<s>
Interval	B-Algorithm
arithmetic	I-Algorithm
(	O
also	O
known	O
as	O
interval	B-Algorithm
mathematics	I-Algorithm
,	O
interval	B-Algorithm
analysis	I-Algorithm
or	O
interval	B-Algorithm
computation	I-Algorithm
)	O
is	O
a	O
mathematical	O
technique	O
used	O
to	O
mitigate	O
rounding	B-Algorithm
and	O
measurement	O
errors	O
in	O
mathematical	B-General_Concept
computation	I-General_Concept
by	O
computing	O
function	O
bounds	O
.	O
</s>
<s>
Numerical	B-General_Concept
methods	I-General_Concept
involving	O
interval	B-Algorithm
arithmetic	I-Algorithm
can	O
guarantee	O
reliable	O
and	O
mathematically	O
correct	O
results	O
.	O
</s>
<s>
Instead	O
of	O
representing	O
a	O
value	O
as	O
a	O
single	O
number	O
,	O
interval	B-Algorithm
arithmetic	I-Algorithm
represents	O
each	O
value	O
as	O
a	O
range	O
of	O
possibilities	O
.	O
</s>
<s>
Mathematically	O
,	O
instead	O
of	O
working	O
with	O
an	O
uncertain	O
real-valued	O
variable	O
,	O
interval	B-Algorithm
arithmetic	I-Algorithm
works	O
with	O
an	O
interval	O
that	O
defines	O
the	O
range	O
of	O
values	O
that	O
can	O
have	O
.	O
</s>
<s>
Interval	B-Algorithm
arithmetic	I-Algorithm
is	O
suitable	O
for	O
a	O
variety	O
of	O
purposes	O
;	O
the	O
most	O
common	O
use	O
is	O
in	O
scientific	O
works	O
,	O
particularly	O
when	O
the	O
calculations	O
are	O
handled	O
by	O
software	O
,	O
where	O
it	O
is	O
used	O
to	O
keep	O
track	O
of	O
rounding	B-Algorithm
errors	I-Algorithm
in	O
calculations	O
and	O
of	O
uncertainties	O
in	O
the	O
knowledge	O
of	O
the	O
exact	O
values	O
of	O
physical	O
and	O
technical	O
parameters	O
.	O
</s>
<s>
Interval	B-Algorithm
arithmetic	I-Algorithm
also	O
helps	O
find	O
guaranteed	O
solutions	O
to	O
equations	O
(	O
such	O
as	O
differential	O
equations	O
)	O
and	O
optimization	O
problems	O
.	O
</s>
<s>
The	O
main	O
objective	O
of	O
interval	B-Algorithm
arithmetic	I-Algorithm
is	O
to	O
provide	O
a	O
simple	O
way	O
of	O
calculating	O
upper	O
and	O
lower	O
bounds	O
of	O
a	O
function	O
's	O
range	O
in	O
one	O
or	O
more	O
variables	O
.	O
</s>
<s>
Suppose	O
a	O
person	O
uses	O
a	O
scale	O
that	O
has	O
a	O
precision	B-Architecture
of	O
one	O
kilogram	O
,	O
where	O
intermediate	O
values	O
cannot	O
be	O
discerned	O
,	O
and	O
the	O
true	O
weight	O
is	O
rounded	O
to	O
the	O
nearest	O
whole	O
number	O
.	O
</s>
<s>
In	O
such	O
a	O
case	O
,	O
the	O
scale	O
's	O
precision	B-Architecture
would	O
be	O
insufficient	O
to	O
make	O
a	O
definitive	O
conclusion	O
.	O
</s>
<s>
But	O
it	O
may	O
still	O
be	O
possible	O
to	O
extend	O
functions	O
to	O
interval	B-Algorithm
arithmetic	I-Algorithm
.	O
</s>
<s>
So	O
,	O
given	O
the	O
fact	O
that	O
an	O
interval	O
number	O
is	O
a	O
real	O
closed	O
interval	O
and	O
a	O
complex	O
number	O
is	O
an	O
ordered	O
pair	O
