/usr/share/octave/packages/interval-2.1.0/@infsup/infsup.m is in octave-interval 2.1.0-2.
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1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 | ## Copyright 2014-2016 Oliver Heimlich
##
## 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
## @deftypeop Constructor {@@infsup} {[@var{X}, @var{ISEXACT}] =} infsup ()
## @deftypeopx Constructor {@@infsup} {[@var{X}, @var{ISEXACT}] =} infsup (@var{M})
## @deftypeopx Constructor {@@infsup} {[@var{X}, @var{ISEXACT}] =} infsup (@var{S})
## @deftypeopx Constructor {@@infsup} {[@var{X}, @var{ISEXACT}] =} infsup (@var{L}, @var{U})
##
## Create an interval (from boundaries). Convert boundaries to double
## precision.
##
## The syntax without parameters creates an (exact) empty interval. The syntax
## with a single parameter @code{infsup (@var{M})} equals
## @code{infsup (@var{M}, @var{M})}. The syntax @code{infsup (@var{S})} parses
## an interval literal in inf-sup form or as a special value, where
## @code{infsup ("[S1, S2]")} is equivalent to @code{infsup ("S1", "S2")}. A
## second, logical output @var{ISEXACT} indicates if @var{X}'s boundaries both
## have been converted without precision loss.
##
## Each boundary can be provided in the following formats: literal constants
## [+-]inf[inity], e, pi; scalar real numeric data types, i. e., double,
## single, [u]int[8,16,32,64]; or decimal numbers as strings of the form
## [+-]d[,.]d[[eE][+-]d]; or hexadecimal numbers as string of the form
## [+-]0xh[,.]h[[pP][+-]d]; or decimal numbers in rational form
## [+-]d/d.
##
## Also it is possible, to construct intervals from the uncertain form in the
## form @code{m?ruE}, where @code{m} is a decimal mantissa,
## @code{r} is empty (= half ULP) or a decimal integer ULP count or a
## second @code{?} character for unbounded intervals, @code{u} is
## empty or a direction character (u: up, d: down), and @code{E} is an
## exponential field.
##
## If decimal or hexadecimal numbers are no binary64 floating point numbers, a
## tight enclosure will be computed. int64 and uint64 numbers of high
## magnitude (> 2^53) can also be affected from precision loss.
##
## For the creation of interval matrices, arguments may be provided as (1) cell
## arrays with arbitrary/mixed types, (2) numeric matrices, or (3) strings.
## Scalar values do broadcast.
##
## Non-standard behavior: This class constructor is not described by IEEE Std
## 1788-2015, IEEE standard for interval arithmetic, however it implements both
## standard functions numsToInterval and textToInterval for bare intervals.
##
## @example
## @group
## infsup ()
## @result{} ans = [Empty]
## infsup ("[1]")
## @result{} ans = [1]
## infsup (2, 3)
## @result{} ans = [2, 3]
## infsup ("0.1")
## @result{} ans ⊂ [0.099999, 0.10001]
## infsup ("0.1", "0.2")
## @result{} ans ⊂ [0.099999, 0.20001]
## infsup ("0xff", "0x1.ffp14")
## @result{} ans = [255, 32704]
## infsup ("1/3")
## @result{} ans ⊂ [0.33333, 0.33334]
## infsup ("[1/9, 47/11]")
## @result{} ans ⊂ [0.11111, 4.2728]
## infsup ("7.3?9u")
## @result{} ans ⊂ [7.2999, 8.2001]
## infsup ("0??")
## @result{} ans = [Entire]
## infsup ("911??de-2")
## @result{} ans ⊂ [-Inf, +9.1101]
## infsup ("10?")
## @result{} ans = [9.5, 10.5]
## @end group
## @end example
## @seealso{exacttointerval}
## @end deftypeop
## Author: Oliver Heimlich
## Keywords: interval
## Created: 2014-09-27
function [x, isexact, overflow, isnai] = infsup (l, u)
persistent scalar_empty_interval = class (struct ("inf", inf, ...
"sup", -inf), ...
"infsup");
## Part 1
##
## Split any arguments into l and u, where l and u denote lower and upper
## boundaries. l and u shall be either numeric arrays or cell arrays with
## strings or numeric entries. The size of l and u will equal the interval
## matrix size of the final result (unless broadcasting is going to be applied
## as a very last step).
##
## Strings in l and u will be normalized, that is, trimmed and converted to
## lower case.
switch nargin
case 0
## empty interval
x = scalar_empty_interval;
isexact = true;
isnai = overflow = false;
return
case 1
if (isa (l, "infsup"))
## already an interval—nothing to be done
x = l;
isexact = true ();
isnai = overflow = false (size (l.inf));
return
endif
if (ischar (l))
## Character string may contain a vector or a matrix of intervals,
## split the interval literals into a cell array of strings.
## Interval literals will be trimmed by the split function.
l = __split_interval_literals__ (lower (l));
char_idx = true (size (l));
elseif (iscell (l))
## Make sure that in a cell array (with possibly mixed types) all
## strings are trimmed. This is required during splitting of
## interval literals into lower and upper bounds.
char_idx = cellfun ("ischar", l);
l(char_idx) = lower (strtrim (l(char_idx)));
else
## Not cell or char, e. g. numeric matrix.
## Syntax infsup (x) has to be equivalent with infsup (x, x).
## No need to trim or normalize character case.
u = l;
