# NAG FL Interfaces21bdf (ellipint_​symm_​3)

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## 1Purpose

s21bdf returns a value of the symmetrised elliptic integral of the third kind, via the function name.

## 2Specification

Fortran Interface
 Function s21bdf ( x, y, z, r,
 Real (Kind=nag_wp) :: s21bdf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x, y, z, r
#include <nag.h>
 double s21bdf_ (const double *x, const double *y, const double *z, const double *r, Integer *ifail)
The routine may be called by the names s21bdf or nagf_specfun_ellipint_symm_3.

## 3Description

s21bdf 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 routine 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: $\mathbf{x}$Real (Kind=nag_wp) Input
2: $\mathbf{y}$Real (Kind=nag_wp) Input
3: $\mathbf{z}$Real (Kind=nag_wp) Input
4: $\mathbf{r}$Real (Kind=nag_wp) 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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
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.
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 ${\mathbf{z}}\ge 0.0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{r}}=0.0$.
Constraint: ${\mathbf{r}}\ne 0.0$.
${\mathbf{ifail}}=3$
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 $L$.
${\mathbf{ifail}}=4$
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 ${\mathbf{z}}\le U$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

In principle the routine 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

s21bdf 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 s21bcf.

## 10Example

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

### 10.1Program Text

Program Text (s21bdfe.f90)

None.

### 10.3Program Results

Program Results (s21bdfe.r)