of	O
real	O
numbers	O
,	O
there	O
is	O
no	O
reason	O
to	O
limit	O
the	O
application	O
of	O
interval	B-Algorithm
arithmetic	I-Algorithm
to	O
the	O
measure	O
of	O
uncertainties	O
in	O
computations	O
with	O
real	O
numbers	O
.	O
</s>
<s>
Interval	B-Algorithm
arithmetic	I-Algorithm
can	O
thus	O
be	O
extended	O
,	O
via	O
complex	O
interval	O
numbers	O
,	O
to	O
determine	O
regions	O
of	O
uncertainty	O
in	O
computing	O
with	O
complex	O
numbers	O
.	O
</s>
<s>
One	O
can	O
either	O
define	O
complex	O
interval	B-Algorithm
arithmetic	I-Algorithm
using	O
rectangles	O
or	O
using	O
disks	O
,	O
both	O
with	O
their	O
respective	O
advantages	O
and	O
disadvantages	O
.	O
</s>
<s>
It	O
is	O
therefore	O
not	O
surprising	O
that	O
complex	O
interval	B-Algorithm
arithmetic	I-Algorithm
is	O
similar	O
to	O
,	O
but	O
not	O
the	O
same	O
as	O
,	O
ordinary	O
complex	O
arithmetic	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
,	O
as	O
it	O
is	O
the	O
case	O
with	O
real	O
interval	B-Algorithm
arithmetic	I-Algorithm
,	O
there	O
is	O
no	O
distributivity	O
between	O
addition	O
and	O
multiplication	O
of	O
complex	O
interval	O
numbers	O
except	O
for	O
certain	O
special	O
cases	O
,	O
and	O
inverse	O
elements	O
do	O
not	O
always	O
exist	O
for	O
complex	O
interval	O
numbers	O
.	O
</s>
<s>
Two	O
other	O
useful	O
properties	O
of	O
ordinary	O
complex	O
arithmetic	O
fail	O
to	O
hold	O
in	O
complex	O
interval	B-Algorithm
arithmetic	I-Algorithm
:	O
the	O
additive	O
and	O
multiplicative	O
properties	O
,	O
of	O
ordinary	O
complex	O
conjugates	O
,	O
do	O
not	O
hold	O
for	O
complex	O
interval	O
conjugates	O
.	O
</s>
<s>
Interval	B-Algorithm
arithmetic	I-Algorithm
can	O
be	O
extended	O
,	O
in	O
an	O
analogous	O
manner	O
,	O
to	O
other	O
multidimensional	O
number	O
systems	O
such	O
as	O
quaternions	O
and	O
octonions	O
,	O
but	O
with	O
the	O
expense	O
that	O
we	O
have	O
to	O
sacrifice	O
other	O
useful	O
properties	O
of	O
ordinary	O
arithmetic	O
.	O
</s>
<s>
The	O
methods	O
of	O
classical	O
numerical	B-General_Concept
analysis	I-General_Concept
cannot	O
be	O
transferred	O
one-to-one	O
into	O
interval-valued	O
algorithms	O
,	O
as	O
dependencies	O
between	O
numerical	O
values	O
are	O
usually	O
not	O
taken	O
into	O
account	O
.	O
</s>
<s>
The	O
earlier	O
operations	O
were	O
based	O
on	O
exact	O
arithmetic	O
,	O
but	O
in	O
general	O
fast	O
numerical	B-General_Concept
solution	I-General_Concept
methods	O
may	O
not	O
be	O
available	O
for	O
it	O
.	O
</s>
<s>
Where	O
the	O
same	O
calculation	O
is	O
done	O
with	O
single	O
digit	O
precision	B-Architecture
,	O
the	O
result	O
would	O
normally	O
be	O
.	O
</s>
<s>
so	O
this	O
approach	O
would	O
contradict	O
the	O
basic	O
principles	O
of	O
interval	B-Algorithm
arithmetic	I-Algorithm
