Mercurial > octave
view scripts/specfun/cosint.m @ 24913:9da779d2f029 stable
cosint improve signed zero imag input near branch cut
Stop special casing the origin, instead have log deal with signed
zeros (real and complex). Ensure we approach the branch cut
(negative real axis) correctly based on signed 0 for imaginary
part.
cosint.m: handle complex signed zero, add and modify BIST.
author | Colin Macdonald <cbm@m.fsf.org> |
---|---|
date | Tue, 23 Jan 2018 23:14:50 -0800 |
parents | 8d6f3941a118 |
children | d7293106945c |
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## Copyright (C) 2017 Michele Ginesi ## ## 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 ## <http://www.gnu.org/licenses/>. ## Author: Michele Ginesi <michele.ginesi@gmail.com> ## -*- 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{expint, cos, sinint} ## ## @end deftypefn function [y] = cosint (x) if (nargin != 1) print_usage (); endif sz = size (x); #x = reshape (x, numel (x), 1); if (iscomplex (x)) ## workaround reshape narrowing to real (#52953) x = complex (real (x)(:), imag (x)(:)); else x = x(:); end y = zeros (size (x), class (x)); tol = eps (class (x)); i_miss = true (length (x), 1); ## special values y(x == Inf) = 0; y((x == -Inf) & !signbit (imag(x))) = 1i * pi; y((x == -Inf) & signbit (imag(x))) = -1i * pi; i_miss = ((i_miss) & (x != Inf) & (x != -Inf)); ## For values large in modulus and not in (-oo,0), we use the relation ## with expint flag_large = (abs (x) > 2 & ((abs (imag (x)) > 1e-15) | real (x) > 0)); 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(i_miss & flag_large) = yy; ## For values small in modulus, use the series expansion (also near (-oo, 0]) i_miss = ((i_miss) & (!flag_large)); 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)(i_miss), imag (x)(i_miss)); else xx = x(i_miss); end ssum = - xx .^ 2 / 4; # First term of the series expansion gma = 0.57721566490153286060651209008; # Euler gamma constant yy = gma + log (complex (xx)) + ssum; # log(complex(...) handles signed zero flag_sum = true (nnz (i_miss), 1); it = 1; 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); it++; endwhile y(i_miss) = yy; y = reshape (y, 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, -2*eps) %!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 = 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"))