/usr/share/octave/packages/interval-2.1.0/@infsup/atan2.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
## @defmethod {@@infsup} atan2 (@var{Y}, @var{X})
##
## Compute the inverse tangent with two arguments.
##
## Accuracy: The result is a tight enclosure.
##
## @example
## @group
## atan2 (infsup (1), infsup (-1))
## @result{} ans ⊂ [2.3561, 2.3562]
## @end group
## @end example
## @seealso{@@infsup/tan}
## @end defmethod
## Author: Oliver Heimlich
## Keywords: interval
## Created: 2014-10-06
function y = atan2 (y, x)
if (nargin ~= 2)
print_usage ();
return
endif
if (not (isa (y, "infsup")))
y = infsup (y);
endif
if (not (isa (x, "infsup")))
x = infsup (x);
endif
## Resize, if scalar × matrix
if (not (size_equal (x.inf, y.inf)))
x.inf = ones (size (y.inf)) .* x.inf;
x.sup = ones (size (y.inf)) .* x.sup;
y.inf = ones (size (x.inf)) .* y.inf;
y.sup = ones (size (x.inf)) .* y.sup;
endif
## Partitionize the function's domain
## y
## ^
## p1 | p2
## ------0------>
## p4 | p3 x
##
persistent pos = infsup (0, inf);
persistent neg = infsup (-inf, 0);
x1 = x4 = intersect (x, neg);
y1 = y2 = intersect (y, pos);
x2 = x3 = intersect (x, pos);
y3 = y4 = intersect (y, neg);
## Intersect each partition with atan2's domain
p1 = not (isempty (x1) | isempty (y1)) & (x1.inf < 0 | y1.sup > 0);
p2 = not (isempty (x2) | isempty (y2)) & (x2.sup > 0 | y2.sup > 0);
p3 = not (isempty (x3) | isempty (y3)) & (x3.sup > 0 | y3.inf < 0);
p4 = not (isempty (x4) | isempty (y4)) & (x4.inf < 0 | y4.inf < 0);
## Prevent wrong limit values of atan2 (0, 0) in cases with y = 0.
select = (p1 & y1.sup == 0);
x1.inf(select) = x1.sup(select) = -1;
select = (p2 & y2.sup == 0);
x2.inf(select) = x2.sup(select) = 1;
select = (p3 & y3.inf == 0);
x3.inf(select) = x3.sup(select) = 1;
p4(y4.inf == 0) = false (); # don't consider y >= 0 in partition 4
## Prevent wrong limit values of atan2 (0, 0) in cases with x = 0.
select = (p1 & x1.inf == 0);
y1.inf(select) = y1.sup(select) = 1;
select = (p2 & x2.sup == 0);
y2.inf(select) = y2.sup(select) = 1;
y3.inf(p3 & x3.sup == 0) = y3.sup(p3 & x3.sup == 0) = -1;
select = (p4 & x4.inf == 0);
y4.inf(select) = y4.sup(select) = -1;
## Fix interval boundaries for y = 0 and x < 0, because atan2 (±0, -eps) = ±pi
y1.inf(p1 & y1.inf == 0) = +0;
y4.sup(p4 & y4.sup == 0) = -0;
## Compute lower boundary (atan2 is increasing from p4 to p1)
l = inf (size (p1));
select = p4;
l(select) = mpfr_function_d ('atan2', -inf, y4.sup(select), x4.inf(select));
select = p3 & not (p4);
l(select) = mpfr_function_d ('atan2', -inf, y3.inf(select), x3.inf(select));
select = p2 & not (p3 | p4);
l(select) = mpfr_function_d ('atan2', -inf, y2.inf(select), x2.sup(select));
select = p1 & not (p2 | p3 | p4);
l(select) = mpfr_function_d ('atan2', -inf, y1.sup(select), x1.sup(select));
## Compute upper boundary (atan2 is decreasing from p1 to p4)
u = -inf (size (p1));
select = p1;
u(select) = mpfr_function_d ('atan2', +inf, y1.inf(select), x1.inf(select));
select = p2 & not (p1);
u(select) = mpfr_function_d ('atan2', +inf, y2.sup(select), x2.inf(select));
select = p3 & not (p1 | p2);
u(select) = mpfr_function_d ('atan2', +inf, y3.sup(select), x3.sup(select));
select = p4 & not (p1 | p2 | p3);
u(select) = mpfr_function_d ('atan2', +inf, y4.inf(select), x4.sup(select));
## Now, we have computed l and u for all cases where p1 | p2 | p3 | p4. In all
## other cases, l and u will produce an empty interval.
l(l == 0) = -0;
y.inf = l;
y.sup = u;
endfunction
%!test "from the documentation string";
%!assert (atan2 (infsup (1), infsup (-1)) == "[0x1.2D97C7F3321D2p1, 0x1.2D97C7F3321D3p1]");
%!# correct use of signed zeros
%!test
%! x = atan2 (0, infsup (1));
%! assert (signbit (inf (x)));
%! assert (not (signbit (sup (x))));
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