,	O
as	O
a	O
part	O
of	O
the	O
domain	O
of	O
would	O
be	O
lost	O
.	O
</s>
<s>
The	O
standard	O
IEEE	O
754	O
for	O
binary	O
floating-point	O
arithmetic	O
also	O
sets	O
out	O
procedures	O
for	O
the	O
implementation	O
of	O
rounding	B-Algorithm
.	O
</s>
<s>
An	O
IEEE	O
754	O
compliant	O
system	O
allows	O
programmers	O
to	O
round	O
to	O
the	O
nearest	O
floating-point	O
number	O
;	O
alternatives	O
are	O
rounding	B-Algorithm
towards	O
0	O
(	O
truncating	O
)	O
,	O
rounding	B-Algorithm
toward	O
positive	O
infinity	O
(	O
i.e.	O
,	O
up	O
)	O
,	O
or	O
rounding	B-Algorithm
towards	O
negative	O
infinity	O
(	O
i.e.	O
,	O
down	O
)	O
.	O
</s>
<s>
The	O
required	O
external	O
rounding	B-Algorithm
for	O
interval	B-Algorithm
arithmetic	I-Algorithm
can	O
thus	O
be	O
achieved	O
by	O
changing	O
the	O
rounding	B-Algorithm
settings	O
of	O
the	O
processor	O
in	O
the	O
calculation	O
of	O
the	O
upper	O
limit	O
(	O
up	O
)	O
and	O
lower	O
limit	O
(	O
down	O
)	O
.	O
</s>
<s>
The	O
so-called	O
"	O
dependency	O
"	O
problem	O
is	O
a	O
major	O
obstacle	O
to	O
the	O
application	O
of	O
interval	B-Algorithm
arithmetic	I-Algorithm
.	O
</s>
<s>
Although	O
interval	B-Algorithm
methods	I-Algorithm
can	O
determine	O
the	O
range	O
of	O
elementary	O
arithmetic	O
operations	O
and	O
functions	O
very	O
accurately	O
,	O
this	O
is	O
not	O
always	O
true	O
with	O
more	O
complicated	O
functions	O
.	O
</s>
<s>
is	O
precisely	O
the	O
line	O
between	O
the	O
points	O
and	O
Using	O
interval	B-Algorithm
methods	I-Algorithm
results	O
in	O
the	O
unit	O
square	O
,	O
This	O
is	O
known	O
as	O
the	O
wrapping	O
effect	O
.	O
</s>
<s>
A	O
rough	O
solution	O
can	O
often	O
be	O
improved	O
by	O
an	O
interval	O
version	O
of	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Since	O
the	O
procedure	O
is	O
more	O
efficient	O
for	O
a	O
diagonally	B-Algorithm
dominant	I-Algorithm
matrix	I-Algorithm
,	O
instead	O
of	O
the	O
system	O
one	O
can	O
often	O
try	O
multiplying	O
it	O
by	O
an	O
appropriate	O
rational	O
matrix	O
with	O
the	O
resulting	O
matrix	O
equation	O
.	O
</s>
<s>
The	O
resulting	O
problems	O
can	O
be	O
resolved	O
using	O
conventional	O
numerical	B-General_Concept
methods	I-General_Concept
.	O
</s>
<s>
Interval	B-Algorithm
arithmetic	I-Algorithm
is	O
still	O
used	O
to	O
determine	O
rounding	B-Algorithm
errors	I-Algorithm
.	O
</s>
<s>
The	O
various	O
interval	B-Algorithm
methods	I-Algorithm
deliver	O
conservative	O
results	O
as	O
dependencies	O
between	O
the	O
sizes	O
of	O
different	O
intervals	O
extensions	O
are	O
not	O
taken	O
into	O
account	O
.	O
</s>
<s>
Interval	B-Algorithm
arithmetic	I-Algorithm
can	O
be	O
used	O
in	O
various	O
areas	O
(	O
such	O
as	O
set	O
inversion	O
,	O
motion	O
planning	O
,	O
set	O
estimation	O
or	O
stability	O
