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view scripts/specfun/gammainc.m @ 30875:5d3faba0342e
doc: Ensure documentation lists output argument when it exists for all m-files.
For new users of Octave it is best to show explicit calling forms
in the documentation and to show a return argument when it exists.
* bp-table.cc, shift.m, accumarray.m, accumdim.m, bincoeff.m, bitcmp.m,
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__actual_axis_position__.m, __pltopt__.m, close.m, graphics_toolkit.m, pan.m,
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arch_rnd.m, arma_rnd.m, autoreg_matrix.m, bartlett.m, blackman.m, detrend.m,
durbinlevinson.m, fftconv.m, fftfilt.m, fftshift.m, fractdiff.m, hamming.m,
hanning.m, hurst.m, ifftshift.m, rectangle_lw.m, rectangle_sw.m, triangle_lw.m,
sinc.m, sinetone.m, sinewave.m, spectral_adf.m, spectral_xdf.m, spencer.m,
ilu.m, __sprand__.m, sprand.m, sprandn.m, sprandsym.m, treelayout.m, beta.m,
betainc.m, betaincinv.m, betaln.m, cosint.m, expint.m, factorial.m, gammainc.m,
gammaincinv.m, lcm.m, nthroot.m, perms.m, reallog.m, realpow.m, realsqrt.m,
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mad.m, mean.m, meansq.m, median.m, mode.m, moment.m, range.m, ranks.m,
run_count.m, skewness.m, spearman.m, statistics.m, std.m, base2dec.m,
bin2dec.m, blanks.m, cstrcat.m, deblank.m, dec2base.m, dec2bin.m, dec2hex.m,
hex2dec.m, index.m, regexptranslate.m, rindex.m, strcat.m, strjust.m,
strtrim.m, strtrunc.m, substr.m, untabify.m, __have_feature__.m,
__prog_output_assert__.m, __run_test_suite__.m, example.m, fail.m, asctime.m,
calendar.m, ctime.m, date.m, etime.m:
Add return arguments to @deftypefn macros where they were missing. Rename
variables in functions (particularly generic "retval") to match documentation.
Rename some return variables for (hopefully) better clarity (e.g., 'ax' to 'hax'
to indicate it is a graphics handle to an axes object).
author | Rik <rik@octave.org> |
---|---|
date | Wed, 30 Mar 2022 20:40:27 -0700 |
parents | 796f54d4ddbf |
children | c8ad083a5802 |
line wrap: on
line source
######################################################################## ## ## Copyright (C) 2016-2022 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 {} {@var{y} =} gammainc (@var{x}, @var{a}) ## @deftypefnx {} {@var{y} =} gammainc (@var{x}, @var{a}, @var{tail}) ## Compute the normalized incomplete gamma function. ## ## This 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 ## with the limiting value of 1 as @var{x} approaches infinity. ## The standard notation is @math{P(a,x)}, e.g., @nospell{Abramowitz} and ## @nospell{Stegun} (6.5.1). ## ## If @var{a} is scalar, then @code{gammainc (@var{x}, @var{a})} is returned ## for each element of @var{x} and vice versa. ## ## If neither @var{x} nor @var{a} is scalar then the sizes of @var{x} and ## @var{a} must agree, and @code{gammainc} is applied element-by-element. ## The elements of @var{a} must be non-negative. ## ## By default, @var{tail} is @qcode{"lower"} and 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 ## calculated. ## ## If @var{tail} is @qcode{"scaledlower"}, then the lower incomplete gamma ## function is multiplied by ## @tex ## $\Gamma(a+1)\exp(x)x^{-a}$. ## @end tex ## @ifnottex ## @math{gamma(a+1)*exp(x)/(x^a)}. ## @end ifnottex ## If @var{tail} is @qcode{"scaledupper"}, then the upper incomplete gamma ## function is multiplied by the same quantity. ## ## References: ## ## @nospell{M. Abramowitz and I.A. Stegun}, ## @cite{Handbook of mathematical functions}, ## @nospell{Dover publications, Inc.}, 1972. ## ## @nospell{W. Gautschi}, ## @cite{A computational procedure for incomplete gamma functions}, ## @nospell{ACM Trans.@: Math Software}, pp.@: 466--481, Vol 5, No.@: 4, 2012. ## ## @nospell{W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery}, ## @cite{Numerical Recipes in Fortran 77}, ch.@: 6.2, Vol 1, 1992. ## ## @seealso{gamma, gammaincinv, gammaln} ## @end deftypefn ## P(a,x) = gamma(a,x)/Gamma(a), upper ## 1-P(a,x)=Q(a,x)=Gamma(a,x)/Gamma(a), lower function y = gammainc (x, a, tail = "lower") if (nargin < 2) print_usage (); endif [err, x, a] = common_size (x, a); if (err > 0) error ("gammainc: X and A must be of common size or scalars"); endif if (iscomplex (x) || iscomplex (a)) error ("gammainc: all inputs must be real"); endif ## Remember original shape of data, but convert to column vector for calcs. x_sz = size (x); x = x(:); a = a(:); if (any (a < 0)) error ("gammainc: A must be non-negative"); endif if (nargin == 3 && ! any (strcmpi (tail, {"lower","upper","scaledlower","scaledupper"}))) error ("gammainc: invalid value for TAIL"); endif tail = tolower (tail); ## If any of the arguments is single then the output should be as well. if (strcmp (class (x), "single") || strcmp (class (a), "single")) x = single (x); a = single (a); endif ## Convert to floating point if necessary if (isinteger (x)) x = double (x); endif if (isinteger (a)) a = double (a); endif ## Initialize output array y = zeros (x_sz, class (x)); ## Different x, a combinations are handled by different subfunctions. todo = true (size (x)); # Track which elements need to be calculated. ## Case 0: x == Inf, a == Inf idx = (x == Inf) & (a == Inf); if (any (idx)) y(idx) = NaN; todo(idx) = false; endif ## Case 1: x == 0, a == 0. idx = (x == 0) & (a == 0); if (any (idx)) y(idx) = gammainc_00 (tail); todo(idx) = false; endif ## Case 2: x == 0. idx = todo & (x == 0); if (any (idx)) y(idx) = gammainc_x0 (tail); todo(idx) = false; endif ## Case 3: x = Inf idx = todo & (x == Inf); if (any (idx)) y(idx) = gammainc_x_inf (tail); todo(idx) = false; endif ## Case 4: a = Inf idx = todo & (a == Inf); if (any (idx)) y(idx) = gammainc_a_inf (tail); todo(idx) = false; endif ## Case 5: a == 0. idx = todo & (a == 0); if (any (idx)) y(idx) = gammainc_a0 (x(idx), tail); todo(idx) = false; endif ## Case 6: a == 1. idx = todo & (a == 1); if (any (idx)) y(idx) = gammainc_a1 (x(idx), tail); todo(idx) = false; endif ## Case 7: positive integer a; exp (x) and a! both under 1/eps. idx = (todo & (a == fix (a)) & (a > 1) & (a <= 18) & (x <= 36) & (abs (x) >= .1)); if (any (idx)) y(idx) = gammainc_an (x(idx), a(idx), tail); todo(idx) = false; endif ## For a < 2, x < 0, we increment a by 2 and use a recurrence formula after ## the computations. flag_a_small = todo & (abs (a) > 0) & (abs (a) < 2) & (x < 0); a(flag_a_small) += 2; flag_s = (((x + 0.25 < a) | (x < 0)) & (x > -20)) | (abs (x) < 1); ## Case 8: x, a relatively small. idx = todo & flag_s; if (any (idx)) y(idx) = gammainc_s (x(idx), a(idx), tail); todo(idx) = false; endif ## Case 9: x positive and large relative to a. idx = todo; if (any (idx)) y(idx) = gammainc_l (x(idx), a(idx), tail); todo(idx) = false; endif if (any (flag_a_small)) if (strcmp (tail, "lower")) y(flag_a_small) += D (x(flag_a_small), a(flag_a_small) - 1) + ... D (x(flag_a_small), a(flag_a_small) - 2); elseif (strcmp (tail, "upper")) y(flag_a_small) -= D (x(flag_a_small), a(flag_a_small) - 1) + ... D (x(flag_a_small), a(flag_a_small) - 2); elseif (strcmp (tail, "scaledlower")) y(flag_a_small) = y(flag_a_small) .* (x(flag_a_small) .^ 2) ./ ... (a(flag_a_small) .* (a(flag_a_small) - 1)) + (x(flag_a_small) ./ ... (a(flag_a_small) - 1)) + 1; elseif (strcmp (tail, "scaledupper")) y(flag_a_small) = y(flag_a_small) .* (x(flag_a_small) .^ 2) ./ ... (a(flag_a_small) .* (a(flag_a_small) - 1)) - (x(flag_a_small) ./ ... (a(flag_a_small) - 1)) - 1; endif endif endfunction ## Subfunctions to handle each case: ## x == 0, a == 0. function y = gammainc_00 (tail) if (strcmp (tail, "upper") || strcmp (tail, "scaledupper")) y = 0; else y = 1; endif endfunction ## x == 0. function y = gammainc_x0 (tail) if (strcmp (tail, "lower")) y = 0; elseif (strcmp (tail, "upper") || strcmp (tail, "scaledlower")) y = 1; else y = Inf; endif endfunction ## x == Inf. function y = gammainc_x_inf (tail) if (strcmp (tail, "lower")) y = 1; elseif (strcmp (tail, "upper") || strcmp (tail, "scaledupper")) y = 0; else y = Inf; endif endfunction ## a == Inf. function y = gammainc_a_inf (tail) if (strcmp (tail, "lower")) y = 0; elseif (strcmp (tail, "upper") || strcmp (tail, "scaledlower")) y = 1; else y = Inf; endif endfunction ## a == 0. function y = gammainc_a0 (x, tail) if (strcmp (tail, "lower")) y = 1; elseif (strcmp (tail, "scaledlower")) y = exp (x); else y = 0; endif endfunction ## a == 1. function y = gammainc_a1 (x, tail) if (strcmp (tail, "lower")) if (abs (x) < 1/2) y = - expm1 (-x); else y = 1 - exp (-x); endif elseif (strcmp (tail, "upper")) y = exp (-x); elseif (strcmp (tail, "scaledlower")) if (abs (x) < 1/2) y = expm1 (x) ./ x; else y = (exp (x) - 1) ./ x; endif else y = 1 ./ x; endif endfunction ## positive integer a; exp (x) and a! both under 1/eps ## uses closed-form expressions for nonnegative integer a ## -- http://mathworld.wolfram.com/IncompleteGammaFunction.html. function y = gammainc_an (x, a, tail) y = t = ones (size (x), class (x)); i = 1; while (any (a(:) > i)) jj = (a > i); t(jj) .*= (x(jj) / i); y(jj) += t(jj); i++; endwhile if (strcmp (tail, "lower")) y = 1 - exp (-x) .* y; elseif (strcmp (tail, "upper")) y .*= exp (-x); elseif (strcmp (tail, "scaledlower")) y = (1 - exp (-x) .* y) ./ D(x, a); elseif (strcmp (tail, "scaledupper")) y .