<s>
In	O
science	O
,	O
engineering	O
,	O
and	O
other	O
quantitative	O
disciplines	O
,	O
order	B-Algorithm
of	I-Algorithm
approximation	I-Algorithm
refers	O
to	O
formal	O
or	O
informal	O
expressions	O
for	O
how	O
accurate	O
an	O
approximation	O
is	O
.	O
</s>
<s>
The	O
expressions	O
:	O
a	O
zeroth-order	O
approximation	O
,	O
a	O
first-order	B-Algorithm
approximation	I-Algorithm
,	O
a	O
second-order	O
approximation	O
,	O
and	O
so	O
forth	O
are	O
used	O
as	O
fixed	O
phrases	O
.	O
</s>
<s>
The	O
expression	O
a	O
zero-order	B-Algorithm
approximation	I-Algorithm
is	O
also	O
common	O
.	O
</s>
<s>
Phrases	O
like	O
first	B-Algorithm
approximation	I-Algorithm
or	O
to	B-Algorithm
a	I-Algorithm
first	I-Algorithm
approximation	I-Algorithm
may	O
refer	O
to	O
a	O
roughly	O
approximate	O
value	O
of	O
a	O
quantity	O
.	O
</s>
<s>
The	O
expression	O
order	B-Algorithm
of	I-Algorithm
approximation	I-Algorithm
is	O
sometimes	O
informally	O
used	O
to	O
mean	O
the	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
,	O
in	O
increasing	O
order	O
of	O
accuracy	O
,	O
or	O
to	O
the	O
order	O
of	O
magnitude	O
.	O
</s>
<s>
The	O
expression	O
order	B-Algorithm
of	I-Algorithm
approximation	I-Algorithm
is	O
expected	O
to	O
indicate	O
progressively	O
more	O
refined	O
approximations	O
of	O
a	O
function	O
in	O
a	O
specified	O
interval	O
.	O
</s>
<s>
The	O
choice	O
of	O
order	B-Algorithm
of	I-Algorithm
approximation	I-Algorithm
depends	O
on	O
the	O
research	O
purpose	O
.	O
</s>
<s>
One	O
may	O
wish	O
to	O
simplify	O
a	O
known	O
analytic	O
expression	O
to	O
devise	O
a	O
new	O
application	O
or	O
,	O
on	O
the	O
contrary	O
,	O
try	O
to	O
fit	O
a	O
curve	O
to	O
data	B-Application
points	I-Application
.	O
</s>
<s>
Higher	O
order	B-Algorithm
of	I-Algorithm
approximation	I-Algorithm
is	O
not	O
always	O
more	O
useful	O
than	O
the	O
lower	O
one	O
.	O
</s>
<s>
The	O
formal	O
usage	O
of	O
order	B-Algorithm
of	I-Algorithm
approximation	I-Algorithm
corresponds	O
to	O
the	O
omission	O
of	O
some	O
terms	O
of	O
the	O
series	O
used	O
in	O
the	O
expansion	O
(	O
usually	O
the	O
higher	O
terms	O
)	O
.	O
</s>
<s>
used	O
formally	O
in	O
the	O
above	O
meaning	O
do	O
not	O
directly	O
give	O
information	O
about	O
percent	B-Algorithm
error	I-Algorithm
or	O
significant	B-Architecture
figures	I-Architecture
.	O
</s>
<s>
Many	O
simplifying	O
assumptions	O
are	O
made	O
,	O
and	O
when	O
a	O
number	O
is	O
needed	O
,	O
an	O
order-of-magnitude	O
answer	O
(	O
or	O
zero	O
significant	B-Architecture
figures	I-Architecture
)	O
is	O
often	O
given	O
.	O
</s>
<s>
A	O
zeroth-order	O
approximation	O
of	O
a	O
function	O
(	O
that	O
is	O
,	O
mathematically	O
determining	O
a	O
formula	O
to	O
fit	O
multiple	O
data	B-Application
points	I-Application
)	O
will	O
be	O
constant	O
,	O
or	O
a	O
flat	O
line	O
with	O
no	O
slope	O
:	O
a	O
polynomial	O
of	O
degree	O
0	O
.	O
</s>
<s>
could	O
be	O
–	O
if	O
data	B-Application
point	I-Application
accuracy	O
were	O
reported	O
–	O
an	O
approximate	O
fit	O
to	O
the	O
data	O
,	O
obtained	O
by	O
simply	O
averaging	O
the	O
x	O
values	O
and	O
the	O
y	O
values	O
.	O
</s>
<s>
However	O
,	O
data	B-Application
points	I-Application
represent	O
results	O
of	O
measurements	O
and	O
they	O
do	O
differ	O
from	O
points	O
in	O
Euclidean	O
geometry	O
.	O
</s>
<s>
Thus	O
