nag_specfun_ellipint_symm_1_degen (s21ba) returns a value of an elementary integral, which occurs as a degenerate case of an elliptic integral of the first kind, via the function name.

Syntax

[result, ifail] = s21ba(x, y)

[result, ifail] = nag_specfun_ellipint_symm_1_degen(x, y)

Description

nag_specfun_ellipint_symm_1_degen (s21ba) calculates an approximate value for the integral

R_{C} (x, y) = \frac{1}{2} \int_{0}^{\infty} \frac{d t}{(t + y) . \sqrt{t + x}}

where

x \geq 0

and

y \neq 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:

\begin{matrix} x_{0} = x & y_{0} = y \\ μ_{n} = (x_{n} + 2 y_{n}) / 3, & S_{n} = (y_{n} - x_{n}) / 3 μ_{n} \\ λ_{n} = y_{n} + 2 \sqrt{x_{n} y_{n}} \\ x_{n + 1} = (x_{n} + λ_{n}) / 4, & y_{n + 1} = (y_{n} + λ_{n}) / 4 . \end{matrix}

The quantity

|S_{n}|

for

n = 0, 1, 2, 3, \dots

decreases with increasing

n

, eventually

|S_{n}| \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

R_{C} (x, y) = (1 + \frac{3 S_{n}^{2}}{10} + \frac{S_{n}^{3}}{7} + \frac{3 S_{n}^{4}}{8} + \frac{9 S_{n}^{5}}{22}) / \sqrt{μ_{n}} .

The truncation error involved in using this approximation is bounded by

16 {|S_{n}|}^{6} / (1 - 2 |S_{n}|)

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.

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

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

Parameters

Compulsory Input Parameters

1: $x$ – double scalar
2: $y$ – double scalar: The arguments $x$ and $y$ of the function, respectively.

Constraint: $x \geq 0.0$ and $y \neq 0.0$ .

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,

x < 0.0

; the function is undefined.

$ifail = 2$

On entry,

y = 0.0

; the function is undefined.

On soft failure the function returns zero.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

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.

Further Comments

You should consult the S Chapter Introduction which shows the relationship of this function to the classical definitions of the elliptic integrals.

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.

Open in the MATLAB editor: s21ba_example

function s21ba_example


fprintf('s21ba example results\n\n');

x = [0.5   1   1.5];
y = [1     1   1  ];
result = x;

for j=1:numel(x)
  [result(j), ifail] = s21ba(x(j), y(j));
end

fprintf('    x      y       R_C(x,y)\n');
fprintf('%7.2f%7.2f%12.4f\n',[x; y; result]);

s21ba example results

    x      y       R_C(x,y)
   0.50   1.00      1.1107
   1.00   1.00      1.0000
   1.50   1.00      0.9312

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox