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annotate scripts/specfun/betainc.m @ 24919:ed6f6bbed604 stable

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betainc: vectorized the Lentz's algorithm -- changed libinterp/corefcn/__betainc_lentz__.cc changed scripts/specfun/betainc.m
author Michele Ginesi <michele.ginesi@gmail.com>
date Wed, 21 Feb 2018 23:13:26 +0100
parents bd89440407aa
children 6b33ee8aad0f
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1 ## Copyright (C) 2017 Michele Ginesi
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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 3 of the License, or
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8 ## (at your option) 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
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13 ## GNU 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, see
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17 ## <http://www.gnu.org/licenses/>.
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18
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19 ## Authors: Michele Ginesi <michele.ginesi@gmail.com>
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20
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21 ## -*- texinfo -*-
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22 ## @deftypefn {} {} betainc (@var{x}, @var{a}, @var{b})
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23 ## @deftypefnx {} {} betainc (@var{x}, @var{a}, @var{b}, @var{tail})
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24 ## Compute the incomplete beta function ratio.
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25 ##
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26 ## This is defined as
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27 ## @tex
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28 ## $$
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29 ## I_x (a, b) = {1 \over {B(a,b)}} \displaystyle{\int_0^x t^{a-1} (1-t)^{b-1} dt}
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30 ## $$
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31 ## @end tex
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32 ## @ifnottex
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33 ##
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34 ## @example
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35 ## @group
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36 ## x
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37 ## /
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38 ## |
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39 ## I_x (a, b) = | t^(a-1) (1-t)^(b-1) dt
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40 ## |
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41 ## /
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42 ## 0
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43 ## @end group
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44 ## @end example
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45 ##
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46 ## @end ifnottex
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47 ##
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48 ## with @var{x} real in [0,1], @var{a} and @var{b} real and strictly positive.
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49 ## If one of the input has more than one components, then the others must be
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50 ## scalar or of compatible dimensions.
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51 ##
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52 ## By default or if @var{tail} is @qcode{"lower"} the incomplete beta function
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53 ## ratio integrated from 0 to @var{x} is computed. If @var{tail} is
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54 ## @qcode{"upper"} then the complementary function integrated from @var{x} to
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55 ## 1 is calculated. The two choices are related as
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56 ##
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57 ## betainc (@var{x}, @var{a}, @var{b}, @qcode{"lower"}) =
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58 ## 1 - betainc (@var{x}, @var{a}, @var{b}, @qcode{"upper"}).
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59 ##
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60 ## Reference
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61 ##
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62 ## @nospell{A. Cuyt, V. Brevik Petersen, B. Verdonk, H. Waadeland, W.B. Jones}
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63 ## @cite{Handbook of Continued Fractions for Special Functions},
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64 ## ch. 18.
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65 ##
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66 ## @seealso{beta, betaincinv, betaln}
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67 ##
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68 ## @end deftypefn
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69
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70 function [y] = betainc (x, a, b, tail = "lower")
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71
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72 if (nargin > 4 || nargin < 3)
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73 print_usage ();
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74 endif
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75
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76 if (! isscalar (x) || ! isscalar (a) || ! isscalar (b))
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77 [err, x, a, b] = common_size (x, a, b);
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78 if (err > 0)
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79 error ("betainc: x, a and b must be of common size or scalars");
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80 endif
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81 endif
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82
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83 if (iscomplex (x) || iscomplex (a) || iscomplex (b))
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84 error ("betainc: inputs must be real or integer");
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85 endif
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86
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87 if (any (a <= 0))
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88 error ("betainc: a must be strictly positive");
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89 endif
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90
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91 if (any (b <= 0))
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92 error ("betainc: b must be strictly positive");
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93 endif
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94
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95 if (any (x > 1 | x < 0))
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96 error ("betainc: x must be between 0 and 1");
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97 endif
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98
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99 if (isinteger (x))
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100 y = double (x);
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101 endif
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102
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103 if (isinteger (a))
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104 a = double (a);
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105 endif
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106
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107 if (isinteger (b))
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108 b = double (b);
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109 endif
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110
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111 sz = size (x);
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112 x = x(:);
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113 a = a(:);
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114 b = b(:);
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115 y = zeros (size (x));
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116
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117 # If any of the argument is single, also the output should be
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118
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119 if (strcmpi (class (y), "single") || strcmpi (class (a), "single") || strcmpi (class (b), "single"))
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120 a = single (a);
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121 b = single (b);
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122 x = single (x);
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123 y = single (y);
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124 endif
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125
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126 # In the following, we use the fact that the continued fraction we will
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127 # use is more efficient when x <= a / (a+b).
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128 # Moreover, to compute the upper version, which is defined as
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129 # I_x(a,b,"upper") = 1 - I_x(a,b) we used the property
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130 # I_x(a,b) + I_(1-x) (b,a) = 1.
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Michele Ginesi <michele.ginesi@gmail.com>
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131
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Michele Ginesi <michele.ginesi@gmail.com>
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132 if (strcmpi (tail, "upper"))
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Michele Ginesi <michele.ginesi@gmail.com>
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133 flag = (x < a./(a+b));
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Michele Ginesi <michele.ginesi@gmail.com>
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134 x(!flag) = 1 - x(!flag);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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135 [a(!flag), b(!flag)] = deal (b(!flag), a(!flag));
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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136 elseif (strcmpi (tail, "lower"))
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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137 flag = (x > a./(a+b));
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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138 x (flag) = 1 - x(flag);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
139 [a(flag), b(flag)] = deal (b(flag), a(flag));
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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140 else
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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141 error ("betainc: invalid value for flag")
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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142 endif
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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143
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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144 f = zeros (size (x), class (x));
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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145
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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146 ## Continued fractions: CPVWJ, formula 18.5.20, modified Lentz algorithm
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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147 ## implemented in a separate .cc file. This particular continued fraction
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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148 ## gives (B(a,b) * I_x(a,b)) / (x^a * (1-x)^b).