endif
## Correct construction of empty intervals is only possible by calling
## this constructor without arguments (see above), or by using one of
## the two interval literals [] or [empty].
## Otherwise, construction of empty intervals must signal
## “UndefinedOperation”.
## In particular, the interval literal [+inf, -inf] is illegal and
## “numsToInterval (+inf, -inf)” would be an illegal function call,
## according to IEEE Std 1788-2015.
##
## 1. isnai defaults to -1 (unknown).
## 2. any non-empty interval (l <= u) is legal and we set isnai to
## false for those in the end.
## 3. for legal empty interval literals we set isnai = false
## explicitly.
## 4. for certain illegal interval literals we give a special warning
## and set isnai = true explicitly.
## 5. it remains isnai = -1 for any illegal intervals which have not
## been handled with, we give a warning and we return an empty
## interval in these cases.
isnai = -ones (size (l), "int8");
illegal_boundary = struct ("inf", false (size (l)), ...
"sup", false (size (l)));
if (iscell (l))
## At this point, only a cell array can contain interval literals
## as string. Split intervals and interval literals into lower and
## upper bounds. Except for empty intervals, strings are not
## converted into numeric values yet.
##
## [] -> (+inf, -inf)
## [empty] -> (+inf, -inf)
## [entire] -> (, )
## [,] -> (, )
## [l, u] -> (l, u)
## [m] -> (m, m)
## m -> (m, m)
## m?r -> (m?r, m?r)
## m?ru -> (m, m?r)
## m?rd -> (m?r, m)
## m?? -> (, )
## m??u -> (m, )
## m??d -> (, m)
## Initialize u such that the syntax infsup (x) is equivalent with
## infsup (x, x) unless interval literals are used.
u = l;
## Find interval literals with square brackets. The strings are
## trimmed already, so they will start at the first character.
square_bracket_idx = strncmp (l, "[", 1);
%!# Verify correct behaviour of strncmp for empty strings and non-string
%!# values within the cell array.
%!assert (strncmp ({"[", double("["), ""}, "[", 1), logical ([1 0 0]));
## A bare interval literal which starts with a square bracket
## must end with a square bracket as well.
square_bracket_idx = ...
find (square_bracket_idx)(...
cellfun (@(s) s(end) == "]", ...
l(square_bracket_idx)));
## Strip square brackets and white space within square brackets.
nobrackets = strtrim (cellfun (@(s) s([2 : (end - 1)]), ...
l(square_bracket_idx), ...
"UniformOutput", false));
## Construction of empty intervals with the correct literal either
## [empty] or [] is legit.
empty_interval_local_idx = strcmp (nobrackets, "empty") ...
| cellfun ("isempty", nobrackets);
empty_interval_idx = square_bracket_idx(empty_interval_local_idx);
isnai(empty_interval_idx) = false;
## We remove the empty interval cases from the current work basket
## and don't need special handling of comma-less strings below
## ([] represents the empty interval, whereas [,] represents the
## set of all real numbers).
square_bracket_idx = ...
square_bracket_idx(not (empty_interval_local_idx));
nobrackets = nobrackets(not (empty_interval_local_idx));
char_idx(empty_interval_idx) = false;
l(empty_interval_idx) = +inf;
u(empty_interval_idx) = -inf;
## [entire] is equivalent to [,]
nobrackets(strcmp (nobrackets, "entire")) = {""};
## Split [l, u] literals into l and u strings at the comma.
nobrackets = cellfun ("strsplit", nobrackets, {","}, ...
"UniformOutput", false);
nobrackets_parts = cellfun ("numel", nobrackets);
## For point intervals [m] we have removed the square brackets,
## trimmed any white space inside the square brackets and must
## store m into both l and u for further parsing below.
## Each boundary will be parsed individually with opposite rounding
## direction.
point_interval_local_idx = (nobrackets_parts == 1);
point_interval_idx = square_bracket_idx(point_interval_local_idx);
l(point_interval_idx) = u(point_interval_idx) = ...
vertcat ({}, nobrackets(point_interval_local_idx){:});
## For infsup intervals [l, u] we can store the trimmed l and u
## strings for further parsing below.
infsup_interval_local_idx = (nobrackets_parts == 2);
infsup_interval_idx = square_bracket_idx(infsup_interval_local_idx);
l(infsup_interval_idx) = strtrim (vertcat ({}, ...
cellindexmat (nobrackets(infsup_interval_local_idx), 1){:}));
u(infsup_interval_idx) = strtrim (vertcat ({}, ...
cellindexmat (nobrackets(infsup_interval_local_idx), 2){:}));
## Find interval literals in uncertain form.
uncertain_idx = char_idx;
uncertain_idx(square_bracket_idx) = false; # already processed
## Find uncertain form with directed uncertainty (down or up)
[~, ~, ~, ~, groups] = regexp (l(uncertain_idx), ...
["([^?]+)", ... # 1: mantissa
"([?])", ... # 2: ?
"(.*)", ... # 3: uncertainty
"([du])", ... # 4: direction
"(.*)"]); # 5: exponent
directed_local_idx = not (cellfun ("isempty", groups));
directed_uncertain_idx = uncertain_idx;
directed_uncertain_idx(uncertain_idx) = directed_local_idx;
groups = vertcat ({}, groups(directed_local_idx){:});
## Remove directed down/up uncertainty.
for direction = ["d", "u"]
direction_local_idx = strcmp (...
vertcat ({}, cellindexmat (groups, 4){:}), ...
direction);
direction_idx = directed_uncertain_idx;
direction_idx(directed_uncertain_idx) = direction_local_idx;
## Remove direction character
persistent join_groups = @(parts) strcat (parts{:});
undirected_uncertain_form = ...
cellfun (...
join_groups, ...
cellindexmat (groups(direction_local_idx), ...
