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update Octave Project Developers copyright for the new year
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update for 2021.
In all .txi and .texi files except gpl.txi and gpl.texi in the
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author | John W. Eaton <jwe@octave.org> |
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date | Tue, 28 Dec 2021 18:22:40 -0500 |
parents | 7854d5752dd2 |
children | 5d3faba0342e |
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######################################################################## ## ## Copyright (C) 2017-2022 The Octave Project Developers ## ## See the file COPYRIGHT.md in the top-level directory of this ## distribution or <https://octave.org/copyright/>. ## ## This file is part of Octave. ## ## Octave is free software: you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## <https://www.gnu.org/licenses/>. ## ######################################################################## ## -*- texinfo -*- ## @deftypefn {} {} gammaincinv (@var{y}, @var{a}) ## @deftypefnx {} {} gammaincinv (@var{y}, @var{a}, @var{tail}) ## Compute the inverse of the normalized incomplete gamma function. ## ## The normalized incomplete gamma function is defined as ## @tex ## $$ ## \gamma (x, a) = {1 \over {\Gamma (a)}}\displaystyle{\int_0^x t^{a-1} e^{-t} dt} ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## x ## 1 / ## gammainc (x, a) = --------- | exp (-t) t^(a-1) dt ## gamma (a) / ## t=0 ## @end group ## @end example ## ## @end ifnottex ## ## and @code{gammaincinv (gammainc (@var{x}, @var{a}), @var{a}) = @var{x}} ## for each non-negative value of @var{x}. If @var{a} is scalar then ## @code{gammaincinv (@var{y}, @var{a})} is returned for each element of ## @var{y} and vice versa. ## ## If neither @var{y} nor @var{a} is scalar then the sizes of @var{y} and ## @var{a} must agree, and @code{gammaincinv} is applied element-by-element. ## The variable @var{y} must be in the interval @math{[0,1]} while @var{a} must ## be real and positive. ## ## By default, @var{tail} is @qcode{"lower"} and the inverse of the incomplete ## gamma function integrated from 0 to @var{x} is computed. If @var{tail} is ## @qcode{"upper"}, then the complementary function integrated from @var{x} to ## infinity is inverted. ## ## The function is computed with Newton's method by solving ## @tex ## $$ ## y - \gamma (x, a) = 0 ## $$ ## @end tex ## @ifnottex ## ## @example ## @var{y} - gammainc (@var{x}, @var{a}) = 0 ## @end example ## ## @end ifnottex ## ## Reference: @nospell{A. Gil, J. Segura, and N. M. Temme}, @cite{Efficient and ## accurate algorithms for the computation and inversion of the incomplete ## gamma function ratios}, @nospell{SIAM J. Sci.@: Computing}, pp.@: ## A2965--A2981, Vol 34, 2012. ## ## @seealso{gammainc, gamma, gammaln} ## @end deftypefn function x = gammaincinv (y, a, tail = "lower") if (nargin < 2) print_usage (); endif [err, y, a] = common_size (y, a); if (err > 0) error ("gammaincinv: Y and A must be of common size or scalars"); endif if (iscomplex (y) || iscomplex (a)) error ("gammaincinv: all inputs must be real"); endif ## Remember original shape of data, but convert to column vector for calcs. orig_sz = size (y); y = y(:); a = a(:); if (any ((y < 0) | (y > 1))) error ("gammaincinv: Y must be in the range [0, 1]"); endif if (any (a <= 0)) error ("gammaincinv: A must be strictly positive"); endif ## If any of the arguments is single then the output should be as well. if (strcmp (class (y), "single") || strcmp (class (a), "single")) y = single (y); a = single (a); endif ## Convert to floating point if necessary if (isinteger (y)) y = double (y); endif if (isinteger (a)) a = double (a); endif ## Initialize output array x = zeros (size (y), class (y)); maxit = 20; tol = eps (class (y)); ## Special cases, a = 1 or y = 0, 1. if (strcmpi (tail, "lower")) x(a == 1) = - log1p (- y(a == 1)); x(y == 0) = 0; x(y == 1) = Inf; p = y; q = 1 - p; elseif (strcmpi (tail, "upper")) x(a == 1) = - log (y(a == 1)); x(y == 0) = Inf; x(y == 1) = 0; q = y; p = 1 - q; else error ("gammaincinv: invalid value for TAIL"); endif todo = (a != 1) & (y != 0) & (y != 1); ## Case 1: p small. i_flag_1 = todo & (p < ((0.2 * (1 + a)) .^ a) ./ gamma (1 + a)); if (any (i_flag_1)) aa = a(i_flag_1); pp = p(i_flag_1); ## Initial guess. r = (pp .* gamma (1 + aa)) .