<s>
Validated	B-Algorithm
numerics	I-Algorithm
,	O
or	O
rigorous	O
computation	O
,	O
verified	O
computation	O
,	O
reliable	O
computation	O
,	O
numerical	O
verification	O
(	O
)	O
is	O
numerics	O
including	O
mathematically	O
strict	O
error	O
(	O
rounding	O
error	O
,	O
truncation	O
error	O
,	O
discretization	O
error	O
)	O
evaluation	O
,	O
and	O
it	O
is	O
one	O
field	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
For	O
computation	O
,	O
interval	B-Algorithm
arithmetic	I-Algorithm
is	O
used	O
,	O
and	O
all	O
results	O
are	O
represented	O
by	O
intervals	O
.	O
</s>
<s>
Validated	B-Algorithm
numerics	I-Algorithm
were	O
used	O
by	O
Warwick	O
Tucker	O
in	O
order	O
to	O
solve	O
the	O
14th	O
of	O
Smale	O
's	O
problems	O
,	O
and	O
today	O
it	O
is	O
recognized	O
as	O
a	O
powerful	O
tool	O
for	O
the	O
study	O
of	O
dynamical	O
systems	O
.	O
</s>
<s>
This	O
is	O
a	O
rare	O
case	O
,	O
but	O
it	O
tells	O
us	O
that	O
when	O
we	O
want	O
to	O
strictly	O
discuss	O
differential	O
equations	O
,	O
numerical	B-General_Concept
solutions	I-General_Concept
must	O
be	O
verified	O
.	O
</s>
<s>
The	O
study	O
of	O
validated	B-Algorithm
numerics	I-Algorithm
is	O
divided	O
into	O
the	O
following	O
fields	O
:	O
</s>