[1 2 3 5]), ...
"UniformOutput", false);
## Also remove uncertainty
undirected_certain_form = ...
cellfun (...
join_groups, ...
cellindexmat (groups(direction_local_idx), ...
[1 5]), ...
"UniformOutput", false);
## Store uncertain boundaries without directed uncertainty
switch direction
case "d"
l(direction_idx) = undirected_uncertain_form;
u(direction_idx) = undirected_certain_form;
case "u"
l(direction_idx) = undirected_certain_form;
u(direction_idx) = undirected_uncertain_form;
endswitch
## Remove parsed uncertain form from work basket to prevent
## double parsing (strings with both d and u uncertainty).
directed_uncertain_idx(direction_idx) = false;
groups = groups(not (direction_local_idx));
endfor
## Remove unbound uncertainty ??
## FIXME We should verify correctness of the interval literal more
## thoroughly. Otherwise we would ignore illegal literals which
## contain the double question mark.
l_unbound_idx = uncertain_idx;
l_unbound_idx(uncertain_idx) = ...
not (cellfun ("isempty", strfind (l(uncertain_idx), "??")));
l(l_unbound_idx) = {""};
u_unbound_idx = uncertain_idx;
u_unbound_idx(uncertain_idx) = ...
not (cellfun ("isempty", strfind (u(uncertain_idx), "??")));
u(u_unbound_idx) = {""};
endif
case 2
## Split interval vectors if supplied as strings.
if (ischar (l))
l = __split_interval_literals__ (l);
endif
if (ischar (u))
u = __split_interval_literals__ (u);
endif
## Check dimensions and whether broadcasting is possible
for dim = 1 : max (ndims (l), ndims (u))
if (size (l, dim) != 1 && size (u, dim) != 1 && ...
size (l, dim) != size (u, dim))
warning ("interval:InvalidOperand", ...
["infsup: Dimensions of lower and upper ", ...
"boundaries are not compatible"]);
## Unable to recover from this kind of error
x = scalar_empty_interval;
isexact = false;
overflow = false;
isnai = true;
return
endif
endfor
## Compute result size after broadcasting and mark any empty intervals
## as illegal (will trigger “UndefinedOperation” signal later on).
## Construction of silent empty intervals is impossible with two args.
isnai = zeros (size (l), "int8") - ones (size (u), "int8"); #-1=unknown
illegal_boundary = struct ("inf", false (size (l)), ...
"sup", false (size (u)));
for argument = ["l", "u"]
switch argument
case "l"
current_arg = l;
case "u"
current_arg = u;
endswitch
if (iscell (current_arg))
## Normalize strings: trim and convert to lower case
## (as is done in the nargin == 1 case).
char_idx = cellfun ("ischar", current_arg);
current_arg(char_idx) = ...
lower (strtrim (current_arg(char_idx)));
## In contrast to the nargin == 1 case we cannot allow interval
## literals here. Only simple boundaries are allowed if two
## arguments are given.
square_bracket_idx = strncmp (current_arg, "[", 1);
uncertain_idx = char_idx;
uncertain_idx(square_bracket_idx) = false;
uncertain_idx(uncertain_idx) = ...
not (cellfun ("isempty", ...
strfind (current_arg(uncertain_idx), "?")));
illegal_literal_idx = (square_bracket_idx | uncertain_idx);
if (any (illegal_literal_idx(:)))
switch argument
case "l"
warning ("interval:UndefinedOperation", ...
"Lower boundary contains an interval literal");
current_arg(illegal_literal_idx) = +inf;
illegal_boundary.inf(illegal_literal_idx) = true;
case "u"
warning ("interval:UndefinedOperation", ...
"Upper boundary contains an interval literal");
current_arg(illegal_literal_idx) = -inf;
illegal_boundary.sup(illegal_literal_idx) = true;
endswitch
endif
switch argument
case "l"
l = current_arg;
case "u"
u = current_arg;
endswitch
endif
endfor
otherwise # nargin >= 3
print_usage ();
return
endswitch
## Part 2
##
## Boundaries have been split into lower and upper boundaries and shall be
## converted to binary64 matrices with string parsing and outward rounding.
##
## l contains a cell array or a matrix of lower boundaries.
## u contains a cell array or a matrix of upper boundaries.
##
## Each of l and u will be converted into binary64 individually and will be
## stored in x.inf and x.sup respectively.
isexact = true ();
x = struct ("inf", inf (size (l)), ...
"sup", -inf (size (u)));
for [boundaries, key] = struct ("inf", {l}, "sup", {u})
if (isfloat (boundaries) && isreal (boundaries))
## Simple case: the boundaries already are binary floating point
## numbers in single or double precision.
## This kind of operation is often used in internal functions and shall
## be fast. We check for NaNs later.
x.(key) = double (boundaries);
possiblyundefined.(key) = overflow.(key) = false (size (boundaries));
continue
endif
if (not (iscell (boundaries)))
if (not (isnumeric (boundaries)))
warning ("interval:InvalidOperand", ...
["infsup: Invalid argument type, only strings, ", ...
