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update Octave Project Developers copyright for the new year In files that have the "Octave Project Developers" copyright notice, update for 2021. In all .txi and .texi files except gpl.txi and gpl.texi in the doc/liboctave and doc/interpreter directories, change the copyright to "Octave Project Developers", the same as used for other source files. Update copyright notices for 2022 (not done since 2019). For gpl.txi and gpl.texi, change the copyright notice to be "Free Software Foundation, Inc." and leave the date at 2007 only because this file only contains the text of the GPL, not anything created by the Octave Project Developers. Add Paul Thomas to contributors.in.
author John W. Eaton <jwe@octave.org>
date Tue, 28 Dec 2021 18:22:40 -0500
parents 168da23530b4
children 5d3faba0342e fe898ae23e0e
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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1 ########################################################################
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2 ##
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3 ## Copyright (C) 2018-2022 The Octave Project Developers
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4 ##
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5 ## See the file COPYRIGHT.md in the top-level directory of this
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6 ## distribution or <https://octave.org/copyright/>.
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7 ##
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8 ## This file is part of Octave.
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9 ##
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10 ## Octave is free software: you can redistribute it and/or modify it
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11 ## under the terms of the GNU General Public License as published by
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12 ## the Free Software Foundation, either version 3 of the License, or
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13 ## (at your option) any later version.
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14 ##
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15 ## Octave is distributed in the hope that it will be useful, but
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16 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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17 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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18 ## GNU General Public License for more details.
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19 ##
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20 ## You should have received a copy of the GNU General Public License
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21 ## along with Octave; see the file COPYING. If not, see
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22 ## <https://www.gnu.org/licenses/>.
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23 ##
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24 ########################################################################
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25
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26 ## -*- texinfo -*-
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27 ## @deftypefn {} {} betainc (@var{x}, @var{a}, @var{b})
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28 ## @deftypefnx {} {} betainc (@var{x}, @var{a}, @var{b}, @var{tail})
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29 ## Compute the incomplete beta function.
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30 ##
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31 ## This is defined as
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32 ## @tex
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33 ## $$
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34 ## I_x (a, b) = {1 \over {B(a,b)}} \displaystyle{\int_0^x t^{a-1} (1-t)^{b-1} dt}
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35 ## $$
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36 ## @end tex
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37 ## @ifnottex
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38 ##
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39 ## @example
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40 ## @group
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41 ## x
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42 ## /
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43 ## 1 |
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44 ## I_x (a, b) = ---------- | t^(a-1) (1-t)^(b-1) dt
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45 ## beta (a,b) |
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46 ## /
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47 ## 0
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48 ## @end group
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49 ## @end example
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50 ##
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51 ## @end ifnottex
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52 ##
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53 ## with real @var{x} in the range [0,1]. The inputs @var{a} and @var{b} must
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54 ## be real and strictly positive (> 0). If one of the inputs is not a scalar
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55 ## then the other inputs must be scalar or of compatible dimensions.
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56 ##
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57 ## By default, @var{tail} is @qcode{"lower"} and the incomplete beta function
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58 ## integrated from 0 to @var{x} is computed. If @var{tail} is @qcode{"upper"}
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59 ## then the complementary function integrated from @var{x} to 1 is calculated.
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60 ## The two choices are related by
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61 ##
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62 ## betainc (@var{x}, @var{a}, @var{b}, @qcode{"upper"}) =
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63 ## 1 - betainc (@var{x}, @var{a}, @var{b}, @qcode{"lower"}).
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64 ##
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65 ## @code{betainc} uses a more sophisticated algorithm than subtraction to
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66 ## get numerically accurate results when the @qcode{"lower"} value is small.
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67 ##
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68 ## Reference: @nospell{A. Cuyt, V. Brevik Petersen, B. Verdonk, H. Waadeland,
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69 ## W.B. Jones}, @cite{Handbook of Continued Fractions for Special Functions},
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70 ## ch.@: 18.
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71 ##
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72 ## @seealso{beta, betaincinv, betaln}
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73 ## @end deftypefn
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74
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75 function y = betainc (x, a, b, tail = "lower")
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76
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77 if (nargin < 3)
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78 print_usage ();
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79 endif
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80
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81 [err, x, a, b] = common_size (x, a, b);
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82 if (err > 0)
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83 error ("betainc: X, A, and B must be of common size or scalars");
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84 endif
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85
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86 if (iscomplex (x) || iscomplex (a) || iscomplex (b))
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87 error ("betainc: all inputs must be real");
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88 endif
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89
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90 ## Remember original shape of data, but convert to column vector for calcs.
