(This ordering, which is possible because of the symmetry of the function, is done for technical reasons related to the avoidance of overflow and underflow.)

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

ε_{n} = \max (|X_{n}|, |Y_{n}|, |Z_{n}|)

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

R_{F} (x, y, z) = \frac{1}{\sqrt{μ_{n}}} (1 - \frac{E_{2}}{10} + \frac{E_{2}^{2}}{24} - \frac{3 E_{2} E_{3}}{44} + \frac{E_{3}}{14})

where

E_{2} = X_{n} Y_{n} + Y_{n} Z_{n} + Z_{n} X_{n}

E_{3} = X_{n} Y_{n} Z_{n}

The truncation error involved in using this approximation is bounded by

ε_{n}^{6} / 4 (1 - ε_{n})

and the recursive process is stopped when this truncation error is negligible compared with the machine precision.

Within the domain of definition, the function value is itself representable for all representable values of its arguments. However, for values of the arguments near the extremes the above algorithm must be modified so as to avoid causing underflows or overflows in intermediate steps. In extreme regions arguments are prescaled away from the extremes and compensating scaling of the result is done before returning to the calling program.

References

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

Parameters

Compulsory Input Parameters

1: $x$ – double scalar
2: $y$ – double scalar
3: $z$ – double scalar: The arguments $x$ , $y$ and $z$ of the function.

Constraint: $x$ , y, $z \geq 0.0$ and only one of x, y and z may be zero.

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: On entry, one or more of x, y and z is negative; the function is undefined.

$ifail = 2$: On entry, two or more of x, y and z are zero; the function is undefined. On soft failure, the function returns zero.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

In principle nag_specfun_ellipint_symm_1 (s21bb) 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.

Further Comments

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

If two arguments are equal, the function reduces to the elementary integral

R_{C}

, computed by nag_specfun_ellipint_symm_1_degen (s21ba).

Example

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

Open in the MATLAB editor: s21bb_example

function s21bb_example


fprintf('s21bb example results\n\n');

x = [0.5   1   1.5];
y = x + 0.5;
z = y + 0.5;
result = x;

for j=1:numel(x)
  [result(j), ifail] = s21bb(x(j), y(j), z(j));
end

fprintf('    x      y      z      R_F(x,y,z)\n');
fprintf('%7.2f%7.2f%7.2f%12.4f\n',[x; y; z; result]);

s21bb example results

    x      y      z      R_F(x,y,z)
   0.50   1.00   1.50      1.0281
   1.00   1.50   2.00      0.8260
   1.50   2.00   2.50      0.7116

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox