s21bd:: Approximations of Special Functions (NAG Toolbox)

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.

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

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Errors or warnings detected by the function:

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.

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 the argument r is equal to any of the other arguments, the function reduces to the integral

R_{D}

, computed by nag_specfun_ellipint_symm_2 (s21bc).

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.

function s21bd_example


fprintf('s21bd example results\n\n');

x = [0.5   0.5   0.5   0.5   0.5   0.5   1    1    1     1.5];
y = [0.5   0.5   0.5   1     1     1.5   1    1    1.5   1.5];
z = [0.5   1     1.5   1     1.5   1.5   1    1.5  1.5   1.5];
r = [2     2     2     2     2     2     2    2    2     2  ];
result = x;

for j=1:numel(x)
  [result(j), ifail] = s21bd(x(j), y(j), z(j), r(j));
end

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

s21bd example results

    x      y      z      r      R_J(x,y,z)
   0.50   0.50   0.50   2.00      1.1184
   0.50   0.50   1.00   2.00      0.9221
   0.50   0.50   1.50   2.00      0.8115
   0.50   1.00   1.00   2.00      0.7671
   0.50   1.00   1.50   2.00      0.6784
   0.50   1.50   1.50   2.00      0.6017
   1.00   1.00   1.00   2.00      0.6438
   1.00   1.00   1.50   2.00      0.5722
   1.00   1.50   1.50   2.00      0.5101
   1.50   1.50   1.50   2.00      0.4561

NAG Toolbox: nag_specfun_ellipint_symm_3 (s21bd)

▸▿ Contents

Purpose

Syntax

Description