analysis	O
)	O
to	O
treat	O
estimates	O
with	O
no	O
exact	O
numerical	O
value	O
.	O
</s>
<s>
Interval	B-Algorithm
arithmetic	I-Algorithm
is	O
used	O
with	O
error	O
analysis	O
,	O
to	O
control	O
rounding	B-Algorithm
errors	I-Algorithm
arising	O
from	O
each	O
calculation	O
.	O
</s>
<s>
The	O
advantage	O
of	O
interval	B-Algorithm
arithmetic	I-Algorithm
is	O
that	O
after	O
each	O
operation	O
there	O
is	O
an	O
interval	O
that	O
reliably	O
includes	O
the	O
true	O
result	O
.	O
</s>
<s>
The	O
distance	O
between	O
the	O
interval	O
boundaries	O
gives	O
the	O
current	O
calculation	O
of	O
rounding	B-Algorithm
errors	I-Algorithm
directly	O
:	O
</s>
<s>
Interval	B-Algorithm
analysis	I-Algorithm
adds	O
to	O
rather	O
than	O
substituting	O
for	O
traditional	O
methods	O
for	O
error	O
reduction	O
,	O
such	O
as	O
pivoting	B-Algorithm
.	O
</s>
<s>
can	O
be	O
found	O
by	O
interval	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
This	O
provides	O
an	O
alternative	O
to	O
traditional	O
propagation	B-General_Concept
of	I-General_Concept
error	I-General_Concept
analysis	O
.	O
</s>
<s>
Unlike	O
point	O
methods	O
,	O
such	O
as	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
,	O
interval	B-Algorithm
arithmetic	I-Algorithm
methodology	O
ensures	O
that	O
no	O
part	O
of	O
the	O
solution	O
area	O
can	O
be	O
overlooked	O
.	O
</s>
<s>
Interval	B-Algorithm
arithmetic	I-Algorithm
can	O
also	O
be	O
used	O
with	O
affiliation	O
functions	O
for	O
fuzzy	O
quantities	O
as	O
they	O
are	O
used	O
in	O
fuzzy	O
logic	O
.	O
</s>
<s>
and	O
can	O
be	O
calculated	O
by	O
interval	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Warwick	O
Tucker	O
used	O
interval	B-Algorithm
arithmetic	I-Algorithm
in	O
order	O
to	O
solve	O
the	O
14th	O
of	O
Smale	O
's	O
problems	O
,	O
that	O
is	O
,	O
to	O
show	O
that	O
the	O
Lorenz	B-Architecture
attractor	I-Architecture
is	O
a	O
strange	O
attractor	O
.	O
</s>
<s>
Thomas	O
Hales	O
used	O
interval	B-Algorithm
arithmetic	I-Algorithm
in	O
order	O
to	O
solve	O
the	O
Kepler	O
conjecture	O
.	O
</s>
<s>
Interval	B-Algorithm
arithmetic	I-Algorithm
is	O
not	O
a	O
completely	O
new	O
phenomenon	O
in	O
mathematics	O
;	O
it	O
has	O
appeared	O
several	O
times	O
under	O
different	O
names	O
in	O
the	O
course	O
of	O
history	O
.	O
</s>
<s>
For	O
example	O
,	O
Archimedes	B-Device
calculated	O
lower	O
and	O
upper	O
bounds	O
223/71	O
<	O
π	O
<	O
22/7	O
in	O
the	O
3rd	O
century	O
BC	O
.	O
</s>
<s>
Arithmetic	O
work	O
on	O
range	O
numbers	O
to	O
improve	O
the	O
reliability	O
of	O
digital	O
systems	O
were	O
then	O
published	O
in	O
a	O
1951	O
textbook	O
on	O
linear	O
algebra	O
by	O
;	O
intervals	O
were	O
used	O
to	O
measure	O
rounding	B-Algorithm
errors	I-Algorithm
associated	O
with	O
floating-point	O
