Mercurial > octave
view scripts/specfun/cosint.m @ 30564:796f54d4ddbf stable
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 | 01de0045b2e3 |
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 {} {} cosint (@var{x}) ## Compute the cosine integral function: ## @tex ## $$ ## {\rm Ci} (x) = - \int_x^\infty {{\cos (t)} \over t} dt ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## +oo ## / ## Ci (x) = - | (cos (t)) / t dt ## / ## x ## @end group ## @end example ## ## @end ifnottex ## An equivalent definition is ## @tex ## $$ ## {\rm Ci} (x) = \gamma + \log (x) + \int_0^x {{\cos (t) - 1} \over t} dt ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## x ## / ## | cos (t) - 1 ## Ci (x) = gamma + log (x) + | ------------- dt ## | t ## / ## 0 ## @end group ## @end example ## ## @end ifnottex ## Reference: ## ## @nospell{M. Abramowitz and I.A. Stegun}, ## @cite{Handbook of Mathematical Functions}, 1964. ## ## @seealso{sinint, expint, cos} ## ## @end deftypefn function y = cosint (x) if (nargin < 1) print_usage (); endif if (! isnumeric (x)) error ("cosint: X must be numeric"); endif ## Convert to floating point if necessary if (isinteger (x)) x = double (x); endif ## Convert to column vector orig_sz = size (x); if (iscomplex (x)) ## Work around reshape which narrows to real (bug #52953) x = complex (real (x)(:), imag (x)(:)); else x = x(:); endif ## Initialize the result y = zeros (size (x), class (x)); tol = eps (class (x)); todo = true (size (x)); ## Special values y(x == Inf) = 0; y((x == -Inf) & ! signbit (imag (x))) = 1i * pi; y((x == -Inf) & signbit (imag (x))) = -1i * pi; todo(isinf (x)) = false; ## For values large in modulus, but not in the range (-oo,0), we use the ## relation with expint. flag_large = (abs (x) > 2); xx = x(flag_large); ## Abramowitz, relation 5.2.20 ii_sw = (real (xx) <= 0 & imag (xx) < 0); xx(ii_sw) = conj (xx(ii_sw)); ii_nw = (real (xx) < 0); xx(ii_nw) *= -1; yy = -0.5 * (expint (1i * xx) + expint (-1i * xx)); yy(ii_nw) += 1i * pi; yy(ii_sw) = conj (yy(ii_sw)); y(todo & flag_large) = yy; todo(flag_large) = false; ## For values small in modulus, use the series expansion (also near (-oo, 0]) if (iscomplex (x)) ## indexing can lose imag part: if it was -0, we could end up on the ## wrong right side of the branch cut along the negative real axis. xx = complex (real (x)(todo), imag (x)(todo)); else xx = x(todo); endif ssum = - xx .^ 2 / 4; # First term of the series expansion ## FIXME: This is way more precision than a double value can hold. gma = 0.57721566490153286060651209008; # Euler gamma constant yy = gma + log (complex (xx)) + ssum; # log (complex (Z)) handles signed zero flag_sum = true (nnz (todo), 1); it = 0; maxit = 300; while (any (flag_sum) && (++it < maxit)) ssum .*= - xx .^ 2 * (2 * it) / ((2 * it + 2) ^ 2 * (2 * it + 1)); yy(flag_sum) += ssum (flag_sum); flag_sum = (abs (ssum) >= tol); endwhile y(todo) = yy; ## Clean up values which are purely real flag_neg_zero_imag = (real (x) < 0) & (imag (x) == 0) & signbit (imag (x)); y(flag_neg_zero_imag) = complex (real (y(flag_neg_zero_imag)), -pi); ## Restore original shape y = reshape (y, orig_sz); endfunction %!assert (cosint (1.1), 0.38487337742465081550, 2 * eps) %!test %! x = [2, 3, pi; exp(1), 5, 6]; %! A = cosint (x); %! B = [0.422980828774864996, 0.119629786008000328, 0.0736679120464254860; ... %! 