"numerics, and cell arrays thereof are allowed"]);
## Unable to recover from this kind of error
x = scalar_empty_interval;
isexact = false;
overflow = false;
isnai = true;
return
endif
boundaries = num2cell (boundaries);
endif
overflow.(key) = true (size (boundaries));
possiblyundefined.(key) = false (size (boundaries));
## Track the entries in cell array boundaries, which haven't been
## converted yet.
todo = true (size (boundaries));
## [,] = [-inf, +inf]
unbound_idx = cellfun ("isempty", boundaries);
switch key
case "inf"
x.inf(unbound_idx) = -inf;
case "sup"
x.sup(unbound_idx) = +inf;
endswitch
overflow.(key)(unbound_idx) = false;
todo(unbound_idx) = false;
## In the cell array each entry must represent a scalar value.
non_scalar_entry_idx = todo & not (...
cellfun ("isscalar", boundaries) | cellfun ("ischar", boundaries));
if (any (non_scalar_entry_idx(:)))
warning ("interval:UndefinedOperation", ...
"Cell arrays of matrix entries do not contain scalar values");
# Use default value of [empty] for these entries and continue
todo(non_scalar_entry_idx) = false;
illegal_boundary.(key)(non_scalar_entry_idx) = true;
endif
## 64 bit integers: approximate in double precision.
integer_idx = find (todo & ( ...
cellfun ("isa", boundaries, {"uint64"}) ...
| cellfun ("isa", boundaries, {"int64"})));
integers = vertcat (boundaries{integer_idx});
converted_integers = double (integers);
exact_integer_local_idx = (converted_integers == integers);
if (any (not (exact_integer_local_idx(:))))
isexact = false;
possiblyundefined.(key)(integer_idx(not (exact_integer_local_idx))) ...
= true;
endif
## Fix conversion in cases where rounding to nearest has resulted in the
## wrong value, i. e., in a value that has not been rounded outward,
## but inward.
switch key
case "inf"
wrong_round_local_idx = (converted_integers > integers);
converted_integers(wrong_round_local_idx) = ...
mpfr_function_d ('minus', -inf, ...
converted_integers(wrong_round_local_idx), pow2 (-1074));
case "sup"
wrong_round_local_idx = (converted_integers < integers);
converted_integers(wrong_round_local_idx) = ...
mpfr_function_d ('plus', +inf, ...
converted_integers(wrong_round_local_idx), pow2 (-1074));
endswitch
x.(key)(integer_idx) = converted_integers;
overflow.(key)(integer_idx) = false;
todo(integer_idx) = false;
## Lossless conversion from binary32, binary64, (u)int8, (u)int16, (u)int32
## and logicals.
real_idx = todo & ...
( ...
( ...
cellfun ("isreal", boundaries) ...
& cellfun ("isfloat", boundaries) ...
) ...
| cellfun ("isinteger", boundaries) ...
| cellfun ("islogical", boundaries)
);
x.(key)(real_idx) = double (vertcat (boundaries{real_idx}));
overflow.(key)(real_idx) = false;
todo(real_idx) = false;
## Complex numbers: not allowed, will be mapped to [empty].
complex_idx = todo & cellfun ("iscomplex", boundaries);
if (any (complex_idx(:)))
warning ("interval:InvalidOperand", ...
"infsup: Complex arguments are not permitted");
todo(complex_idx) = false;
illegal_boundary.(key)(complex_idx) = true;
endif
## Other kinds of parameters that are not strings
char_idx = todo & cellfun ("ischar", boundaries);
if (any ((todo & not (char_idx))(:)))
warning ("interval:InvalidOperand", ...
"infsup: Illegal boundary: must be numeric or string");
illegal_boundary.(key)(todo & not (char_idx)) = true;
endif
todo = char_idx;
## Hex strings
hex_idx = find (todo & (strncmp (boundaries, "0x", 2) ...
| strncmp (boundaries, "+0x", 3) ...
| strncmp (boundaries, "-0x", 3)));
switch key
case "inf"
direction = -inf;
case "sup"
direction = inf;
endswitch
for i = vec (hex_idx, 2)
try
[x.(key)(i), exact_conversion] = ...
hex2double (boundaries{i}, direction);
possiblyundefined.(key)(i) = not (exact_conversion);
isexact = isexact && exact_conversion;
catch
warning ("interval:UndefinedOperation", lasterr ());
illegal_boundary.(key)(i) = true;
end_try_catch
endfor
overflow.(key)(hex_idx) = false;
todo(hex_idx) = false;
## Selected boundary literals
persistent boundary_const = struct (...
"inf", struct (...
"-inf", -inf, ...
"-infinity", -inf, ...
"inf", inf, ...
"+inf", inf, ...
"infinity", inf, ...
"+infinity", inf, ...
"e", 0x56FC2A2 * pow2 (-25) ...
+ 0x628AED2 * pow2 (-52), ...
"pi", 0x6487ED5 * pow2 (-25) ...
+ 0x442D180 * pow2 (-55)), ...
"sup", struct (...
"-inf", -inf, ...
"-infinity", -inf, ...
"inf", inf, ...
"+inf", inf, ...
"infinity", inf, ...
"+infinity", inf, ...
"e", 0x56FC2A2 * pow2 (-25) ...
+ 0x628AED4 * pow2 (-52), ...
"pi", 0x6487ED5 * pow2 (-25) ...