numbers	O
.	O
</s>
<s>
A	O
comprehensive	O
paper	O
on	O
interval	O
algebra	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
was	O
published	O
by	O
Teruo	O
Sunaga	O
(	O
1958	O
)	O
.	O
</s>
<s>
The	O
birth	O
of	O
modern	O
interval	B-Algorithm
arithmetic	I-Algorithm
was	O
marked	O
by	O
the	O
appearance	O
of	O
the	O
book	O
Interval	B-Algorithm
Analysis	I-Algorithm
by	O
Ramon	O
E	O
.	O
Moore	O
in	O
1966	O
.	O
</s>
<s>
He	O
had	O
the	O
idea	O
in	O
spring	O
1958	O
,	O
and	O
a	O
year	O
later	O
he	O
published	O
an	O
article	O
about	O
computer	O
interval	B-Algorithm
arithmetic	I-Algorithm
.	O
</s>
<s>
Its	O
merit	O
was	O
that	O
starting	O
with	O
a	O
simple	O
principle	O
,	O
it	O
provided	O
a	O
general	O
method	O
for	O
automated	O
error	O
analysis	O
,	O
not	O
just	O
errors	O
resulting	O
from	O
rounding	B-Algorithm
.	O
</s>
<s>
In	O
the	O
following	O
twenty	O
years	O
,	O
German	O
groups	O
of	O
researchers	O
carried	O
out	O
pioneering	O
work	O
around	O
Ulrich	O
W	O
.	O
Kulisch	O
and	O
at	O
the	O
University	O
of	O
Karlsruhe	B-Algorithm
and	O
later	O
also	O
at	O
the	O
Bergische	O
University	O
of	O
Wuppertal	O
.	O
</s>
<s>
Classical	O
methods	O
in	O
this	O
often	O
have	O
the	O
problem	O
of	O
determining	O
the	O
largest	O
(	O
or	O
smallest	O
)	O
global	O
value	O
,	O
but	O
could	O
only	O
find	O
a	O
local	O
optimum	O
and	O
could	O
not	O
find	O
better	O
values	O
;	O
Helmut	O
Ratschek	O
and	O
Jon	O
George	O
Rokne	O
developed	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
methods	O
,	O
which	O
until	O
then	O
had	O
only	O
applied	O
to	O
integer	O
values	O
,	O
by	O
using	O
intervals	O
to	O
provide	O
applications	O
for	O
continuous	O
values	O
.	O
</s>
<s>
In	O
1988	O
,	O
Rudolf	O
Lohner	O
developed	O
Fortran-based	O
software	O
for	O
reliable	O
solutions	O
for	O
initial	O
value	O
problems	O
using	O
ordinary	O
differential	O
equations	O
.	O
</s>
<s>
The	O
journal	O
Reliable	O
Computing	O
(	O
originally	O
Interval	B-Algorithm
Computations	I-Algorithm
)	O
has	O
been	O
published	O
since	O
the	O
1990s	O
,	O
dedicated	O
to	O
the	O
reliability	O
of	O
computer-aided	O
computations	O
.	O
</s>
<s>
As	O
lead	O
editor	O
,	O
R	O
.	O
Baker	O
Kearfott	O
,	O
in	O
addition	O
to	O
his	O
work	O
on	O
global	O
optimization	O
,	O
has	O
contributed	O
significantly	O
to	O
the	O
unification	O
of	O
notation	O
and	O
terminology	O
used	O
in	O
interval	B-Algorithm
arithmetic	I-Algorithm
.	O
</s>
<s>
There	O
are	O
many	O
software	O
packages	O
that	O
permit	O
the	O
development	O
of	O
numerical	O
applications	O
using	O
interval	B-Algorithm
arithmetic	I-Algorithm
.	O
</s>
<s>
There	O
are	O
also	O
C++	B-Language
and	O
Fortran	B-Application
compilers	B-Language
that	O
handle	O
interval	O
data	O
types	O
and	O
suitable	O
