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1 # Copyright (C) 1996 John W. Eaton |
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2 # |
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3 # This file is part of Octave. |
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4 # |
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5 # Octave is free software; you can redistribute it and/or modify it |
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6 # under the terms of the GNU General Public License as published by the |
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7 # Free Software Foundation; either version 2, or (at your option) any |
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8 # later version. |
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9 # |
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10 # Octave is distributed in the hope that it will be useful, but WITHOUT |
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11 # ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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12 # FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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13 # for more details. |
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14 # |
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15 # You should have received a copy of the GNU General Public License |
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16 # along with Octave; see the file COPYING. If not, write to the Free |
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17 # Software Foundation, 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. |
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18 |
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19 function y = gammai (a, x) |
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20 |
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21 # usage: gammai (a, x) |
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22 # |
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23 # Computes the incomplete gamma function |
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24 # |
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25 # gammai (a, x) |
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26 # = (integral from 0 to x of exp(-t) t^(a-1) dt) / gamma(a). |
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27 # |
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28 # If a is scalar, then gammai(a, x) is returned for each element of x |
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29 # and vice versa. |
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30 # |
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31 # If neither a nor x is scalar, the sizes of a and x must agree, and |
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32 # gammai is applied pointwise. |
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33 |
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34 # Written by KH (Kurt.Hornik@ci.tuwien.ac.at) on Aug 13, 1994 |
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35 |
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36 if (nargin != 2) |
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37 usage ("gammai (a, x)"); |
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38 endif |
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39 |
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40 [r_a, c_a] = size (a); |
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41 [r_x, c_x] = size (x); |
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42 e_a = r_a * c_a; |
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43 e_x = r_x * c_x; |
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44 |
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45 # The following code is rather ugly. We want the function to work |
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46 # whenever a and x have the same size or a or x is scalar. |
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47 # We do this by reducing the latter cases to the former. |
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48 |
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49 if (e_a == 0 || e_x == 0) |
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50 error ("gammai: both a and x must be nonempty"); |
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51 endif |
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52 if (r_a == r_x && c_a == c_x) |
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53 n = e_a; |
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54 a = reshape (a, 1, n); |
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55 x = reshape (x, 1, n); |
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56 r_y = r_a; |
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57 c_y = c_a; |
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58 elseif (e_a == 1) |
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59 n = e_x; |
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60 a = a * ones (1, n); |
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61 x = reshape (x, 1, n); |
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62 r_y = r_x; |
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63 c_y = c_x; |
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64 elseif (e_x == 1) |
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65 n = e_a; |
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66 a = reshape (a, 1, n); |
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67 x = x * ones (1, n); |
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68 r_y = r_a; |
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69 c_y = c_a; |
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70 else |
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71 error ("gammai: a and x must have the same size if neither is scalar"); |
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72 endif |
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73 |
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74 # Now we can do sanity checking ... |
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75 |
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76 if (any (a <= 0) || any (a == Inf)) |
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77 error ("gammai: all entries of a must be positive anf finite"); |
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78 endif |
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79 if (any (x < 0)) |
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80 error ("gammai: all entries of x must be nonnegative"); |
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81 endif |
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82 |
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83 y = zeros (1, n); |
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84 |
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85 # For x < a + 1, use summation. The below choice of k should ensure |
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86 # that the overall error is less than eps ... |
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87 |
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88 S = find ((x > 0) & (x < a + 1)); |
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89 s = length (S); |
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90 if (s > 0) |
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91 k = ceil (- max ([a(S), x(S)]) * log (eps)); |
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92 K = (1:k)'; |
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93 M = ones (k, 1); |
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94 A = cumprod ((M * x(S)) ./ (M * a(S) + K * ones(1, s))); |
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95 y(S) = exp (-x(S) + a(S) .* log (x(S))) .* (1 + sum (A)) ./ gamma (a(S)+1); |
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96 endif |
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97 |
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98 # For x >= a + 1, use the continued fraction. |
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99 # Note, however, that this converges MUCH slower than the series |
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100 # expansion for small a and x not too large! |
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101 |
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102 S = find ((x >= a + 1) & (x < Inf)); |
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103 s = length (S); |
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104 if (s > 0) |
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105 t1 = zeros (1, s); |
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106 t2 = ones (1, s); |
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107 u = [t1; t2]; |
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108 v = [t2; x(S)]; |
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109 c_old = 0; |
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110 c_new = v(1,:) ./ v(2,:); |
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111 n = 1; |
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112 while (max (abs (c_old ./ c_new - 1)) > 10 * eps) |
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113 c_old = c_new; |
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114 u = v + u .* (ones (2, 1) * (n - a(S))); |
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115 v = u .* (ones (2, 1) * x(S)) + n * v; |
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116 c_new = v(1,:) ./ v(2,:); |
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117 n = n + 1; |
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118 endwhile |
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119 y(S) = 1 - exp (-x(S) + a(S) .* log (x(S))) .* c_new ./ gamma (a(S)); |
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120 endif |
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121 |
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122 y (find (x == Inf)) = ones (1, sum (x == Inf)); |
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123 |
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124 y = reshape (y, r_y, c_y); |
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125 |
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126 endfunction |