+ 0x442D190 * pow2 (-55)));
for [val, lit] = boundary_const.(key)
const_idx = todo & strcmp (boundaries, lit);
x.(key)(const_idx) = val;
overflow.(key)(const_idx) = false;
possiblyundefined.(key)(const_idx) = isfinite (val);
todo(const_idx) = false;
endfor
## It remains the decimal boundary strings
for i = vec (find (todo), 2)
try
boundary = boundaries{i};
## We have to parse a decimal string boundary and round the
## result up or down depending on the boundary
## (inf = down, sup = up).
## str2double will produce the correct answer in 50 % of
## all cases, because it uses rounding mode “to nearest”.
## The input and a double format approximation can be
## compared in a decimal floating point format without
## precision loss.
if (strfind (boundary, "?"))
## Special case: uncertain-form
[boundary, uncertain] = uncertainsplit (boundary);
else
uncertain = [];
endif
if (strfind (boundary, "/"))
## Special case: rational form
boundary = strsplit (boundary, "/");
if (length (boundary) ~= 2)
warning ("interval:UndefinedOperation", ...
["illegal " key " boundary: ", ...
"rational form must have single slash"]);
illegal_boundary.(key)(i) = true;
continue;
endif
[decimal, remainder] = decimaldivide (...
str2decimal (boundary{1}), ...
str2decimal (boundary{2}), 18);
if (not (isempty (remainder.m)))
## This will guarantee the enclosure of the exact
## value
decimal.m(19, 1) = 1;
isexact = false ();
if (key == "inf")
possiblyundefined.(key)(i) = true;
endif
endif
## Write result back into boundary for conversion to
## double
boundary = ["0.", num2str(decimal.m)', ...
"e", num2str(decimal.e)];
if (decimal.s)
boundary = ["-", boundary];
endif
else
decimal = str2decimal (boundary);
endif
## Parse and add uncertainty
if (not (isempty (uncertain)))
uncertain = str2decimal (uncertain);
if ((key == "inf") == decimal.s)
uncertain.s = decimal.s;
else
uncertain.s = not (decimal.s);
endif
decimal = decimaladd (decimal, uncertain);
## Write result back into boundary for conversion to
## double
boundary = ["0.", num2str(decimal.m)', ...
"e", num2str(decimal.e)];
if (decimal.s)
boundary = ["-", boundary];
endif
endif
clear uncertain;
## Check if number is outside of range
## Realmax == 1.7...e308 == 0.17...e309
if (decimal.e > 309 || ...
(decimal.e == 309 && ...
decimal.s && ...
decimalcompare (double2decimal (-realmax ()), ...
decimal) > 0) || ...
(decimal.e == 309 && ...
not (decimal.s) && ...
decimalcompare (double2decimal (realmax ()), ...
decimal) < 0))
switch key
case "inf"
if (decimal.s) # -inf ... -realmax
x.inf(i) = -inf;
else # realmax ... inf
x.inf(i) = realmax ();
overflow.inf(i) = false;
endif
case "sup"
if (decimal.s) # -inf ... -realmax
x.sup(i) = -realmax ();
overflow.sup(i) = false;
else # realmax ... inf
x.sup(i) = inf;
endif
endswitch
possiblyundefined.(key)(i) = true;
isexact = false;
continue
endif
overflow.(key)(i) = false;
## Compute approximation, this only works between ± realmax
binary = str2double (strrep (boundary, ",", "."));
## Check approximation value
comparison = decimalcompare (double2decimal (binary), ...
decimal);
if (comparison ~= 0)
possiblyundefined.(key)(i) = true;
isexact = false;
endif
if (comparison == 0 || ... # approximation is exact
(comparison < 0 && key == "inf") || ... # lower bound
(comparison > 0 && key == "sup")) # upper bound
x.(key)(i) = binary;
else
## Approximation is not exact and not rounded as needed
## However, because of faithful rounding the
## approximation is right next to the desired number.
switch key
case "inf"
x.inf(i) = mpfr_function_d ('minus', -inf, ...
binary, pow2 (-1074));
case "sup"
x.sup(i) = mpfr_function_d ('plus', +inf, ...
binary, pow2 (-1074));
endswitch
endif
catch
warning ("interval:UndefinedOperation", lasterr ());
illegal_boundary.(key)(i) = true;
end_try_catch
endfor
endfor
## Part 3
##
## Check boundaries individually after conversion to double
## NaNs are illegal values
for [boundary, key] = struct ("inf", {x.inf}, "sup", {x.sup})
nanvalue = isnan (boundary);
if (any (nanvalue(:)))
warning ("interval:UndefinedOperation", ...
"input contains NaN values");
illegal_boundary.(key)(nanvalue) = true;
endif
endfor
## normalize boundaries:
## representation of the set containing only zero is always [-0,+0]
x.inf(x.inf == 0) = -0;
x.sup(x.sup == 0) = +0;
## Part 4
##
## Broadcast boundaries and final checks
if (nargout >= 3)
overflow = overflow.inf | overflow.sup;
overflow(x.inf > -inf & x.sup < inf) = false;
endif
x.inf = x.inf - zeros (size (x.sup));
x.sup = x.sup + zeros (size (x.inf));
possiblyundefined = possiblyundefined.inf & possiblyundefined.sup;
if (any (possiblyundefined(:)))
if (not (iscell (l)) && not (iscell (u)))
## infsup (x, x) or infsup (x) is not possibly undefined
possiblyundefined(l == u) = false;
else
## infsup ("x", "x") or infsup ("x") is not possibly undefined
if (not (iscell (l)))
l = num2cell (l);
endif
if (not (iscell (u)))
u = num2cell (u);
endif
possiblyundefined(cellfun ("isequal", l, u)) = false;
endif
endif
## Non-empty intervals are always legal.
isnai(x.inf <= x.sup) = false;
## Check for illegal boundaries [inf,inf] and [-inf,-inf].
illegal_inf_idx = not (isfinite (x.inf (x.inf == x.sup)));
if (any (illegal_inf_idx(:)))
warning ("interval:UndefinedOperation", ...
