipint_symm_2)

▸▿ 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 Routine Document

s21bcf (ellipint_symm_2)

▸▿ 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