(i)	if $x$ is too large, there is a danger of overflow since $Γ (x)$ could become too large to be represented in the machine;
(ii)	if $x$ is too large and negative, there is a danger of underflow;
(iii)	if $x$ is equal to a negative integer, $Γ (x)$ would overflow since it has poles at such points;
(iv)	if $x$ is too near zero, there is again the danger of overflow on some machines. For small $x$ , $Γ (x) ≃ 1 / x$ , and on some machines there exists a range of nonzero but small values of $x$ for which $1 / x$ is larger than the greatest representable value.

References

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

Parameters

Compulsory Input Parameters

1: $x$ – double scalar: The argument $x$ of the function.

Constraint: $x$ must not be zero or a negative integer.

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:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$: The argument is too large. On soft failure the function returns the approximate value of $Γ (x)$ at the nearest valid argument.

$ifail = 2$: The argument is too large and negative. On soft failure the function returns zero.

W $ifail = 3$: The argument is too close to zero. On soft failure the function returns the approximate value of $Γ (x)$ at the nearest valid argument.

W $ifail = 4$: The argument is a negative integer, at which value $Γ (x)$ is infinite. On soft failure the function returns a large positive value.

$ifail = - 99$: 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

Let

δ

and

ε

be the relative errors in the argument and the result respectively. If

δ

is somewhat larger than the machine precision (i.e., is due to data errors etc.), then

ε

and

δ

are approximately related by:

ε ≃ |x Ψ (x)| δ

(provided

ε

is also greater than the representation error). Here

Ψ (x)

is the digamma function

\frac{Γ^{'} (x)}{Γ (x)}

. Figure 1 shows the behaviour of the error amplification factor

|x Ψ (x)|

δ

is of the same order as machine precision, then rounding errors could make

ε

slightly larger than the above relation predicts.

There is clearly a severe, but unavoidable, loss of accuracy for arguments close to the poles of

Γ (x)

at negative integers. However relative accuracy is preserved near the pole at

x = 0

right up to the point of failure arising from the danger of overflow.

Also accuracy will necessarily be lost as

x

becomes large since in this region

ε ≃ δ x \ln x .

However since

Γ (x)

increases rapidly with

x

, the function must fail due to the danger of overflow before this loss of accuracy is too great. (For example, for

x = 20

, the amplification factor

≃ 60

Figure 1

Further Comments

None.

Example

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

Open in the MATLAB editor: s14aa_example

function s14aa_example


fprintf('s14aa example results\n\n');

x = [ 1    1.25     1.5     1.75     2       5       10     -1.5];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s14aa(x(j));
end

disp('      x        Gamma(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s14aa_plot;



function s14aa_plot
  x = {[-3.99:0.01:-3.01]; [-2.99:0.05:-2.04]; [-1.9:0.1:-1.1];
	  [-0.9:0.1:-0.1]; [0.1:0.2:3.9]};

  for k = 1:5
    for j=1:numel(x{k})
      [g{k}(j), ifail] = s14aa(x{k}(j));
    end
  end

  fig1 = figure;
  hold on;
  for k = 1:5
    plot(x{k},g{k},'-r');
  end
  xlabel('x');
  ylabel('\Gamma(x)');
  title('Gamma Function \Gamma(x)');
  axis([-4 4 -5 5]);
  hold off;

s14aa example results

      x        Gamma(x)
   1.000e+00   1.000e+00
   1.250e+00   9.064e-01
   1.500e+00   8.862e-01
   1.750e+00   9.191e-01
   2.000e+00   1.000e+00
   5.000e+00   2.400e+01
   1.000e+01   3.629e+05
  -1.500e+00   2.363e+00

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

Chapter Contents

Chapter Introduction

NAG Toolbox