"illegal interval boundaries: infimum = supremum = +/- infinity");
isnai(find (x.inf == x.sup)(illegal_inf_idx)) = true;
endif
## Illegal boundaries make illegal intervals,
## we have triggered a warning already (see above).
isnai(illegal_boundary.inf | illegal_boundary.sup) = true;
## Check boundary order
## isnai has been initialized with -1. Any non-empty or legal empty intervals
## have been set to 0. Any illegal interval literals or illegal boundaries
## have been set to +1. Any intervals with inf > sup still have their initial
## value of -1.
wrong_boundary_order_idx = signbit (isnai);
if (any (wrong_boundary_order_idx(:)))
warning ("interval:UndefinedOperation", ...
"illegal interval boundaries: infimum greater than supremum");
isnai(wrong_boundary_order_idx) = true;
endif
isnai = logical(isnai);
## Return [empty] for any illegal intervals.
x.inf(isnai) = +inf;
x.sup(isnai) = -inf;
possiblyundefined(isnai) = false;
isexact = isexact && not (any (isnai(:)));
## Check for possibly wrong boundary order.
if (any (possiblyundefined(:)))
## Let a, b, and c be three consecutive floating point numbers.
##
## If a < u < b < l < c, then u and l will both be mapped to the same
## number b by outward rounding.
##
## If a < u < l < b, then u and l will both be mapped to consecutive
## floating point numbers a and b.
if (any ((x.inf(possiblyundefined) == x.sup(possiblyundefined))(:)) ...
|| ...
any ((max (-realmax, ...
mpfr_function_d ('plus',...
+inf, ...
x.inf(possiblyundefined), ...
pow2 (-1074))) == x.sup(possiblyundefined))(:)))
warning ("interval:PossiblyUndefinedOperation", ...
"infimum may be greater than supremum");
endif
endif
x = class (x, "infsup");
endfunction
%!# Empty intervals
%!test
%! x = infsup ();
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!test
%! x = infsup ("[]");
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!test
%! x = infsup ("[ ]");
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!test
%! x = infsup ("[\t]");
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!test
%! x = infsup ("[empty]");
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!test
%! x = infsup ("[EMPTY]");
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!test
%! x = infsup ("[ empty ]");
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!test
%! x = infsup ("\t[\t Empty\t]\t");
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!# Entire interval
%!test
%! x = infsup ("[,]");
%! assert (inf (x), -inf);
%! assert (sup (x), +inf);
%!test
%! x = infsup ("[entire]");
%! assert (inf (x), -inf);
%! assert (sup (x), +inf);
%!test
%! x = infsup ("[ENTIRE]");
%! assert (inf (x), -inf);
%! assert (sup (x), +inf);
%!test
%! x = infsup ("[ entire ]");
%! assert (inf (x), -inf);
%! assert (sup (x), +inf);
%!test
%! x = infsup (" [Entire \t] ");
%! assert (inf (x), -inf);
%! assert (sup (x), +inf);
%!test
%! x = infsup ("[-inf,+inf]");
%! assert (inf (x), -inf);
%! assert (sup (x), +inf);
%!test
%! x = infsup ("[-infinity, +infinity]");
%! assert (inf (x), -inf);
%! assert (sup (x), +inf);
%!test
%! x = infsup ("[-INF, +INFinitY]");
%! assert (inf (x), -inf);
%! assert (sup (x), +inf);
%!# double boundaries
%!test
%! x = infsup (0);
%! assert (inf (x), 0);
%! assert (sup (x), 0);
%! assert (signbit (inf (x)));
%! assert (not (signbit (sup (x))));
%!test
%! x = infsup (2, 3);
%! assert (inf (x), 2);
%! assert (sup (x), 3);
%!test
%! x = infsup (-inf, 0.1);
%! assert (inf (x), -inf);
%! assert (sup (x), 0.1);
%!test
%! x = infsup (-inf, +inf);
%! assert (inf (x), -inf);
%! assert (sup (x), +inf);
%!# NaN values
%!warning id=interval:UndefinedOperation
%! x = infsup (nan);
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!warning id=interval:UndefinedOperation
%! x = infsup (nan, 2);
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!warning id=interval:UndefinedOperation
%! x = infsup (3, nan);
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!# illegal numeric boundaries
%!warning id=interval:UndefinedOperation
%! x = infsup (+inf, -inf);
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!warning id=interval:UndefinedOperation
%! x = infsup (+inf, +inf);
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!warning id=interval:UndefinedOperation
%! x = infsup (-inf, -inf);
