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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_ellipint_symm_2 (s21bc)

## Purpose

nag_specfun_ellipint_symm_2 (s21bc) returns a value of the symmetrised elliptic integral of the second kind, via the function name.

## Syntax

[result, ifail] = s21bc(x, y, z)
[result, ifail] = nag_specfun_ellipint_symm_2(x, y, z)

## Description

nag_specfun_ellipint_symm_2 (s21bc) calculates an approximate value for the integral
 $RDx,y,z=32∫0∞dt t+xt+y t+z 3$
where $x$, $y\ge 0$, 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:
 $x0 = x,y0=y,z0=z μn = xn+yn+3zn/5 Xn = 1-xn/μn Yn = 1-yn/μn Zn = 1-zn/μn λn = xnyn+ynzn+znxn xn+1 = xn+λn/4 yn+1 = yn+λn/4 zn+1 = zn+λn/4$
For $n$ sufficiently large,
 $εn=maxXn,Yn,Zn∼ 14 n$
and the function may be approximated adequately by a fifth order power series
 $RDx,y,z= 3∑m= 0 n- 1 4-mzm+λnzm + 4-nμn3 1+ 37Sn 2 + 13Sn 3 + 322Sn 2 2+ 311Sn 4 + 313Sn 2 Sn 3 + 313Sn 5$
where ${S}_{n}^{\left(m\right)}=\left({X}_{n}^{m}+{Y}_{n}^{m}+3{Z}_{n}^{m}\right)/2m\text{.}$ The truncation error in this expansion is bounded by $\frac{3{\epsilon }_{n}^{6}}{\sqrt{{\left(1-{\epsilon }_{n}\right)}^{3}}}$ 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:  ${R}_{D}\left(x,x,x\right)={x}^{-3/2}$, 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

1:     $\mathrm{x}$ – double scalar
2:     $\mathrm{y}$ – double scalar
3:     $\mathrm{z}$ – double scalar
The arguments $x$, $y$ and $z$ of the function.
Constraint: ${\mathbf{x}}$, ${\mathbf{y}}\ge 0.0$, ${\mathbf{z}}>0.0$ and only one of x and y may be zero.

None.

### Output Parameters

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

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
On entry, either x or y is negative, or both x and y are zero; the function is undefined.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{z}}\le 0.0$; the function is undefined.
${\mathbf{ifail}}=3$
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.
${\mathbf{ifail}}=4$
On entry, at least one of x, y and z is too large: there is a danger of setting underflow. On soft failure the function returns zero.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## 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.

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

## 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 s21bc_example

fprintf('s21bc example results\n\n');

x = [0.5   0.5   0.5   1    1   1.5];
y = [0.5   1.0   1.5   1    1.5   1.5];
z = [1     1     1     1    1   1];
result = x;

for j=1:numel(x)
[result(j), ifail] = s21bc(x(j), y(j), z(j));
end

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

```
```s21bc example results

x      y      z      R_D(x,y,z)
0.50   0.50   1.00      1.4787
0.50   1.00   1.00      1.2108
0.50   1.50   1.00      1.0611
1.00   1.00   1.00      1.0000
1.00   1.50   1.00      0.8805
1.50   1.50   1.00      0.7775
```