# NAG CL Interfaces21bdc (ellipint_​symm_​3)

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## 1Purpose

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

## 2Specification

 #include
 double s21bdc (double x, double y, double z, double r, NagError *fail)
The function may be called by the names: s21bdc, nag_specfun_ellipint_symm_3 or nag_elliptic_integral_rj.

## 3Description

s21bdc calculates an approximation to the integral
 $RJ(x,y,z,ρ)=32∫0∞dt (t+ρ)(t+x)(t+y)(t+z)$
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:
 $x0 = x,y0=y,z0=z,ρ0=ρ μn = (xn+yn+zn+2ρn)/5 Xn = 1-xn/μn Yn = 1-yn/μn Zn = 1-zn/μn Pn = 1-ρn/μn λn = xnyn+ynzn+znxn xn+1 = (xn+λn)/4 yn+1 = (yn+λn)/4 zn+1 = (zn+λn)/4 ρn+1 = (ρn+λn)/4 αn = [ρn(xn,+yn,+zn)+xnynzn] 2 βn = ρn (ρn+λn) 2$
For $n$ sufficiently large,
 $εn=max(|Xn|,|Yn|,|Zn|,|Pn|)∼14n$
and the function may be approximated by a fifth order power series
 $RJ(x,y,z,ρ)= 3∑m= 0 n- 14-m RC(αm,βm) + 4-nμn3 [1+ 37Sn (2) + 13Sn (3) + 322(Sn (2) )2+ 311Sn (4) + 313Sn (2) Sn (3) + 313Sn (5) ]$
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.

## 4References

NIST Digital Library of Mathematical Functions
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

## 5Arguments

1: $\mathbf{x}$double Input
2: $\mathbf{y}$double Input
3: $\mathbf{z}$double Input
4: $\mathbf{r}$double Input
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 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface 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: $|{\mathbf{r}}|\le U$ 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: $|{\mathbf{r}}|\ge L$ 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 ${\mathbf{z}}\ge 0.0$.

## 7Accuracy

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.

## 8Parallelism and Performance

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 s21bcc.

## 10Example

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

### 10.1Program Text

Program Text (s21bdce.c)

None.

### 10.3Program Results

Program Results (s21bdce.r)