s Chapter Contents
s Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_elliptic_integral_rj (s21bdc)

## 1  Purpose

nag_elliptic_integral_rj (s21bdc) returns a value of the symmetrised elliptic integral of the third kind.

## 2  Specification

 #include #include
 double nag_elliptic_integral_rj (double x, double y, double z, double r, NagError *fail)

## 3  Description

nag_elliptic_integral_rj (s21bdc) calculates an approximation to the integral
 $RJx,y,z,ρ=32∫0∞dt t+ρt+xt+yt+z$
where $x$, $y$, $z\ge 0$, $\rho \ne 0$ and at most one of $x$, $y$ and $z$ is zero.
If $\rho <0$, the result computed is the Cauchy principal value of the integral.
The basic algorithm, which is due to Carlson (1979) and Carlson (1988), is to reduce the arguments recursively towards their mean by the rule:
 $x0 = x,y0=y,z0=z,ρ0=ρ μn = xn+yn+zn+2ρn/5 Xn = 1-xn/μn Yn = 1-yn/μn Zn = 1-zn/μn Pn = 1-ρn/μn λn = xnyn+ynzn+znxn xn+1 = xn+λn/4 yn+1 = yn+λn/4 zn+1 = zn+λn/4 ρn+1 = ρn+λn/4 αn = ρnxn,+yn,+zn+xnynzn 2 βn = ρn ρn+λn 2$
For $n$ sufficiently large,
 $εn=maxXn,Yn,Zn,Pn∼14n$
and the function may be approximated by a fifth order power series
 $RJx,y,z,ρ= 3∑m= 0 n- 14-m RCαm,βm + 4-nμn3 1+ 37Sn 2 + 13Sn 3 + 322Sn 2 2+ 311Sn 4 + 313Sn 2 Sn 3 + 313Sn 5$
where ${S}_{n}^{\left(m\right)}=\left({X}_{n}^{m}+{Y}_{n}^{m}+{Z}_{n}^{m}+2{P}_{n}^{m}\right)/2m$.
The truncation error in this expansion is bounded by $3{\epsilon }_{n}^{6}/\sqrt{{\left(1-{\epsilon }_{n}\right)}^{3}}$ 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:  ${R}_{J}\left(x,x,x,x\right)={x}^{-\frac{3}{2}}$, so there exists a region of extreme arguments for which the function value is not representable.

## 4  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

## 5  Arguments

1:    $\mathbf{x}$doubleInput
2:    $\mathbf{y}$doubleInput
3:    $\mathbf{z}$doubleInput
4:    $\mathbf{r}$doubleInput
On entry: the arguments $x$, $y$, $z$ and $\rho$ of the function.
Constraint: ${\mathbf{x}}$, y, ${\mathbf{z}}\ge 0.0$, ${\mathbf{r}}\ne 0.0$ and at most one of x, y and z may be zero.
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARG_EQ
On entry, ${\mathbf{r}}=0.0$.
Constraint: ${\mathbf{r}}\ne 0.0$.
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$, ${\mathbf{y}}=〈\mathit{\text{value}}〉$ and ${\mathbf{z}}=〈\mathit{\text{value}}〉$.
Constraint: at most one of x, y and z is $0.0$.
The function is undefined.
NE_REAL_ARG_GT
On entry, $U=〈\mathit{\text{value}}〉$, ${\mathbf{r}}=〈\mathit{\text{value}}〉$, ${\mathbf{x}}=〈\mathit{\text{value}}〉$, ${\mathbf{y}}=〈\mathit{\text{value}}〉$ and ${\mathbf{z}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{r}}\right|\le U$ and ${\mathbf{x}}\le U$ and ${\mathbf{y}}\le U$ and ${\mathbf{z}}\le U$.
NE_REAL_ARG_LT
On entry, $L=〈\mathit{\text{value}}〉$, ${\mathbf{r}}=〈\mathit{\text{value}}〉$, ${\mathbf{x}}=〈\mathit{\text{value}}〉$, ${\mathbf{y}}=〈\mathit{\text{value}}〉$ and ${\mathbf{z}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{r}}\right|\ge L$ and at most one of x, y and z is less than $L$.
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$, ${\mathbf{y}}=〈\mathit{\text{value}}〉$ and ${\mathbf{z}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\ge 0.0$ and ${\mathbf{y}}\ge 0.0$ and ${\mathbf{z}}\ge 0.0$.

## 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.

## 8  Parallelism and Performance

nag_elliptic_integral_rj (s21bdc) 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.
If the argument r is equal to any of the other arguments, the function reduces to the integral ${R}_{D}$, computed by nag_elliptic_integral_rd (s21bcc).

## 10  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.

### 10.1  Program Text

Program Text (s21bdce.c)

None.

### 10.3  Program Results

Program Results (s21bdce.r)