view scripts/specfun/cosint.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, bitget.m, bitset.m, blkdiag.m, celldisp.m, cplxpair.m, dblquad.m, flip.m, fliplr.m, flipud.m, idivide.m, int2str.m, interpft.m, logspace.m, num2str.m, polyarea.m, postpad.m, prepad.m, randi.m, repmat.m, rng.m, rot90.m, rotdim.m, structfun.m, triplequad.m, uibuttongroup.m, uicontrol.m, uipanel.m, uipushtool.m, uitoggletool.m, uitoolbar.m, waitforbuttonpress.m, help.m, __additional_help_message__.m, hsv.m, im2double.m, im2frame.m, javachk.m, usejava.m, argnames.m, char.m, formula.m, inline.m, __vectorize__.m, findstr.m, flipdim.m, strmatch.m, vectorize.m, commutation_matrix.m, cond.m, cross.m, duplication_matrix.m, expm.m, orth.m, rank.m, rref.m, trace.m, vech.m, cast.m, compare_versions.m, delete.m, dir.m, fileattrib.m, grabcode.m, gunzip.m, inputname.m, license.m, list_primes.m, ls.m, mexext.m, movefile.m, namelengthmax.m, nargoutchk.m, nthargout.m, substruct.m, swapbytes.m, ver.m, verLessThan.m, what.m, fminunc.m, fsolve.m, fzero.m, optimget.m, __fdjac__.m, matlabroot.m, savepath.m, campos.m, camroll.m, camtarget.m, camup.m, camva.m, camzoom.m, clabel.m, diffuse.m, legend.m, orient.m, rticks.m, specular.m, thetaticks.m, xlim.m, xtickangle.m, xticklabels.m, xticks.m, ylim.m, ytickangle.m, yticklabels.m, yticks.m, zlim.m, ztickangle.m, zticklabels.m, zticks.m, ellipsoid.m, isocolors.m, isonormals.m, stairs.m, surfnorm.m, __actual_axis_position__.m, __pltopt__.m, close.m, graphics_toolkit.m, pan.m, print.m, printd.m, __ghostscript__.m, __gnuplot_print__.m, __opengl_print__.m, rotate3d.m, subplot.m, zoom.m, compan.m, conv.m, poly.m, polyaffine.m, polyder.m, polyint.m, polyout.m, polyreduce.m, polyvalm.m, roots.m, prefdir.m, prefsfile.m, profexplore.m, profexport.m, profshow.m, powerset.m, unique.m, 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, sinint.m, hadamard.m, hankel.m, hilb.m, invhilb.m, magic.m, pascal.m, rosser.m, toeplitz.m, vander.m, wilkinson.m, center.m, corr.m, cov.m, discrete_cdf.m, discrete_inv.m, discrete_pdf.m, discrete_rnd.m, empirical_cdf.m, empirical_inv.m, empirical_pdf.m, empirical_rnd.m, kendall.m, kurtosis.m, 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 597f3ee61a48
line wrap: on
line source

########################################################################
##
## 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 {} {@var{y} =} 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")