NAG FL Interface
s21bdf (ellipint_​symm_​3)

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1 Purpose

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

2 Specification

Fortran Interface
Function s21bdf ( x, y, z, r, ifail)
Real (Kind=nag_wp) :: s21bdf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x, y, z, r
C Header Interface
#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.

3 Description

s21bdf calculates an approximation to the integral
RJ(x,y,z,ρ)=320dt (t+ρ)(t+x)(t+y)(t+z)  
where x, y, z0, ρ0 and at most one of x, y and z is zero.
If ρ<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,
and the function may be approximated by a fifth order power series
RJ(x,y,z,ρ)= 3m= 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 Sn (m) =(Xnm+Ynm+Znm+2Pnm)/2m.
The truncation error in this expansion is bounded by 3εn6/ (1-εn) 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:  RJ(x,x,x,x)=x-32, so there exists a region of extreme arguments for which the function value is not representable.

4 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

5 Arguments

1: x Real (Kind=nag_wp) Input
2: y Real (Kind=nag_wp) Input
3: z Real (Kind=nag_wp) Input
4: r Real (Kind=nag_wp) Input
On entry: the arguments x, y, z and ρ of the function.
Constraint: x, y, z0.0, r0.0 and at most one of x, y and z may be zero.
5: 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 -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry 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:
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.
On entry, x=value, y=value and z=value.
Constraint: x0.0 and y0.0 and z0.0.
On entry, r=0.0.
Constraint: r0.0.
On entry, L=value, r=value, x=value, y=value and z=value.
Constraint: |r|L and at most one of x, y and z is less than L.
On entry, U=value, r=value, x=value, y=value and z=value.
Constraint: |r|U and xU and yU and zU.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s21bdf is not threaded in any implementation.

9 Further Comments

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 RD, computed by s21bcf.

10 Example

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

10.1 Program Text

Program Text (s21bdfe.f90)

10.2 Program Data


10.3 Program Results

Program Results (s21bdfe.r)