0.213958001340379779, -0.190029749656643879, -0.0680572438932471262]; %! assert (A, B, -5e-15); %!assert (cosint (0), - Inf) %!assert (cosint (-0), -inf + 1i*pi) %!assert (cosint (complex (-0, 0)), -inf + 1i*pi) %!assert (cosint (complex (-0, -0)), -inf - 1i*pi) %!assert (cosint (inf), 0) %!assert (cosint (-inf), 1i * pi) %!assert (cosint (complex (-inf, -0)), -1i * pi) %!assert (isnan (cosint (nan))) %!assert (class (cosint (single (1))), "single") ## tests against maple %!assert (cosint (1), 0.337403922900968135, -2*eps) %!assert (cosint (-1), 0.337403922900968135 + 3.14159265358979324*I, -2*eps) %!assert (cosint (pi), 0.0736679120464254860, -4e-15) %!assert (cosint (-pi), 0.0736679120464254860 + 3.14159265358979324*I, -2*eps) %!assert (cosint (300), -0.00333219991859211178, -2*eps) %!assert (cosint (1e4), -0.0000305519167244852127, -2*eps) %!assert (cosint (20i), 1.28078263320282944e7 + 1.57079632679489662*I, -2*eps) %!test %! x = (0:4).'; %! y_ex = [-Inf %! 0.337403922900968135 %! 0.422980828774864996 %! 0.119629786008000328 %! -0.140981697886930412]; %! assert (cosint (x), y_ex, -3e-15); %!test %! x = -(1:4).'; %! y_ex = [0.337403922900968135 + pi*1i %! 0.422980828774864996 + pi*1i %! 0.119629786008000328 + pi*1i %! -0.140981697886930412 + pi*1i]; %! assert (cosint (x), y_ex, -4*eps); %!test %! x = complex (-(1:4).', 0); %! y_ex = [0.337403922900968135 + pi*1i %! 0.422980828774864996 + pi*1i %! 0.119629786008000328 + pi*1i %! -0.140981697886930412 + pi*1i]; %! assert (cosint (x), y_ex, -4*eps); %!test %! x = complex (-(1:4).', -0); %! y_ex = [0.337403922900968135 - pi*1i %! 0.422980828774864996 - pi*1i %! 0.119629786008000328 - pi*1i %! -0.140981697886930412 - pi*1i]; %! assert (cosint (x), y_ex, -4*eps); %!test %! x = 1i * (0:4).'; %! y_ex = [-Inf %! 0.837866940980208241 + 1.57079632679489662*I %! 2.45266692264691452 + 1.57079632679489662*I %! 4.96039209476560976 + 1.57079632679489662*I %! 9.81354755882318556 + 1.57079632679489662*I]; %! assert (cosint (x), y_ex, -4*eps); %!test %! x = -1i * (1:4).'; %! y_ex = [0.837866940980208241 - 1.57079632679489662*I %! 2.45266692264691452 - 1.57079632679489662*I %! 4.96039209476560976 - 1.57079632679489662*I %! 9.81354755882318556 - 1.57079632679489662*I]; %! assert (cosint (x), y_ex, -4*eps); %!test %! x = [1+2i; -2+5i; 2-5i; 100; 10i; -1e-4 + 1e-6*1i; -20-1i]; %! A = [ 2.03029639329172164 - 0.151907155175856884*I %! 1.61538963829107749 + 19.7257540553382650*I %! 1.61538963829107749 + 16.5841614017484717*I %! -0.00514882514261049214 %! 1246.11448604245441 + 1.57079632679489662*I %! -8.63307471207423322 + 3.13159298695312800*I %! 0.0698222284673061493 - 3.11847446254772946*I ]; %! B = cosint (x); %! assert (A, B, -3*eps); %! B = cosint (single (x)); %! assert (A, B, -3*eps ("single")); ## Fails along negative real axis %!test %! x = [-25; -100; -1000]; %! yex = [-0.0068485971797025909189 + pi*1i %! -0.0051488251426104921444 + pi*1i %! 0.000826315511090682282 + pi*1i]; %! y = cosint (x); %! assert (y, yex, -5*eps); ## FIXME: Need a test for bug #52953 %#!test <*52953> ## Test input validation %!error <Invalid call> cosint () %!error <X must be numeric> cosint ("1")