/usr/share/octave/packages/interval-2.1.0/@infsup/tanrev.m is in octave-interval 2.1.0-2.
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##
## This program 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.
##
## This program 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 this program; if not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @documentencoding UTF-8
## @deftypemethod {@@infsup} {@var{X} =} tanrev (@var{C}, @var{X})
## @deftypemethodx {@@infsup} {@var{X} =} tanrev (@var{C})
##
## Compute the reverse tangent function.
##
## That is, an enclosure of all @code{x ∈ @var{X}} where
## @code{tan (x) ∈ @var{C}}.
##
## Accuracy: The result is a valid enclosure.
##
## @example
## @group
## tanrev (infsup (0), infsup (2, 4))
## @result{} ans ⊂ [3.1415, 3.1416]
## @end group
## @end example
## @seealso{@@infsup/tan}
## @end deftypemethod
## Author: Oliver Heimlich
## Keywords: interval
## Created: 2014-10-19
function result = tanrev (c, x)
if (nargin > 2)
print_usage ();
return
endif
if (nargin < 2)
x = infsup (-inf, inf);
endif
if (not (isa (c, "infsup")))
c = infsup (c);
endif
if (not (isa (x, "infsup")))
x = infsup (x);
endif
arctangent = atan (c);
result = x;
## Resize, if scalar × matrix
if (not (size_equal (arctangent.inf, result.inf)))
arctangent.inf = ones (size (result.inf)) .* arctangent.inf;
arctangent.sup = ones (size (result.inf)) .* arctangent.sup;
result.inf = ones (size (arctangent.inf)) .* result.inf;
result.sup = ones (size (arctangent.inf)) .* result.sup;
endif
result.inf(isempty (arctangent)) = inf;
result.sup(isempty (arctangent)) = -inf;
idx.type = '()';
persistent pi = infsup ("pi");
select = not (isempty (result)) ...
& not (subset (infsup (-pi.sup / 2, pi.sup / 2), arctangent));
if (any (select(:)))
## Find a smaller upper bound for x, if the restriction from c allows it
u = inf (size (result.inf));
select_u = select & result.sup < inf;
## Find n, such that result.sup is within a distance of pi/2 around n * pi.
n = result.sup;
n(select_u) = ceil (floor (sup (n(select_u) ./ (pi ./ 2))) ./ 2);
arctangentshifted = arctangent;
idx.subs = {select_u};
arctangentshifted = subsasgn (arctangentshifted, idx, ...
subsref (arctangentshifted, idx) + n(select_u) .* pi);
overlapping = not (isempty (intersect (result, arctangentshifted)));
u(select_u & overlapping) = ...
min (result.sup(select_u & overlapping), ...
arctangentshifted.sup(select_u & overlapping));
n(select_u & ~overlapping) = ...
mpfr_function_d ('minus', +inf, n(select_u & ~overlapping), 1);
idx.subs = {(select_u & ~overlapping)};
u(idx.subs{1}) = ...
sup (subsref (arctangent, idx) + subsref (n, idx) .* pi);
## Find a larger lower bound for x, if the restriction from c allows it
l = -inf (size (result.inf));
select_l = select & result.inf > -inf;
## Find n, such that result.inf is within a distance of pi/2 around n * pi.
n = result.inf;
n(select_l) = floor (ceil (inf (n(select_l) ./ (pi ./ 2))) ./ 2);
arctangentshifted = arctangent;
idx.subs = {select_l};
arctangentshifted = subsasgn (arctangentshifted, idx, ...
subsref (arctangentshifted, idx) + n(select_l) .* pi);
overlapping = not (isempty (intersect (result, arctangentshifted)));
l(select_l & overlapping) = ...
max (result.inf (select_l & overlapping), ...
arctangentshifted.inf(select_l & overlapping));
n(select_l & ~overlapping) = ...
mpfr_function_d ('plus', -inf, n(select_l & ~overlapping), 1);
idx.subs = {(select_l & ~overlapping)};
l(idx.subs {1}) = ...
inf (subsref (arctangent, idx) + subsref (n, idx) .* pi);
result.inf(select) = max (l(select), result.inf(select));
result.sup(select) = min (u(select), result.sup(select));
result.inf(result.inf > result.sup) = inf;
result.sup(result.inf > result.sup) = -inf;
endif
endfunction
%!# from the documentation string
%!assert (tanrev (infsup (0), infsup (2, 4)) == infsup ("pi"));
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