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91 orig_sz = size (x);
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92 x = x(:);
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93 a = a(:);
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94 b = b(:);
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95
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96 if (any ((x < 0) | (x > 1)))
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97 error ("betainc: X must be in the range [0, 1]");
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98 endif
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99
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100 if (any (a <= 0))
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101 error ("betainc: A must be strictly positive");
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102 endif
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103
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104 if (any (b <= 0))
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105 error ("betainc: B must be strictly positive");
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106 endif
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107
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108 ## If any of the arguments is single then the output should be as well.
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109 if (strcmp (class (x), "single") || strcmp (class (a), "single")
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110 || strcmp (class (b), "single"))
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111 a = single (a);
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112 b = single (b);
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113 x = single (x);
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114 endif
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115
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116 ## Convert to floating point if necessary
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117 if (isinteger (x))
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118 y = double (x);
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119 endif
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120 if (isinteger (a))
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121 a = double (a);
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122 endif
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123 if (isinteger (b))
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124 b = double (b);
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125 endif
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126
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127 ## Initialize output array
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128 y = zeros (size (x), class (x));
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129
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130 ## Trivial cases (long code here trades memory for speed)
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131 a_one = (a == 1);
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132 b_one = (b == 1);
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133 a_b_one = a_one & b_one;
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134 a_not_one = ! a_one;
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135 b_not_one = ! b_one;
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136 non_trivial = a_not_one & b_not_one;
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137 a_one &= b_not_one;
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138 b_one &= a_not_one;
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139
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140 if (strcmpi (tail, "lower"))
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141 y(a_b_one) = x(a_b_one);
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142 y(a_one) = 1 - (1 - x(a_one)) .^ b(a_one);
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143 y(b_one) = x(b_one) .^ a(b_one);
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144 elseif (strcmpi (tail, "upper"))
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145 y(a_b_one) = 1 - x(a_b_one);
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146 y(a_one) = (1 - x(a_one)) .^ b(a_one);
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147 y(b_one) = 1 - x(b_one) .^ a(b_one);
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148 endif
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149
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150 ## Non-Trivial cases
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151 ## In the following, we use the fact that the continued fraction Octave uses
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152 ## is more efficient when x <= a / (a + b). Moreover, to compute the upper
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153 ## version, which is defined as I_x(a,b,"upper") = 1 - I_x(a,b) we use the
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154 ## property I_x(a,b) + I_(1-x) (b,a) = 1.
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155
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156 x = x(non_trivial);
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157 a = a(non_trivial);
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158 b = b(non_trivial);
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159
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160 if (strcmpi (tail, "lower"))
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161 fflag = (x > a./(a + b));
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162 x(fflag) = 1 - x(fflag);
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163 [a(fflag), b(fflag)] = deal (b(fflag), a(fflag));
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164 elseif (strcmpi (tail, "upper"))
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165 fflag = (x < (a ./ (a + b)));
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166 x(! fflag) = 1 - x(! fflag);
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167 [a(! fflag), b(! fflag)] = deal (b(! fflag), a(! fflag));
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168 else
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169 error ("betainc: invalid value for TAIL");
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170 endif
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171
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172 f = zeros (size (x), class (x));
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173
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174 ## Continued fractions: CPVWJ, formula 18.5.20, modified Lentz algorithm
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175 ## implemented in a separate .cc file. This particular continued fraction
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176 ## gives (B(a,b) * I_x(a,b)) / (x^a * (1-x)^b).
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177
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178 f = __betainc__ (x, a, b);
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179
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180 ## Divide continued fraction by B(a,b) / (x^a * (1-x)^b) to obtain I_x(a,b).
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181 y_nt = a .* log (x) + b .* log1p (-x) ...