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!warning id=interval:UndefinedOperation
%! x = infsup (3, 2);
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!warning id=interval:UndefinedOperation
%! x = infsup (3, -inf);
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!# double matrix
%!test
%! x = infsup (magic (4));
%! assert (inf (x), magic (4));
%! assert (sup (x), magic (4));
%!test
%! x = infsup (magic (3), magic (3) + 1);
%! assert (inf (x), magic (3));
%! assert (sup (x), magic (3) + 1);
%!warning id=interval:UndefinedOperation
%! x = infsup (nan (3));
%! assert (inf (x), +inf (3));
%! assert (sup (x), -inf (3));
%!test
%! x = infsup (-inf (3), +inf (3));
%! assert (inf (x), -inf (3));
%! assert (sup (x), +inf (3));
%!# decimal boundaries
%!test
%! x = infsup ("0.1");
%! assert (inf (x), 0.1 - eps / 16);
%! assert (sup (x), 0.1);
%!test
%! x = infsup ("0.1e1");
%! assert (inf (x), 1);
%! assert (sup (x), 1);
%!# hexadecimal boundaries
%!test
%! x = infsup ("0xff");
%! assert (inf (x), 255);
%! assert (sup (x), 255);
%!test
%! x = infsup ("0xff.1");
%! assert (inf (x), 255.0625);
%! assert (sup (x), 255.0625);
%!test
%! x = infsup ("0xff.1p-1");
%! assert (inf (x), 127.53125);
%! assert (sup (x), 127.53125);
%!# named constants
%!test
%! x = infsup ("pi");
%! assert (inf (x), pi);
%! assert (sup (x), pi + 2 * eps);
%!test
%! x = infsup ("e");
%! assert (inf (x), e);
%! assert (sup (x), e + eps);
%!# uncertain form
%!test
%! x = infsup ("32?");
%! assert (inf (x), 31.5);
%! assert (sup (x), 32.5);
%!test
%! x = infsup ("32?8");
%! assert (inf (x), 24);
%! assert (sup (x), 40);
%!test
%! x = infsup ("32?u");
%! assert (inf (x), 32);
%! assert (sup (x), 32.5);
%!test
%! x = infsup ("32?d");
%! assert (inf (x), 31.5);
%! assert (sup (x), 32);
%!test
%! x = infsup ("32??");
%! assert (inf (x), -inf);
%! assert (sup (x), +inf);
%!test
%! x = infsup ("32??d");
%! assert (inf (x), -inf);
%! assert (sup (x), 32);
%!test
%! x = infsup ("32??u");
%! assert (inf (x), 32);
%! assert (sup (x), +inf);
%!test
%! x = infsup ("32?e5");
%! assert (inf (x), 3150000);
%! assert (sup (x), 3250000);
%!# rational form
%!test
%! x = infsup ("6/9");
%! assert (inf (x), 2 / 3);
%! assert (sup (x), 2 / 3 + eps / 2);
%!test
%! x = infsup ("6e1/9");
%! assert (inf (x), 20 / 3 - eps * 2);
%! assert (sup (x), 20 / 3);
%!test
%! x = infsup ("6/9e1");
%! assert (inf (x), 2 / 30);
%! assert (sup (x), 2 / 30 + eps / 16);
%!test
%! x = infsup ("-6/9");
%! assert (inf (x), -(2 / 3 + eps / 2));
%! assert (sup (x), -2 / 3);
%!test
%! x = infsup ("6/-9");
%! assert (inf (x), -(2 / 3 + eps / 2));
%! assert (sup (x), -2 / 3);
%!test
%! x = infsup ("-6/-9");
%! assert (inf (x), 2 / 3);
%! assert (sup (x), 2 / 3 + eps / 2);
%!test
%! x = infsup ("6.6/9.9");
%! assert (inf (x), 2 / 3);
%! assert (sup (x), 2 / 3 + eps / 2);
%!# inf-sup interval literal
%!test
%! x = infsup ("[2, 3]");
%! assert (inf (x), 2);
%! assert (sup (x), 3);
%!test
%! x = infsup ("[0.1]");
%! assert (inf (x), 0.1 - eps / 16);
%! assert (sup (x), 0.1);
%!test
%! x = infsup ("[0xff, 0xff.1]");
%! assert (inf (x), 255);
%! assert (sup (x), 255.0625);
%!test
%! x = infsup ("[e, pi]");
%! assert (inf (x), e);
%! assert (sup (x), pi + 2 * eps);
%!test
%! x = infsup ("[6/9, 6e1/9]");
%! assert (inf (x), 2 / 3);
%! assert (sup (x), 20 / 3);
%!# corner cases
%!test
%! x = infsup (",");
%! assert (inf (x), -inf);
%! assert (sup (x), +inf);
%!test
%! x = infsup ("[, 3]");
%! assert (inf (x), -inf);
%! assert (sup (x), 3);
%!test
%! x = infsup ("", "3");
%! assert (inf (x), -inf);
%! assert (sup (x), 3);
%!test
%! x = infsup ("[2, ]");
%! assert (inf (x), 2);
%! assert (sup (x), inf);
%!test
%! x = infsup ("2", "");
%! assert (inf (x), 2);
%! assert (sup (x), inf);
%!# decimal vector
%!test
%! x = infsup (["0.1"; "0.2"; "0.3"]);
%! assert (inf (x), [0.1 - eps / 16; 0.2 - eps / 8; 0.3]);
%! assert (sup (x), [0.1; 0.2; 0.3 + eps / 8]);
%!test
%! x = infsup ("0.1; 0.2; 0.3");
%! assert (inf (x), [0.1 - eps / 16; 0.2 - eps / 8; 0.3]);
%! assert (sup (x), [0.1; 0.2; 0.3 + eps / 8]);
%!test
%! x = infsup ("0.1\n0.2\n0.3");
%! assert (inf (x), [0.1 - eps / 16; 0.2 - eps / 8; 0.3]);
%! assert (sup (x), [0.1; 0.2; 0.3 + eps / 8]);
