/usr/share/octave/packages/interval-1.4.1/@infsup/nthroot.m is in octave-interval 1.4.1-1.
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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} nthroot (@var{X}, @var{N})
##
## Compute the real n-th root of @var{X}.
##
## Accuracy: The result is a valid enclosure. The result is a tight
## enclosure for @var{n} ≥ -2. The result also is a tight enclosure if the
## reciprocal of @var{n} can be computed exactly in double-precision.
## @seealso{@@infsup/pown, @@infsup/pownrev, @@infsup/realsqrt, @@infsup/cbrt}
## @end defmethod
## Author: Oliver Heimlich
## Keywords: interval
## Created: 2015-02-20
function result = nthroot (x, n)
if (nargin ~= 2)
print_usage ();
return
endif
if (not (isa (x, "infsup")))
x = infsup (x);
endif
if (not (isnumeric (n)) || fix (n) ~= n)
error ("interval:InvalidOperand", "nthroot: degree is not an integer");
endif
even = mod (n, 2) == 0;
if (even)
x = intersect (x, infsup (0, inf));
endif
switch sign (n)
case +1
emptyresult = isempty (x);
l = mpfr_function_d ('nthroot', -inf, x.inf, n);
u = mpfr_function_d ('nthroot', +inf, x.sup, n);
l (emptyresult) = inf;
u (emptyresult) = -inf;
result = infsup (l, u);
case -1
emptyresult = isempty (x) ...
| (x.sup <= 0 & even) | (x.inf == 0 & x.sup == 0);
if (even)
l = zeros (size (x.inf));
u = inf (size (x.inf));
select = x.inf > 0 & isfinite (x.inf);
if (any (any (select)))
u (select) = invrootrounded (x.inf (select), -n, +inf);
endif
select = x.sup > 0 & isfinite (x.sup);
if (any (any (select)))
l (select) = invrootrounded (x.sup (select), -n, -inf);
endif
l (emptyresult) = inf;
u (emptyresult) = -inf;
result = infsup (l, u);
else # uneven
l = zeros (size (x.inf));
u = inf (size (x.inf));
select = x.inf > 0 & isfinite (x.inf);
if (any (any (select)))
u (select) = invrootrounded (x.inf (select), -n, +inf);
endif
select = x.sup > 0 & isfinite (x.sup);
if (any (any (select)))
l (select) = invrootrounded (x.sup (select), -n, -inf);
endif
notpositive = x.sup <= 0;
l (emptyresult | notpositive) = inf;
u (emptyresult | notpositive) = -inf;
result = infsup (l, u); # this is only the positive part
l = zeros (size (x.inf));
u = inf (size (x.inf));
select = x.sup < 0 & isfinite (x.sup);
if (any (any (select)))
u (select) = invrootrounded (-x.sup (select), -n, +inf);
endif
select = x.inf < 0 & isfinite (x.inf);
if (any (any (select)))
l (select) = invrootrounded (-x.inf (select), -n, -inf);
endif
notnegative = x.inf >= 0;
l (emptyresult | notnegative) = inf;
u (emptyresult | notnegative) = -inf;
result = union (result, infsup (-u, -l));
endif
otherwise
error ("interval:InvalidOperand", "nthroot: degree must not be zero");
endswitch
endfunction
function x = invrootrounded (z, n, direction)
## We cannot compute the inverse of the n-th root of z in a single step.
## Thus, we use three different ways for computation, each of which has an
## intermediate result with possible rounding errors and can't guarantee to
## produce a correctly rounded result.
## When we finally merge the 3 results, it is still not guaranteed to be
## correctly rounded. However, chances are good that one of the three ways
## produced a “relatively good” result.
##
## x1: z ^ (- 1 / n)
## x2: 1 / root (z, n)
## x3: root (1 / z, n)
inv_n = 1 ./ infsup (n);
if (direction > 0)
x1 = z;
select = z > 1;
x1 (select) = mpfr_function_d ('pow', direction, z (select), -inv_n.inf);
select = z < 1;
x1 (select) = mpfr_function_d ('pow', direction, z (select), -inv_n.sup);
else
x1 = z;
select = z > 1;
x1 (select) = mpfr_function_d ('pow', direction, z (select), -inv_n.sup);
select = z < 1;
x1 (select) = mpfr_function_d ('pow', direction, z (select), -inv_n.inf);
endif
if (issingleton (inv_n))
## We are lucky: The result is correctly rounded
x = x1;
return
endif
x2 = mpfr_function_d ('rdivide', direction, 1, ...
mpfr_function_d ('nthroot', -direction, z, n));
x3 = mpfr_function_d ('nthroot', direction, ...
mpfr_function_d ('rdivide', direction, 1, z), n);
## Choose the most accurate result
if (direction > 0)
x = min (min (x1, x2), x3);
else
x = max (max (x1, x2), x3);
endif
endfunction
%!assert (nthroot (infsup (25, 36), 2) == infsup (5, 6));
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