^ (1 ./ aa); c2 = 1 ./ (aa + 1); c3 = (3 * aa + 5) ./ (2 * (aa + 1) .^2 .* (aa + 2)); c4 = (8 * aa .^ 2 + 33 * aa + 31) ./ (3 * (aa + 1) .^ 3 .* (aa + 2) .* ... (aa + 3)); c5 = (125 * aa .^ 4 + 1179 * aa .^ 3 + 3971 * aa.^2 + 5661 * aa + 2888) ... ./ (24 * (1 + aa) .^4 .* (aa + 2) .^ 2 .* (aa + 3) .* (aa + 4)); ## FIXME: Would polyval() be better here for more accuracy? x0 = r + c2 .* r .^ 2 + c3 .* r .^ 3 + c4 .* r .^4 + c5 .* r .^ 5; ## For this case we invert the lower version. F = @(p, a, x) p - gammainc (x, a, "lower"); JF = @(a, x) - exp (- gammaln (a) - x + (a - 1) .* log (x)); x(i_flag_1) = newton_method (F, JF, pp, aa, x0, tol, maxit); endif todo(i_flag_1) = false; ## Case 2: q small. i_flag_2 = (q < exp (- 0.5 * a) ./ gamma (1 + a)) & (a > 0) & (a < 10); i_flag_2 &= todo; if (any (i_flag_2)) aa = a(i_flag_2); qq = q(i_flag_2); ## Initial guess. x0 = (-log (qq) - gammaln (aa)); ## For this case, we invert the upper version. F = @(q, a, x) q - gammainc (x, a, "upper"); JF = @(a, x) exp (- gammaln (a) - x) .* x .^ (a - 1); x(i_flag_2) = newton_method (F, JF, qq, aa, x0, tol, maxit); endif todo(i_flag_2) = false; ## Case 3: a small. i_flag_3 = todo & ((a > 0) & (a < 1)); if (any (i_flag_3)) aa = a(i_flag_3); pp = p(i_flag_3); ## Initial guess xl = (pp .* gamma (aa + 1)) .^ (1 ./ aa); x0 = xl; ## For this case, we invert the lower version. F = @(p, a, x) p - gammainc (x, a, "lower"); JF = @(a, x) - exp (-gammaln (a) - x) .* x .^ (a - 1); x(i_flag_3) = newton_method (F, JF, pp, aa, x0, tol, maxit); endif todo(i_flag_3) = false; ## Case 4: a large. i_flag_4 = todo; if (any (i_flag_4)) aa = a(i_flag_4); qq = q(i_flag_4); ## Initial guess d = 1 ./ (9 * aa); t = 1 - d + sqrt (2) * erfcinv (2 * qq) .* sqrt (d); x0 = aa .* (t .^ 3); ## For this case, we invert the upper version. F = @(q, a, x) q - gammainc (x, a, "upper"); JF = @(a, x) exp (- gammaln (a) - x + (a - 1) .* log (x)); x(i_flag_4) = newton_method (F, JF, qq, aa, x0, tol, maxit); endif ## Restore original shape x = reshape (x, orig_sz); endfunction ## subfunction: Newton's Method function x = newton_method (F, JF, y, a, x0, tol, maxit); l = numel (y); res = -F (y, a, x0) ./ JF (a, x0); todo = (abs (res) >= tol * abs (x0)); x = x0; it = 0; while (any (todo) && (it++ < maxit)) x(todo) += res(todo); res(todo) = -F (y(todo), a(todo), x(todo)) ./ JF (a(todo), x(todo)); todo = (abs (res) >= tol * abs (x)); endwhile x += res; endfunction %!test %! x = [1e-10, 1e-09, 1e-08, 1e-07]; %! a = [2, 3, 4]; %! [x, a] = ndgrid (x, a); %! xx = gammainc (gammaincinv (x, a), a); %! assert (xx, x, -3e-14); %!test %! x = [1e-10, 1e-09, 1e-08, 1e-07]; %! a = [2, 3, 4]; %! [x, a] = ndgrid (x, a); %! xx = gammainc (gammaincinv (x, a, "upper"), a, "upper"); %! assert (xx, x, -3e-14); %!test %! x = linspace (0, 1)'; %! a = [linspace(0.1, 1, 10), 2:5]; %! [x, a] = ndgrid (x, a); %! xx = gammainc (gammaincinv (x, a), a); %! assert (xx, x, -1e-13); %!test %! x = linspace (0, 1)'; %! a = [linspace(0.1, 1, 10), 2:5]; %! [x, a] = ndgrid (x, a); %! xx = gammainc (gammaincinv (x, a, "upper"), a, "upper"); %! assert (xx, x, -1e-13); %!test <*56453> %! assert (gammaincinv (1e-15, 1) * 2, 2e-15, -1e-15); %! assert (gammaincinv (1e-16, 1) * 2, 2e-16, -1e-15); ## Test the conservation of the input class %!assert (class (gammaincinv (0.5, 1)), "double") %!assert (class (gammaincinv (single (0.5), 1)), "single") %!assert (class (gammaincinv (0.5, single (1))), "single") %!assert (class (gammaincinv (int8 (0), 1)), "double") %!assert (class (gammaincinv (0.5, int8 (1))), "double") %!assert (class (gammaincinv (int8 (0), single (1))), "single") %!assert (class (gammaincinv (single (0.5), int8 (1))), "single") ## Test input validation %!error <Invalid call> gammaincinv () %!error <Invalid call> gammaincinv (1) %!error <must be of common size or scalars> %! gammaincinv (ones (2,2), ones (1,2), 1); %!error <all inputs must be real> gammaincinv (0.5i, 1) %!error <all inputs must be real> gammaincinv (0, 1i) %!error <Y must be in the range \[0, 1\]> gammaincinv (-0.1,1) %!error <Y must be in the range \[0, 1\]> gammaincinv (1.1,1) %!error <Y must be in the range \[0, 1\]> %! y = ones (1, 1, 2); %! y(1,1,2) = -1; %! gammaincinv (y,1); %!error <A must be strictly positive> gammaincinv (0.5, 0) %!error <A must be strictly positive> %! a = ones (1, 1, 2); %! a(1,1,2) = 0; %! gammaincinv (1,a,1); %!error <invalid value for TAIL> gammaincinv (1,2, "foobar")