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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_gamma (s14aa)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_gamma (s14aa) returns the value of the gamma function Γx, via the function name.


[result, ifail] = s14aa(x)
[result, ifail] = nag_specfun_gamma(x)


nag_specfun_gamma (s14aa) evaluates an approximation to the gamma function Γx. The function is based on the Chebyshev expansion:
Γ1+u=r=0arTrt,   where ​ 0u<1,t=2u-1,  
and uses the property Γ1+x=xΓx. If x=N+1+u where N is integral and 0u<1 then it follows that:
There are four possible failures for this function:
(i) if x is too large, there is a danger of overflow since Γx 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, Γx 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, Γx1/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.


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


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


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.

The argument is too large. On soft failure the function returns the approximate value of Γx at the nearest valid argument.
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 Γx at the nearest valid argument.
W  ifail=4
The argument is a negative integer, at which value Γx is infinite. On soft failure the function returns a large positive value.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


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:
(provided ε is also greater than the representation error). Here Ψx is the digamma function Γx Γx . Figure 1 shows the behaviour of the error amplification factor xΨx.
If 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 Γx 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
However since Γx 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
Figure 1

Further Comments



This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.
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));

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


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));

  fig1 = figure;
  hold on;
  for k = 1:5
  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

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