*= exp (-x) ./ D(x, a); endif endfunction ## x + 0.25 < a | x < 0 | abs(x) < 1. ## Numerical Recipes in Fortran 77 (6.2.5) ## series function y = gammainc_s (x, a, tail) if (strcmp (tail, "scaledlower") || strcmp (tail, "scaledupper")) y = ones (size (x), class (x)); term = x ./ (a + 1); else ## Of course it is possible to scale at the end, but some tests fail. ## And try gammainc (1,1000), it take 0 iterations if you scale now. y = D (x,a); term = y .* x ./ (a + 1); endif n = 1; while (any (abs (term(:)) > (abs (y(:)) * eps))) ## y can be zero from the beginning (gammainc (1,1000)) jj = abs (term) > abs (y) * eps; n += 1; y(jj) += term(jj); term(jj) .*= x(jj) ./ (a(jj) + n); endwhile if (strcmp (tail, "upper")) y = 1 - y; elseif (strcmp (tail, "scaledupper")) y = 1 ./ D (x,a) - y; endif endfunction ## x positive and large relative to a ## NRF77 (6.2.7) ## Gamma (a,x)/Gamma (a) ## Lentz's algorithm ## __gammainc__ in libinterp/corefcn/__gammainc__.cc function y = gammainc_l (x, a, tail) y = __gammainc__ (x, a); if (strcmp (tail, "lower")) y = 1 - y .* D (x, a); elseif (strcmp (tail, "upper")) y .*= D (x, a); elseif (strcmp (tail, "scaledlower")) y = 1 ./ D (x, a) - y; endif endfunction ## Compute exp(-x)*x^a/Gamma(a+1) in a stable way for x and a large. ## ## L. Knusel, Computation of the Chi-square and Poisson distribution, ## SIAM J. Sci. Stat. Comput., 7(3), 1986 ## which quotes Section 5, Abramowitz&Stegun 6.1.40, 6.1.41. function y = D (x, a) athresh = 10; # FIXME: can this be better tuned? y = zeros (size (x), class (x)); todo = true (size (x)); todo(x == 0) = false; ii = todo & (x > 0) & (a > athresh) & (a >= x); if (any (ii)) lnGa = log (2 * pi * a(ii)) / 2 + 1 ./ (12 * a(ii)) - ... 1 ./ (360 * a(ii) .^ 3) + 1 ./ (1260 * a(ii) .^ 5) - ... 1 ./ (1680 * a(ii) .^ 7) + 1 ./ (1188 * a(ii) .^ 9)- ... 691 ./ (87360 * a(ii) .^ 11) + 1 ./ (156 * a(ii) .^ 13) - ... 3617 ./ (122400 * a(ii) .^ 15) + ... 43867 ./ (244188 * a(ii) .^ 17) - 174611 ./ (125400 * a(ii) .^ 19); lns = log1p ((a(ii) - x(ii)) ./ x(ii)); y(ii) = exp ((a(ii) - x(ii)) - a(ii) .* lns - lnGa); todo(ii) = false; endif ii = todo & (x > 0) & (a > athresh) & (a < x); if (any (ii)) lnGa = log (2 * pi * a(ii)) / 2 + 1 ./ (12 * a(ii)) - ... 1 ./ (360 * a(ii) .^ 3) + 1 ./ (1260 * a(ii) .^ 5) - ... 1 ./ (1680 * a(ii) .^ 7) + 1 ./ (1188 * a(ii) .^ 9)- ... 691 ./ (87360 * a(ii) .^ 11) + 1 ./ (156 * a(ii) .^ 13) - ... 3617 ./ (122400 * a(ii) .^ 15) + ... 43867 ./ (244188 * a(ii) .^ 17) - 174611 ./ (125400 * a(ii) .^ 19); lns = -log1p ((x(ii) - a(ii)) ./ a(ii)); y(ii) = exp ((a(ii) - x(ii)) - a(ii) .* lns - lnGa); todo(ii) = false; endif ii = todo & ((x <= 0) | (a <= athresh)); if (any (ii)) # standard formula for a not so large. y(ii) = exp (a(ii) .