NAG CL Interface
s21bac (ellipint_symm_1_degen)
1
Purpose
s21bac returns a value of an elementary integral, which occurs as a degenerate case of an elliptic integral of the first kind.
2
Specification
double 
s21bac (double x,
double y,
NagError *fail) 

The function may be called by the names: s21bac, nag_specfun_ellipint_symm_1_degen or nag_elliptic_integral_rc.
3
Description
s21bac calculates an approximate value for the integral
where
$x\ge 0$ and
$y\ne 0$.
This function, which is related to the logarithm or inverse hyperbolic functions for $y<x$ and to inverse circular functions if $x<y$, arises as a degenerate form of the elliptic integral of the first kind. If $y<0$, 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 system:
The quantity
$\left{S}_{n}\right$ for
$n=0,1,2,3,\dots \text{}$ decreases with increasing
$n$, eventually
$\left{S}_{n}\right\sim 1/{4}^{n}$. For small enough
${S}_{n}$ the required function value can be approximated by the first few terms of the Taylor series about the mean. That is
The truncation error involved in using this approximation is bounded by
$16{\left{S}_{n}\right}^{6}/\left(12\left{S}_{n}\right\right)$ and the recursive process is stopped when
${S}_{n}$ is small enough for this truncation error to be negligible compared to the
machine precision.
Within the domain of definition, the function value is itself representable for all representable values of its arguments. However, for values of the arguments near the extremes the above algorithm must be modified so as to avoid causing underflows or overflows in intermediate steps. In extreme regions arguments are prescaled away from the extremes and compensating scaling of the result is done before returning to the calling program.
4
References
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:
$\mathbf{x}$ – double
Input

2:
$\mathbf{y}$ – double
Input

On entry: the arguments $x$ and $y$ of the function, respectively.
Constraint:
${\mathbf{x}}\ge 0.0$ and ${\mathbf{y}}\ne 0.0$.

3:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error 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, ${\mathbf{y}}=0.0$.
Constraint: ${\mathbf{y}}\ne 0.0$.
The function is undefined and returns zero.
 NE_REAL_ARG_LT

On entry, ${\mathbf{x}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{x}}\ge 0.0$.
The function is undefined.
7
Accuracy
In principle the function is capable of producing full machine precision. However, roundoff 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 roundoff error. It is reasonable to assume that the result is accurate to within a small multiple of the machine precision.
8
Parallelism and Performance
s21bac 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.
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
10.2
Program Data
None.
10.3
Program Results