operations	O
as	O
a	O
language	O
extension	O
,	O
so	O
interval	B-Algorithm
arithmetic	I-Algorithm
is	O
supported	O
directly	O
.	O
</s>
<s>
Since	O
1967	O
,	O
Extensions	O
for	O
Scientific	O
Computation	O
(	O
XSC	O
)	O
have	O
been	O
developed	O
in	O
the	O
University	O
of	O
Karlsruhe	B-Algorithm
for	O
various	O
programming	O
languages	O
,	O
such	O
as	O
C++	B-Language
,	O
Fortran	B-Application
and	O
Pascal	B-Application
.	O
</s>
<s>
The	O
first	O
platform	O
was	O
a	O
Zuse	O
Z23	B-Device
,	O
for	O
which	O
a	O
new	O
interval	O
data	O
type	O
with	O
appropriate	O
elementary	O
operators	O
was	O
made	O
available	O
.	O
</s>
<s>
There	O
followed	O
in	O
1976	O
,	O
Pascal-SC	O
,	O
a	O
Pascal	B-Application
variant	O
on	O
a	O
Zilog	B-General_Concept
Z80	I-General_Concept
that	O
it	O
made	O
possible	O
to	O
create	O
fast	O
,	O
complicated	O
routines	O
for	O
automated	O
result	O
verification	O
.	O
</s>
<s>
Then	O
came	O
the	O
Fortran	B-Application
77-based	O
ACRITH-XSC	O
for	O
the	O
System/370	B-Device
architecture	O
(	O
FORTRAN-SC	O
)	O
,	O
which	O
was	O
later	O
delivered	O
by	O
IBM	O
.	O
</s>
<s>
Starting	O
from	O
1991	O
one	O
could	O
produce	O
code	O
for	O
C	B-Language
compilers	B-Language
with	O
Pascal-XSC	O
;	O
a	O
year	O
later	O
the	O
C++	B-Language
class	O
library	O
supported	O
C-XSC	O
on	O
many	O
different	O
computer	O
systems	O
.	O
</s>
<s>
In	O
1997	O
,	O
all	O
XSC	O
variants	O
were	O
made	O
available	O
under	O
the	O
GNU	B-License
General	I-License
Public	I-License
License	I-License
.	O
</s>
<s>
At	O
the	O
beginning	O
of	O
2000	O
C-XSC	O
2.0	O
was	O
released	O
under	O
the	O
leadership	O
of	O
the	O
working	O
group	O
for	O
scientific	O
computation	O
at	O
the	O
Bergische	O
University	O
of	O
Wuppertal	O
to	O
correspond	O
to	O
the	O
improved	O
C++	B-Language
standard	O
.	O
</s>
<s>
Another	O
C++	B-Language
-class	O
library	O
was	O
created	O
in	O
1993	O
at	O
the	O
Hamburg	O
University	O
of	O
Technology	O
called	O
Profil/BIAS	O
(	O
Programmer	O
's	O
Runtime	O
Optimized	O
Fast	O
Interval	O
Library	O
,	O
Basic	O
Interval	B-Algorithm
Arithmetic	I-Algorithm
)	O
,	O
which	O
made	O
the	O
usual	O
interval	O
operations	O
more	O
user	O
friendly	O
.	O
</s>
<s>
The	O
Boost	B-Language
collection	I-Language
of	O
C++	B-Language
libraries	O
contains	O
a	O
template	O
class	O
for	O
intervals	O
.	O
</s>
<s>
Its	O
authors	O
are	O
aiming	O
to	O
have	O
interval	B-Algorithm
arithmetic	I-Algorithm
in	O
the	O
standard	O
C++	B-Language
language	I-Language
.	O
</s>
<s>
The	O
Frink	B-Language
programming	O
language	O
has	O
an	O
implementation	O
of	O
interval	B-Algorithm
arithmetic	I-Algorithm
that	O
handles	O
arbitrary-precision	B-Algorithm
numbers	O
.	O
</s>
<s>
Programs	O
written	O
in	O
Frink	B-Language
can	O