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182 + (gammaln (a + b) - gammaln (a) - gammaln (b)) + log (f);
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183 y_nt = real (exp (y_nt));
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184 y_nt(fflag) = 1 - y_nt(fflag);
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185
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186 y(non_trivial) = y_nt;
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187
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188 ## Restore original shape
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189 y = reshape (y, orig_sz);
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190
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191 endfunction
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192
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193
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194 ## Double precision
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195 %!test
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196 %! a = [1, 1.5, 2, 3];
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197 %! b = [4, 3, 2, 1];
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198 %! v1 = betainc (1, a, b);
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199 %! v2 = [1,1,1,1];
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200 %! x = [.2, .4, .6, .8];
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201 %! v3 = betainc (x, a, b);
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202 %! v4 = 1 - betainc (1-x, b, a);
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203 %! assert (v1, v2, sqrt (eps));
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204 %! assert (v3, v4, sqrt (eps));
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205
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206 ## Single precision
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207 %!test
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208 %! a = single ([1, 1.5, 2, 3]);
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209 %! b = single ([4, 3, 2, 1]);
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210 %! v1 = betainc (1, a, b);
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211 %! v2 = single ([1,1,1,1]);
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212 %! x = single ([.2, .4, .6, .8]);
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213 %! v3 = betainc (x, a, b);
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214 %! v4 = 1 - betainc (1-x, b, a);
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215 %! assert (v1, v2, sqrt (eps ("single")));
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216 %! assert (v3, v4, sqrt (eps ("single")));
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217
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218 ## Mixed double/single precision
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219 %!test
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220 %! a = single ([1, 1.5, 2, 3]);
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221 %! b = [4, 3, 2, 1];
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222 %! v1 = betainc (1,a,b);
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223 %! v2 = single ([1,1,1,1]);
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224 %! x = [.2, .4, .6, .8];
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225 %! v3 = betainc (x, a, b);
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226 %! v4 = 1 - betainc (1. - x, b, a);
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227 %! assert (v1, v2, sqrt (eps ("single")));
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228 %! assert (v3, v4, sqrt (eps ("single")));
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229
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230 %!test <*51157>
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231 %! y = betainc ([0.00780;0.00782;0.00784],250.005,49750.995);
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232 %! y_ex = [0.999999999999989; 0.999999999999992; 0.999999999999995];
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233 %! assert (y, y_ex, -1e-14);
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234
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235 %!assert (betainc (0.001, 20, 30), 2.750687665855991e-47, -3e-14)
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236 %!assert (betainc (0.0001, 20, 30), 2.819953178893307e-67, -7e-14)
30330
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237 %!assert <*54383> (betainc (0.99, 20, 30, "upper"),
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238 %! 1.5671643161872703e-47, -7e-14)
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239 %!assert (betainc (0.999, 20, 30, "upper"), 1.850806276141535e-77, -7e-14)
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240 %!assert (betainc (0.5, 200, 300), 0.9999964565197356, -1e-15)
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241 %!assert (betainc (0.5, 200, 300, "upper"), 3.54348026439253e-06, -3e-13)
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242
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243 ## Test trivial values
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244 %!test
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245 %! [a,b] = ndgrid (linspace (1e-4, 100, 20), linspace (1e-4, 100, 20));
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246 %! assert (betainc (0, a, b), zeros (20));
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247 %! assert (betainc (1, a, b), ones (20));
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248
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249 %!test <*34405>
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250 %! assert (betainc (NaN, 1, 2), NaN);
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ad6a57b215e8 betainc.m: Improve accuracy for certain special integer inputs (bug #60682).
Michele Ginesi <michele.ginesi@gmail.com>
parents: 29665
diff changeset
251 %! assert (betainc (0.5, 1, Inf), 1);
25015
baa7e37453b1 Added more tests for betainc and expint.
Michele Ginesi <michele.ginesi@gmail.com>
parents: 24996
diff changeset
252
24907
bd89440407aa Incomplete beta function moved to a .m file, fixing accuracy and
Michele Ginesi <michele.ginesi@gmail.com>
parents:
diff changeset
253 ## Test input validation
28886
d8318c12d903 test: remove unnecessary BIST tests in m-files checking for excessive number of inputs.
Rik <rik@octave.org>
parents: 28789
diff changeset
254 %!error <Invalid call> betainc ()
d8318c12d903 test: remove unnecessary BIST tests in m-files checking for excessive number of inputs.
Rik <rik@octave.org>
parents: 28789
diff changeset
255 %!error <Invalid call> betainc (1)
d8318c12d903 test: remove unnecessary BIST tests in m-files checking for excessive number of inputs.
Rik <rik@octave.org>
parents: 28789
diff changeset
256 %!error <Invalid call> betainc (1,2)
24927
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
257 %!error <must be of common size or scalars> betainc (ones (2,2), ones (1,2), 1)
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
258 %!error <all inputs must be real> betainc (0.5i, 1, 2)
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
259 %!error <all inputs must be real> betainc (0, 1i, 1)
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
260 %!error <all inputs must be real> betainc (0, 1, 1i)
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
261 %!error <X must be in the range \[0, 1\]> betainc (-0.1,1,1)
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
262 %!error <X must be in the range \[0, 1\]> betainc (1.1,1,1)
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
263 %!error <X must be in the range \[0, 1\]>
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
264 %! x = ones (1, 1, 2);
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
265 %! x(1,1,2) = -1;
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
266 %! betainc (x,1,1);
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
267 %!error <A must be strictly positive> betainc (0.5,0,1)
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
268 %!error <A must be strictly positive>
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
269 %! a = ones (1, 1, 2);
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
270 %! a(1,1,2) = 0;
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
271 %! betainc (1,a,1);
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
272 %!error <B must be strictly positive> betainc (0.5,1,0)
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
273 %!error <B must be strictly positive>
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
274 %! b = ones (1, 1, 2);
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
275 %! b(1,1,2) = 0;
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
276 %! betainc (1,1,b);
c280560d9c96 Overhaul special functions modified by GSOC2018 project.
Rik <rik@octave.org>
parents: 24923
diff changeset
277 %!error <invalid value for TAIL> betainc (1,2,3, "foobar")