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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 |
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7 ## the Free Software Foundation; either version 2, or (at your option) |
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8 ## any later version. |
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9 ## |
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10 ## Octave is distributed in the hope that it will be useful, but |
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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13 ## General Public License 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 |
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18 ## 02111-1307, USA. |
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19 |
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20 ## usage: gammai (a, x) |
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21 ## |
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22 ## Computes the incomplete gamma function |
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23 ## |
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24 ## gammai (a, x) |
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25 ## = (integral from 0 to x of exp(-t) t^(a-1) dt) / gamma(a). |
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26 ## |
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27 ## If a is scalar, then gammai(a, x) is returned for each element of x |
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28 ## and vice versa. |
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29 ## |
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30 ## If neither a nor x is scalar, the sizes of a and x must agree, and |
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31 ## gammai is applied pointwise. |
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32 |
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33 ## Author: KH <Kurt.Hornik@ci.tuwien.ac.at> |
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34 ## Created: 13 August 1994 |
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35 ## Adapted-By: jwe |
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36 |
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37 function y = gammai (a, x) |
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38 |
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39 if (nargin != 2) |
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40 usage ("gammai (a, x)"); |
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41 endif |
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42 |
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43 [r_a, c_a] = size (a); |
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44 [r_x, c_x] = size (x); |
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45 e_a = r_a * c_a; |
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46 e_x = r_x * c_x; |
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47 |
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48 ## The following code is rather ugly. We want the function to work |
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49 ## whenever a and x have the same size or a or x is scalar. |
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50 ## We do this by reducing the latter cases to the former. |
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51 |
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52 if (e_a == 0 || e_x == 0) |
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53 error ("gammai: both a and x must be nonempty"); |
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54 endif |
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55 if (r_a == r_x && c_a == c_x) |
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56 n = e_a; |
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57 a = reshape (a, 1, n); |
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58 x = reshape (x, 1, n); |
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59 r_y = r_a; |
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60 c_y = c_a; |
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61 elseif (e_a == 1) |
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62 n = e_x; |
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63 a = a * ones (1, n); |
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64 x = reshape (x, 1, n); |
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65 r_y = r_x; |
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66 c_y = c_x; |
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67 elseif (e_x == 1) |
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68 n = e_a; |
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69 a = reshape (a, 1, n); |
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70 x = x * ones (1, n); |
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71 r_y = r_a; |
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72 c_y = c_a; |
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73 else |
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74 error ("gammai: a and x must have the same size if neither is scalar"); |
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75 endif |
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76 |
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77 ## Now we can do sanity checking ... |
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78 |
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79 if (any (a <= 0) || any (a == Inf)) |
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80 error ("gammai: all entries of a must be positive anf finite"); |
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81 endif |
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82 if (any (x < 0)) |
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83 error ("gammai: all entries of x must be nonnegative"); |
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84 endif |
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85 |
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86 y = zeros (1, n); |
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87 |
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88 ## For x < a + 1, use summation. The below choice of k should ensure |
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89 ## that the overall error is less than eps ... |
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90 |
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91 S = find ((x > 0) & (x < a + 1)); |
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92 s = length (S); |
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93 if (s > 0) |
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94 k = ceil (- max ([a(S), x(S)]) * log (eps)); |
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95 K = (1:k)'; |
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96 M = ones (k, 1); |
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97 A = cumprod ((M * x(S)) ./ (M * a(S) + K * ones(1, s))); |
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98 y(S) = exp (-x(S) + a(S) .* log (x(S))) .* (1 + sum (A)) ./ gamma (a(S)+1); |
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99 endif |
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100 |
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101 ## For x >= a + 1, use the continued fraction. |
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102 ## Note, however, that this converges MUCH slower than the series |
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103 ## expansion for small a and x not too large! |
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104 |
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105 S = find ((x >= a + 1) & (x < Inf)); |
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106 s = length (S); |
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107 if (s > 0) |
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108 t1 = zeros (1, s); |
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109 t2 = ones (1, s); |
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110 u = [t1; t2]; |
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111 v = [t2; x(S)]; |
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112 c_old = 0; |
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113 c_new = v(1,:) ./ v(2,:); |
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114 n = 1; |
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115 while (max (abs (c_old ./ c_new - 1)) > 10 * eps) |
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116 c_old = c_new; |
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117 u = v + u .* (ones (2, 1) * (n - a(S))); |
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118 v = u .* (ones (2, 1) * x(S)) + n * v; |
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119 c_new = v(1,:) ./ v(2,:); |
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120 n = n + 1; |
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121 endwhile |
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122 y(S) = 1 - exp (-x(S) + a(S) .* log (x(S))) .* c_new ./ gamma (a(S)); |
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123 endif |
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124 |
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125 y (find (x == Inf)) = ones (1, sum (x == Inf)); |
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126 |
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127 y = reshape (y, r_y, c_y); |
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128 |
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129 endfunction |