* log (x(ii)) - x(ii) - gammaln (a(ii) + 1)); todo(ii) = false; endif ii = (x < 0) & (a == fix (a)); if (any (ii)) # remove spurious imaginary part. y(ii) = real (y(ii)); endif endfunction ## Test: case 1,2,5 %!assert (gammainc ([0, 0, 1], [0, 1, 0]), [1, 0, 1]) %!assert (gammainc ([0, 0, 1], [0, 1, 0], "upper"), [0, 1, 0]) %!assert (gammainc ([0, 0, 1], [0, 1, 0], "scaledlower"), [1, 1, exp(1)]) %!assert (gammainc ([0, 0, 1], [0, 1, 0], "scaledupper"), [0, Inf, 0]) ## Test: case 3,4 %!assert (gammainc ([2, Inf], [Inf, 2]), [0, 1]) %!assert (gammainc ([2, Inf], [Inf, 2], "upper"), [1, 0]) %!assert (gammainc ([2, Inf], [Inf, 2], "scaledlower"), [1, Inf]) %!assert (gammainc ([2, Inf], [Inf, 2], "scaledupper"), [Inf, 0]) ## Test: case 5 ## Matlab fails for this test %!assert (gammainc (-100,1,"upper"), exp (100), -eps) ## Test: case 6 %!assert (gammainc ([1, 2, 3], 1), 1 - exp (-[1, 2, 3])) %!assert (gammainc ([1, 2, 3], 1, "upper"), exp (- [1, 2, 3])) %!assert (gammainc ([1, 2, 3], 1, "scaledlower"), ... %! (exp ([1, 2, 3]) - 1) ./ [1, 2, 3]) %!assert (gammainc ([1, 2, 3], 1, "scaledupper"), 1 ./ [1, 2, 3]) ## Test: case 7 %!assert (gammainc (2, 2, "lower"), 0.593994150290162, -2e-15) %!assert (gammainc (2, 2, "upper"), 0.406005849709838, -2e-15) %!assert (gammainc (2, 2, "scaledlower"), 2.194528049465325, -2e-15) %!assert (gammainc (2, 2, "scaledupper"), 1.500000000000000, -2e-15) %!assert (gammainc ([3 2 36],[2 3 18], "upper"), ... %! [4/exp(3) 5*exp(-2) (4369755579265807723 / 2977975)/exp(36)], -eps) %!assert (gammainc (10, 10), 1 - (5719087 / 567) * exp (-10), -eps) %!assert (gammainc (10, 10, "upper"), (5719087 / 567) * exp (-10), -eps) ## Test: case 8 %!assert (gammainc (-10, 10), 3.112658265341493126871617e7, -2*eps) ## Matlab fails this next one%! %! %!assert (isreal (gammainc (-10, 10)), true) %!assert (gammainc (-10, 10.1, "upper"), ... %! -2.9582761911890713293e7-1i * 9.612022339061679758e6, -30*eps) %!assert (gammainc (-10, 10, "upper"), -3.112658165341493126871616e7, ... %! -2*eps) %!assert (gammainc (-10, 10, "scaledlower"), 0.5128019364747265, -1e-14) %!assert (gammainc (-10, 10, "scaledupper"), -0.5128019200000000, -1e-14) %!assert (gammainc (200, 201, "upper"), 0.518794309678684497, -2 * eps) %!assert (gammainc (200, 201, "scaledupper"), %! 18.4904360746560462660798514, -eps) ## Here we are very good (no D (x,a)) involved %!assert (gammainc (1000, 1000.5, "scaledlower"), 39.48467539583672271, -2*eps) %!assert (gammainc (709, 1000, "upper"), 0.99999999999999999999999954358, -eps) ## Test: case 9 %!test <*47800> %! assert (gammainc (60, 6, "upper"), 6.18022358081160257327264261e-20, %! -10*eps); ## Matlab is better here than Octave %!assert (gammainc (751, 750, "upper"), 0.4805914320558831327179457887, -12*eps) %!assert (gammainc (200, 200, "upper"), 0.49059658199276367497217454, -6*eps) %!assert (gammainc (200, 200), 0.509403418007236325027825459574527043, -5*eps) %!assert (gammainc (200, 200, "scaledupper"), 17.3984438553791505135122900, %! -3*eps) %!assert (gammainc (200, 200, "scaledlower"), 18.065406676779221643065, -8*eps) %!assert (gammainc (201, 200, "upper"), 0.46249244908276709524913736667, %! -7*eps) %!assert <*54550> (gammainc (77, 2), 1) %!assert (gammainc (77, 2, "upper"), 0, -eps) %!assert (gammainc (1000, 3.1), 1) %!assert (gammainc (1000, 3.1, "upper"), 0) ## Test small argument %!assert (gammainc ([1e-05, 1e-07,1e-10,1e-14], 0.1), ... %! [0.33239840504050, 0.20972940370977, 0.10511370061022, ... %! 