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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_specfun_gamma_incomplete (s14ba)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_specfun_gamma_incomplete (s14ba) computes values for the incomplete gamma functions

P (a, x)

and

Q (a, x)

Syntax

[p, q, ifail] = s14ba(a, x, tol)

[p, q, ifail] = nag_specfun_gamma_incomplete(a, x, tol)

Description

nag_specfun_gamma_incomplete (s14ba) evaluates the incomplete gamma functions in the normalized form

P (a, x) = \frac{1}{Γ (a)} \int_{0}^{x} t^{a - 1} e^{- t} d t,

Q (a, x) = \frac{1}{Γ (a)} \int_{x}^{\infty} t^{a - 1} e^{- t} d t,

with

x \geq 0

and

a > 0

, to a user-specified accuracy. With this normalization,

P (a, x) + Q (a, x) = 1

Several methods are used to evaluate the functions depending on the arguments

a

and

x

, the methods including Taylor expansion for

P (a, x)

, Legendre's continued fraction for

Q (a, x)

, and power series for

Q (a, x)

. When both

a

and

x

are large, and

a ≃ x

, the uniform asymptotic expansion of Temme (1987) is employed for greater efficiency – specifically, this expansion is used when

a \geq 20

and

0.7 a \leq x \leq 1.4 a

Once either

P

Q

is computed, the other is obtained by subtraction from

1

. In order to avoid loss of relative precision in this subtraction, the smaller of

P

and

Q

is computed first.

This function is derived from the function GAM in Gautschi (1979b).

References

Gautschi W (1979a) A computational procedure for incomplete gamma functions ACM Trans. Math. Software 5 466–481

Gautschi W (1979b) Algorithm 542: Incomplete gamma functions ACM Trans. Math. Software 5 482–489

Temme N M (1987) On the computation of the incomplete gamma functions for large values of the parameters Algorithms for Approximation (eds J C Mason and M G Cox) Oxford University Press

Parameters

Compulsory Input Parameters

1: $a$ – double scalar: The argument $a$ of the functions.

Constraint: $a > 0.0$ .
2: $x$ – double scalar: The argument $x$ of the functions.

Constraint: $x \geq 0.0$ .
3: $tol$ – double scalar: The relative accuracy required by you in the results. If nag_specfun_gamma_incomplete (s14ba) is entered with tol greater than $1.0$ or less than machine precision, then the value of machine precision is used instead.

Optional Input Parameters

None.

Output Parameters

1: $p$ – double scalar
2: $q$ – double scalar: The values of the functions $P (a, x)$ and $Q (a, x)$ respectively.
3: $ifail$ – 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,

a \leq 0.0

$ifail = 2$

On entry,

x < 0.0

$ifail = 3$: Convergence of the Taylor series or Legendre continued fraction fails within $600$ iterations. This error is extremely unlikely to occur; if it does, contact NAG.

$ifail = - 99$: 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

There are rare occasions when the relative accuracy attained is somewhat less than that specified by argument tol. However, the error should never exceed more than one or two decimal places. Note also that there is a limit of

18

decimal places on the achievable accuracy, because constants in the function are given to this precision.

Further Comments

The time taken for a call of nag_specfun_gamma_incomplete (s14ba) depends on the precision requested through tol, and also varies slightly with the input arguments

a

and

x

Example

This example reads values of the argument

a

and

x

from a file, evaluates the function and prints the results.

Open in the MATLAB editor: s14ba_example

function s14ba_example


fprintf('s14ba example results\n\n');

a = [ 2  7   0.5  20  21];
x = [ 3  1  99.0  21  20];
n = size(x,2);
p = x;
q = x;
tol = x02aj;

for j=1:n
  [p(j), q(j), ifail] = s14ba(a(j), x(j), tol);
end

disp('     a      x     P(a,x)    Q(a,x)');
fprintf('%7.1f%7.1f%10.4f%10.4f\n',[a; x; p; q]);

s14ba_plot;



function s14ba_plot
  x = [0:0.5:20];
  a = [0.1:0.1:0.4,0.5:0.5:20];
  p = zeros(numel(a),numel(x));
  q = p;

  tol = x02aj;
  for i=1:numel(a)
    for j=1:numel(x)
      [p(i,j), q(i,j), ifail] = s14ba(a(i), x(j), tol);
    end
  end

  fig1 = figure;
  hold on
  mesh(x, a, p, 'FaceColor','r');
  mesh(x, a, q, 'FaceColor','g');
  xlabel('x');
  ylabel('a');
  title('Incomplete Gamma Functions');
  legend('P(a,x)','Q(a,x)');
  view(-35, 46);
  hold off;

s14ba example results

     a      x     P(a,x)    Q(a,x)
    2.0    3.0    0.8009    0.1991
    7.0    1.0    0.0001    0.9999
    0.5   99.0    1.0000    0.0000
   20.0   21.0    0.6157    0.3843
   21.0   20.0    0.4409    0.5591

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Chapter Contents

Chapter Introduction

NAG Toolbox