<s>
Affine	B-Algorithm
arithmetic	I-Algorithm
(	O
AA	O
)	O
is	O
a	O
model	O
for	O
self-validated	B-Algorithm
numerical	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
Affine	B-Algorithm
arithmetic	I-Algorithm
is	O
meant	O
to	O
be	O
an	O
improvement	O
on	O
interval	B-Algorithm
arithmetic	I-Algorithm
(	O
IA	O
)	O
,	O
and	O
is	O
similar	O
to	O
generalized	O
interval	B-Algorithm
arithmetic	I-Algorithm
,	O
first-order	O
Taylor	O
arithmetic	O
,	O
the	O
center-slope	O
model	O
,	O
and	O
ellipsoid	O
calculus	O
in	O
the	O
sense	O
that	O
it	O
is	O
an	O
automatic	O
method	O
to	O
derive	O
first-order	O
guaranteed	O
approximations	O
to	O
general	O
formulas	O
.	O
</s>
<s>
Affine	B-Algorithm
arithmetic	I-Algorithm
is	O
potentially	O
useful	O
in	O
every	O
numeric	O
problem	O
where	O
one	O
needs	O
guaranteed	O
enclosures	O
to	O
smooth	O
functions	O
,	O
such	O
as	O
solving	O
systems	O
of	O
non-linear	O
equations	O
,	O
analyzing	O
dynamical	O
systems	O
,	O
integrating	O
functions	O
,	O
differential	O
equations	O
,	O
etc	O
.	O
</s>
<s>
Applications	O
include	O
ray	B-Algorithm
tracing	I-Algorithm
,	O
plotting	O
curves	O
,	O
intersecting	O
implicit	B-Algorithm
and	O
parametric	O
surfaces	O
,	O
error	B-Algorithm
analysis	I-Algorithm
(	O
mathematics	O
)	O
,	O
process	O
control	O
,	O
worst-case	O
analysis	O
of	O
electric	O
circuits	O
,	O
and	O
more	O
.	O
</s>
<s>
then	O
the	O
individual	O
ranges	O
of	O
X	O
and	O
Y	O
are	O
 [ 2 , 18 ] 	O
and	O
[13,27],	O
but	O
the	O
joint	O
range	O
of	O
the	O
pair	O
(	O
X	O
,	O
Y	O
)	O
is	O
the	O
hexagon	B-Application
with	O
corners	O
(	O
2	O
,	O
27	O
)	O
,	O
(	O
6	O
,	O
27	O
)	O
,	O
(	O
18	O
,	O
19	O
)	O
,	O
(	O
18	O
,	O
13	O
)	O
,	O
(	O
14	O
,	O
13	O
)	O
,	O
(	O
2	O
,	O
21	O
)	O
which	O
is	O
a	O
proper	O
subset	O
of	O
the	O
rectangle	O
 [ 2 , 18 ] × [ 13 , 27 ] 	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
affine	B-Algorithm
arithmetic	I-Algorithm
will	O
often	O
yield	O
much	O
tighter	O
bounds	O
than	O
standard	O
interval	B-Algorithm
arithmetic	I-Algorithm
(	O
whose	O
errors	O
are	O
proportional	O
to	O
h	O
)	O
.	O
</s>
<s>
In	O
order	O
to	O
provide	O
guaranteed	O
enclosure	O
,	O
affine	B-Algorithm
arithmetic	I-Algorithm
operations	O
must	O
account	O
for	O
the	O
roundoff	O
errors	O
in	O
the	O
computation	O
of	O
the	O
resulting	O
coefficients	O
.	O
</s>
<s>
Affine	B-Algorithm
arithmetic	I-Algorithm
can	O
be	O
viewed	O
in	O
matrix	O
form	O
as	O
follows	O
.	O
</s>
<s>
Affine	B-Algorithm
arithmetic	I-Algorithm
can	O
be	O
implemented	O
by	O
a	O
global	O
array	O
A	O
and	O
a	O
global	O
vector	O
b	O
,	O
as	O
described	O
above	O
.	O
</s>
<s>
In	O
such	O
situations	O
,	O
one	O
should	O
use	O
a	O
sparse	B-Algorithm
implementation	O
.	O
</s>