0.041846517936723], 1e-13); %!assert (gammainc ([1e-05, 1e-07,1e-10,1e-14], 0.2), ... %! [0.10891226058559, 0.043358823442178, 0.010891244210402, ... %! 0.0017261458806785], 1e-13); %!test %!assert (gammainc ([1e-02, 1e-03, 1e-5, 1e-9, 1e-14], 0.9), ... %! [0.016401189184068, 0.0020735998660840, 0.000032879756964708, ... %! 8.2590606569241e-9, 2.6117443021738e-13], -1e-12); %!test %!assert (gammainc ([1e-02, 1e-03, 1e-5, 1e-9, 1e-14], 2), ... %! [0.0000496679133402659, 4.99666791633340e-7, 4.99996666679167e-11, ... %! 4.99999999666667e-19, 4.99999999999997e-29], -1e-12); %!test <*53543> %! y_exp = 9.995001666250085e-04; %! assert (gammainc (1/1000, 1), y_exp, -eps); %!test <53612> %! assert (gammainc (-20, 1.1, "upper"), ... %! 6.50986687074979e8 + 2.11518396291149e8*i, -1e-13); ## Test conservation of the class (five tests for each subroutine). %!assert (class (gammainc (0, 1)) == "double") %!assert (class (gammainc (single (0), 1)) == "single") %!assert (class (gammainc (int8 (0), 1)) == "double") %!assert (class (gammainc (0, single (1))) == "single") %!assert (class (gammainc (0, int8 (1))) == "double") %!assert (class (gammainc (1, 0)) == "double") %!assert (class (gammainc (single (1), 0)) == "single") %!assert (class (gammainc (int8 (1), 0)) == "double") %!assert (class (gammainc (1, single (0))) == "single") %!assert (class (gammainc (1, int8 (0))) == "double") %!assert (class (gammainc (1, 1)) == "double") %!assert (class (gammainc (single (1), 1)) == "single") %!assert (class (gammainc (int8 (1), 1)) == "double") %!assert (class (gammainc (1, single (1))) == "single") %!assert (class (gammainc (1, int8 (1))) == "double") %!assert (class (gammainc (1, 2)) == "double") %!assert (class (gammainc (single (1), 2)) == "single") %!assert (class (gammainc (int8 (1), 2)) == "double") %!assert (class (gammainc (1, single (2))) == "single") %!assert (class (gammainc (1, int8 (2))) == "double") %!assert (class (gammainc (-1, 0.5)) == "double") %!assert (class (gammainc (single (-1), 0.5)) == "single") %!assert (class (gammainc (int8 (-1), 0.5)) == "double") %!assert (class (gammainc (-1, single (0.5))) == "single") %!assert (class (gammainc (-1, int8 (0.5))) == "double") %!assert (class (gammainc (1, 0.5)) == "double") %!assert (class (gammainc (single (1), 0.5)) == "single") %!assert (class (gammainc (int8 (1), 0.5)) == "double") %!assert (class (gammainc (1, single (0.5))) == "single") %!assert (class (gammainc (1, int8 (0.5))) == "double") ## Test input validation %!error <Invalid call> gammainc () %!error <Invalid call> gammainc (1) %!error <must be of common size or scalars> gammainc ([0, 0],[0; 0]) %!error <must be of common size or scalars> gammainc ([1 2 3], [1 2]) %!error <all inputs must be real> gammainc (2+i, 1) %!error <all inputs must be real> gammainc (1, 2+i) %!error <A must be non-negative> gammainc (1, [0, -1, 1]) %!error <A must be non-negative> %! a = ones (2,2,2); %! a(1,1,2) = -1; %! gammainc (1, a); %!error <invalid value for TAIL> gammainc (1,2, "foobar")