naginterfaces.library.specfun.ellipint_​symm_​3

naginterfaces.library.specfun.ellipint_symm_3(x, y, z, r)

ellipint_symm_3 returns a value of the symmetrised elliptic integral of the third kind.

For full information please refer to the NAG Library document for s21bd

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s21bdf.html

Parameters

xfloat

The arguments $x$, $y$, $z$ and $\rho$ of the function.

yfloat

The arguments $x$, $y$, $z$ and $\rho$ of the function.

zfloat

The arguments $x$, $y$, $z$ and $\rho$ of the function.

rfloat

The arguments $x$, $y$, $z$ and $\rho$ of the function.

Returns

rjfloat

The value of the symmetrised elliptic integral of the third kind.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$, $y=⟨value⟩$ and $z=⟨value⟩$.

Constraint: at most one of $x$, $y$ and $z$ is $0.0$.

The function is undefined.

(errno $1$)

On entry, $x=⟨value⟩$, $y=⟨value⟩$ and $z=⟨value⟩$.

Constraint: $x\ge 0.0$ and $y\ge 0.0$ and $z\ge 0.0$.

(errno $2$)

On entry, $r=0.0$.

Constraint: $r\ne 0.0$.

(errno $3$)

On entry, $L=⟨value⟩$, $r=⟨value⟩$, $x=⟨value⟩$, $y=⟨value⟩$ and $z=⟨value⟩$.

Constraint: $\left|r\right|\ge L$ and at most one of $x$, $y$ and $z$ is less than $L$.

(errno $4$)

On entry, $U=⟨value⟩$, $r=⟨value⟩$, $x=⟨value⟩$, $y=⟨value⟩$ and $z=⟨value⟩$.

Constraint: $\left|r\right|\le U$ and $x\le U$ and $y\le U$ and $z\le U$.

Notes

ellipint_symm_3 calculates an approximation to the integral

$$R_J(x,y,z,\rho )=\frac{3}{2}{\int }_0^{\infty }\frac{dt}{(t+\rho )\sqrt{(t+x)(t+y)(t+z)}}$$

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:

$$\begin{array}{c}\begin{array}{ccc}{x}_0& =& x,y_0=y,z_0=z,{\rho }_0=\rho \\ {\mu }_n& =& (x_n+y_n+z_n+2{\rho }_n)/5\\ X_n& =& 1-x_n/{\mu }_n\\ Y_n& =& 1-y_n/{\mu }_n\\ Z_n& =& 1-z_n/{\mu }_n\\ P_n& =& 1-{\rho }_n/{\mu }_n\\ {\lambda }_n& =& \sqrt{x_n{y}_n}+\sqrt{y_n{z}_n}+\sqrt{z_n{x}_n}\\ x_{n+1}& =& (x_n+{\lambda }_n)/4\\ y_{n+1}& =& (y_n+{\lambda }_n)/4\\ z_{n+1}& =& (z_n+{\lambda }_n)/4\\ {\rho }_{n+1}& =& ({\rho }_n+{\lambda }_n)/4\\ {\alpha }_n& =& {[{\rho }_n(\sqrt{x_n},+\sqrt{y_n},+\sqrt{z_n})+\sqrt{x_n{y}_n{z}_n}]}^2\\ {\beta }_n& =& {\rho }_n{({\rho }_n+{\lambda }_n)}^2\end{array}\end{array}$$

For $n$ sufficiently large,

$${ϵ}_n=max(\left|X_n\right|,\left|Y_n\right|,\left|Z_n\right|,\left|P_n\right|)\sim \frac{1}{4^n}$$

and the function may be approximated by a fifth order power series

$$\begin{array}{c}\begin{array}{cc}{R}_J(x,y,z,\rho )=& 3{\sum }_{m=0}^{n-1}{4}^{-m}{R}_C({\alpha }_m,{\beta }_m)\\ & \\ & \\ & +\frac{4^{-n}}{\sqrt{{\mu }_n^3}}[1+\frac{3}{7}{S}_n^{\left(2\right)}+\frac{1}{3}{S}_n^{\left(3\right)}+\frac{3}{22}{\left(S_n^{\left(2\right)}\right)}^2+\frac{3}{11}{S}_n^{\left(4\right)}+\frac{3}{13}{S}_n^{\left(2\right)}{S}_n^{\left(3\right)}+\frac{3}{13}{S}_n^{\left(5\right)}]\end{array}\end{array}$$

where $S_n^{\left(m\right)}=(X_n^m+Y_n^m+Z_n^m+2P_n^m)/2m$.

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

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