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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149
24919
ed6f6bbed604 betainc: vectorized the Lentz's algorithm
Michele Ginesi <michele.ginesi@gmail.com>
parents: 24907
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150 f = __betainc_lentz__ (x, a, b, strcmpi (class (x), "single"));
ed6f6bbed604 betainc: vectorized the Lentz's algorithm
Michele Ginesi <michele.ginesi@gmail.com>
parents: 24907
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151
24907
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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152 # We divide the continued fraction by B(a,b) / (x^a * (1-x)^b)
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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153 # to obtain I_x(a,b).
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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154 y = a .* log (x) + b .* log1p (-x) + gammaln (a + b) - ...
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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155 gammaln (a) - gammaln (b) + log (f);
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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156 y = real (exp (y));
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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157 y(flag) = 1 - y(flag);
24919
ed6f6bbed604 betainc: vectorized the Lentz's algorithm
Michele Ginesi <michele.ginesi@gmail.com>
parents: 24907
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158 y = reshape (y, sz);
24907
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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159
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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160 endfunction
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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161
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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162 ## Tests from betainc.cc
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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163
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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164 # Double precision
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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165 %!test
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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166 %! a = [1, 1.5, 2, 3];
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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167 %! b = [4, 3, 2, 1];
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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168 %! v1 = betainc (1,a,b);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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169 %! v2 = [1,1,1,1];
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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170 %! x = [.2, .4, .6, .8];
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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171 %! v3 = betainc (x, a, b);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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172 %! v4 = 1 - betainc (1.-x, b, a);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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173 %! assert (v1, v2, sqrt (eps));
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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174 %! assert (v3, v4, sqrt (eps));
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Michele Ginesi <michele.ginesi@gmail.com>
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175
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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176 # Single precision
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Michele Ginesi <michele.ginesi@gmail.com>
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177 %!test
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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178 %! a = single ([1, 1.5, 2, 3]);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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179 %! b = single ([4, 3, 2, 1]);
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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180 %! v1 = betainc (1,a,b);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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181 %! v2 = single ([1,1,1,1]);
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
182 %! x = single ([.2, .4, .6, .8]);
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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183 %! v3 = betainc (x, a, b);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
184 %! v4 = 1 - betainc (1.-x, b, a);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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185 %! assert (v1, v2, sqrt (eps ("single")));
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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186 %! assert (v3, v4, sqrt (eps ("single")));
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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187
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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188 # Mixed double/single precision
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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189 %!test
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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190 %! a = single ([1, 1.5, 2, 3]);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
191 %! b = [4, 3, 2, 1];
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
192 %! v1 = betainc (1,a,b);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
193 %! v2 = single ([1,1,1,1]);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
194 %! x = [.2, .4, .6, .8];
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
195 %! v3 = betainc (x, a, b);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
196 %! v4 = 1-betainc (1.-x, b, a);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
197 %! assert (v1, v2, sqrt (eps ("single")));
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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198 %! assert (v3, v4, sqrt (eps ("single")));
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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199
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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200 ## New test
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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201
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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202 %!test #<51157>
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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203 %! y = betainc([0.00780;0.00782;0.00784],250.005,49750.995);
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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204 %! y_ex = [0.999999999999989; 0.999999999999992; 0.999999999999995];
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
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205 %! assert (y, y_ex, -1e-14);
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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206
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
207 %!assert (betainc (0.001, 20, 30), 2.750687665855991e-47, -3e-14);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
208 %!assert (betainc (0.0001, 20, 30), 2.819953178893307e-67, -3e-14);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
209 %!assert (betainc (0.99, 20, 30, "upper"), 1.5671643161872703e-47, -3e-14);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
210 %!assert (betainc (0.999, 20, 30, "upper"), 1.850806276141535e-77, -3e-14);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
211 %!assert (betainc (0.5, 200, 300), 0.9999964565197356, -1e-15);
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
212 %!assert (betainc (0.5, 200, 300, "upper"), 3.54348026439253e-06, -1e-13);
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
213
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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214 # Test trivial values
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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215
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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216 %!test
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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217 %! [a,b] = ndgrid (linspace(1e-4, 100, 20), linspace(1e-4, 100, 20));
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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218 %! assert (betainc (0,a,b), zeros(20));
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
219 %! assert (betainc (1,a,b), ones(20));
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
220
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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221 ## Test input validation
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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222
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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223 %!error betainc ()
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
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224 %!error betainc (1)
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
225 %!error betainc (1,2)
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
226 %!error betainc (1.1,1,1)
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
227 %!error betainc (-0.1,1,1)
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
228 %!error betainc (0.5,0,1)
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
229 %!error betainc (0.5,1,0)
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
230 %!error betainc (1,2,3,4)
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
231 %!error betainc (1,2)
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
232 %!error betainc (1,2,3,4,5)
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
233 %!error betainc (0.5i, 1, 2)
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Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
234 %!error betainc (0, 1i, 1)
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
235 %!error betainc (0, 1, 1i)