use	O
intervals	O
without	O
rewriting	O
or	O
recompilation	O
.	O
</s>
<s>
GAOL	O
is	O
another	O
C++	B-Language
interval	B-Algorithm
arithmetic	I-Algorithm
library	O
that	O
is	O
unique	O
in	O
that	O
it	O
offers	O
the	O
relational	O
interval	O
operators	O
used	O
in	O
interval	O
constraint	B-Application
programming	I-Application
.	O
</s>
<s>
The	O
Moore	O
library	O
is	O
an	O
efficient	O
implementation	O
of	O
interval	B-Algorithm
arithmetic	I-Algorithm
in	O
C++	B-Language
.	O
</s>
<s>
It	O
provides	O
intervals	O
with	O
endpoints	O
of	O
arbitrary	B-Algorithm
precision	I-Algorithm
and	O
is	O
based	O
on	O
the	O
concepts	O
feature	O
of	O
C++	B-Language
.	O
</s>
<s>
The	O
Julia	B-Application
programming	I-Application
language	I-Application
has	O
an	O
implementation	O
of	O
interval	B-Algorithm
arithmetics	I-Algorithm
along	O
with	O
high-level	O
features	O
,	O
such	O
as	O
root-finding	O
(	O
for	O
both	O
real	O
and	O
complex-valued	O
functions	O
)	O
and	O
interval	O
constraint	B-Application
programming	I-Application
,	O
via	O
the	O
ValidatedNumerics.jl	O
package	O
.	O
</s>
<s>
In	O
addition	O
,	O
computer	O
algebra	O
systems	O
,	O
such	O
as	O
FriCAS	B-Language
,	O
Mathematica	B-Language
,	O
Maple	B-Language
,	O
Maxima	B-Language
(	O
software	O
)	O
and	O
MuPAD	B-Language
,	O
can	O
handle	O
intervals	O
.	O
</s>
<s>
A	O
Matlab	B-Language
extension	O
Intlab''	O
builds	O
on	O
BLAS	B-Application
routines	O
,	O
and	O
the	O
Toolbox	O
b4m	O
makes	O
a	O
Profil/BIAS	O
interface	O
.	O
</s>
<s>
Moreover	O
,	O
the	O
Software	O
Euler	B-Language
Math	I-Language
Toolbox	I-Language
includes	O
an	O
interval	B-Algorithm
arithmetic	I-Algorithm
.	O
</s>
<s>
A	O
library	O
for	O
the	O
functional	O
language	O
OCaml	B-Language
was	O
written	O
in	O
assembly	O
language	O
and	O
C	B-Language
.	O
</s>
<s>
A	O
standard	O
for	O
interval	B-Algorithm
arithmetic	I-Algorithm
,	O
IEEE	O
Std	O
1788-2015	O
,	O
has	O
been	O
approved	O
in	O
June	O
2015	O
.	O
</s>
<s>
These	O
have	O
been	O
developed	O
by	O
members	O
of	O
the	O
standard	O
's	O
working	O
group	O
:	O
The	O
libieeep1788	O
library	O
for	O
C++	B-Language
,	O
and	O
the	O
interval	O
package	O
for	O
GNU	B-Language
Octave	I-Language
.	O
</s>
<s>
The	O
main	O
conference	O
is	O
probably	O
SCAN	O
(	O
International	O
Symposium	O
on	O
Scientific	O
Computing	O
,	O
Computer	O
Arithmetic	O
,	O
and	O
Verified	O
Numerical	B-General_Concept
Computation	I-General_Concept
)	O
,	O
but	O
there	O
is	O
also	O
SWIM	O
(	O
Small	O
Workshop	O
on	O
Interval	B-Algorithm
Methods	I-Algorithm
)	O
,	O
PPAM	O
(	O
International	O
Conference	O
on	O
Parallel	O
Processing	O
and	O
Applied	O
Mathematics	O
)	O
,	O
REC	O
(	O
International	O
Workshop	O
on	O
Reliable	O
Engineering	O
Computing	O
)	O
.	O
</s>