%!# cell array with mixed boundaries
%!test
%! x = infsup ({"0.1", 42; "e", "3.2/8"}, {"0xffp2", "42e1"; "pi", 2});
%! assert (inf (x), [0.1 - eps / 16, 42; e, 0.4 - eps / 4]);
%! assert (sup (x), [1020, 420; pi + 2 * eps, 2]);
%!test
%! x = infsup ({"[2, 3]", "3/4", "[Entire]", "42?3", 1, "0xf"});
%! assert (inf (x), [2, 0.75, -inf, 39, 1, 15]);
%! assert (sup (x), [3, 0.75, +inf, 45, 1, 15]);
%!# broadcasting
%!test
%! x = infsup (magic (3), 10);
%! assert (inf (x), magic (3));
%! assert (sup (x), 10 .* ones (3));
%!test
%! x = infsup (zeros (1, 20), ones (20, 1));
%! assert (inf (x), zeros (20, 20));
%! assert (sup (x), ones (20, 20));
%!# nai
%!warning id=interval:UndefinedOperation
%! x = infsup ("[nai]");
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!warning id=interval:UndefinedOperation
%! x = infsup ("Ausgeschnitzel");
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!# interval literals vs. two arguments
%!warning id=interval:UndefinedOperation
%! x = infsup ("[empty]", 42);
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!warning id=interval:UndefinedOperation
%! x = infsup ("0?", 42);
%! assert (inf (x), +inf);
%! assert (sup (x), -inf);
%!# extraction of single errors
%!warning id=interval:UndefinedOperation
%! x = infsup ("0 1 2 [xxx] 3 4");
%! assert (inf (x), [0 1 2 +inf 3 4]);
%! assert (sup (x), [0 1 2 -inf 3 4]);
%!warning id=interval:UndefinedOperation
%! x = infsup ({1 2; 3 "[xxx]"});
%! assert (inf (x), [1 2; 3 +inf]);
%! assert (sup (x), [1 2; 3 -inf]);
%!# complex values
%!warning id=interval:InvalidOperand
%! x = infsup ([1 2 3+i 4+0i]);
%! assert (inf (x), [1 2 +inf 4]);
%! assert (sup (x), [1 2 -inf 4]);
%!# inaccurate conversion
%!warning id=interval:PossiblyUndefinedOperation
%! x = infsup ("1.000000000000000000002", "1.000000000000000000001");
%! assert (inf (x), 1);
%! assert (sup (x), 1 + eps);
%!test
%! n = uint64(2 ^ 53);
%! x = infsup (n, n + 1);
%! assert (inf (x), double (n));
%! assert (sup (x), double (n + 2));
%!test
%! n = uint64(2 ^ 53);
%! x = infsup ({n}, n + 1);
%! assert (inf (x), double (n));
%! assert (sup (x), double (n + 2));
%!test
%! n = uint64(2 ^ 53);
%! x = infsup (n + 1, n + 1);
%! assert (inf (x), double (n));
%! assert (sup (x), double (n + 2));
%!test
%! n = uint64(2 ^ 54);
%! x = infsup (n, n + 1);
%! assert (inf (x), double (n));
%! assert (sup (x), double (n + 4));
%!warning id=interval:PossiblyUndefinedOperation
%! n = uint64(2 ^ 54);
%! x = infsup (n + 1, n + 2);
%! assert (inf (x), double (n));
%! assert (sup (x), double (n + 4));
%!warning id=interval:PossiblyUndefinedOperation
%! x = infsup ("pi", "3.141592653589793");
%! assert (inf (x), pi);
%! assert (sup (x), pi);
%!warning id=interval:PossiblyUndefinedOperation
%! x = infsup ("pi", "3.1415926535897932");
%! assert (inf (x), pi);
%! assert (sup (x), pi + 2 * eps);
%!# isexact flag
%!test
%! [~, isexact] = infsup ();
%! assert (isexact);
%!test
%! [~, isexact] = infsup (0);
%! assert (isexact);
%!test
%! [~, isexact] = infsup ("1 2 3");
%! assert (isexact, true);
%!test
%! [~, isexact] = infsup ("1 2 3.1");
%! assert (isexact, false);
%!warning
%! [~, isexact] = infsup ("[nai]");
%! assert (not (isexact));
%!# overflow flag
%!test
%! [~, ~, overflow] = infsup ();
%! assert (not (overflow));
%!test
%! [~, ~, overflow] = infsup (0);
%! assert (not (overflow));
%!test
%! [~, ~, overflow] = infsup ([1 2 3]);
%! assert (overflow, false (1, 3));
%!warning
%! [~, ~, overflow] = infsup ("[nai]");
%! assert (not (overflow));
%!test
%! [~, ~, overflow] = infsup ("1e3000");
%! assert (overflow);
%!test
%! [~, ~, overflow] = infsup ("[1, inf]");
%! assert (not (overflow));
%!# isnai flag
%!test
%! [~, ~, ~, isnai] = infsup ();
%! assert (not (isnai));
%!test
%! [~, ~, ~, isnai] = infsup (0);
%! assert (not (isnai));
%!test
%! [~, ~, ~, isnai] = infsup ([1 2 3]);
%! assert (isnai, false (1, 3));
%!warning
%! [~, ~, ~, isnai] = infsup ("[nai]");
%! assert (isnai);
%!warning
%! [~, ~, ~, isnai] = infsup ("xxx");
%! assert (isnai);
%!warning
%! [~, ~, ~, isnai] = infsup ("1 2 xxx 4");
%! assert (isnai, [false, false, true, false]);
%!warning
%! [~, ~, ~, isnai] = infsup ("[-inf, inf] [inf, inf]");
%! assert (isnai, [false, true]);
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