quoting	O
an	O
average	O
value	O
containing	O
three	O
significant	B-Architecture
digits	I-Architecture
in	O
the	O
output	O
with	O
just	O
one	O
significant	B-Architecture
digit	I-Architecture
in	O
the	O
input	O
data	O
could	O
be	O
recognized	O
as	O
an	O
example	O
of	O
false	B-Algorithm
precision	I-Algorithm
.	O
</s>
<s>
With	O
the	O
implied	O
accuracy	O
of	O
the	O
data	B-Application
points	I-Application
of	O
±	O
0.5	O
,	O
the	O
zeroth	O
order	O
approximation	O
could	O
at	O
best	O
yield	O
the	O
result	O
for	O
y	O
of	O
~	O
3.7	O
±	O
2.0	O
in	O
the	O
interval	O
of	O
x	O
from	O
−	O
0.5	O
to	O
2.5	O
,	O
considering	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
If	O
only	O
three	O
data	B-Application
points	I-Application
are	O
available	O
,	O
one	O
has	O
no	O
knowledge	O
about	O
the	O
rest	O
of	O
the	O
interval	O
,	O
which	O
may	O
be	O
a	O
large	O
part	O
of	O
it	O
.	O
</s>
<s>
First-order	B-Algorithm
approximation	I-Algorithm
is	O
the	O
term	O
scientists	O
use	O
for	O
a	O
slightly	O
better	O
answer	O
.	O
</s>
<s>
Some	O
simplifying	O
assumptions	O
are	O
made	O
,	O
and	O
when	O
a	O
number	O
is	O
needed	O
,	O
an	O
answer	O
with	O
only	O
one	O
significant	B-Architecture
figure	I-Architecture
is	O
often	O
given	O
(	O
"	O
the	O
town	O
has	O
,	O
or	O
four	O
thousand	O
,	O
residents	O
"	O
)	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
first-order	B-Algorithm
approximation	I-Algorithm
,	O
at	O
least	O
one	O
number	O
given	O
is	O
exact	O
.	O
</s>
<s>
A	O
first-order	B-Algorithm
approximation	I-Algorithm
of	O
a	O
function	O
(	O
that	O
is	O
,	O
mathematically	O
determining	O
a	O
formula	O
to	O
fit	O
multiple	O
data	B-Application
points	I-Application
)	O
will	O
be	O
a	O
linear	O
approximation	O
,	O
straight	O
line	O
with	O
a	O
slope	O
:	O
a	O
polynomial	O
of	O
degree1	O
.	O
</s>
<s>
Few	O
simplifying	O
assumptions	O
are	O
made	O
,	O
and	O
when	O
a	O
number	O
is	O
needed	O
,	O
an	O
answer	O
with	O
two	O
or	O
more	O
significant	B-Architecture
figures	I-Architecture
(	O
"	O
the	O
town	O
has	O
,	O
or	O
thirty-nine	O
hundred	O
,	O
residents	O
"	O
)	O
is	O
generally	O
given	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
"	O
3	O
"	O
and	O
"	O
9	O
"	O
are	O
given	O
as	O
the	O
two	O
successive	O
levels	O
of	O
precision	B-Architecture
,	O
instead	O
of	O
simply	O
the	O
"	O
4	O
"	O
from	O
the	O
first	O
order	O
,	O
or	O
"	O
a	O
few	O
"	O
from	O
the	O
zeroth	O
order	O
found	O
in	O
the	O
examples	O
above	O
.	O
</s>
<s>
A	O
second-order	O
approximation	O
of	O
a	O
function	O
(	O
that	O
is	O
,	O
mathematically	O
determining	O
a	O
formula	O
to	O
fit	O
multiple	O
data	B-Application
points	I-Application
)	O
will	O
be	O
a	O
quadratic	O
polynomial	O
,	O
geometrically	O
,	O
a	O
parabola	O
:	O
a	O
polynomial	O
of	O
degree2	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
with	O
only	O
three	O
data	B-Application
points	I-Application
,	O
a	O
parabola	O
is	O
an	O
exact	O
fit	O
based	O
on	O
the	O
data	O
provided	O
.	O
</s>
<s>
However	O
,	O
the	O
data	B-Application
points	I-Application
for	O
most	O
of	O
the	O
interval	O
are	O
not	O
available	O
,	O
which	O
advises	O
caution	O
(	O
see	O
"	O
zeroth	O
order	O
"	O
)	O
.	O
</s>
<s>
Continuing	O
the	O
above	O
,	O
a	O
third-order	O
approximation	O
would	O
be	O
required	O
to	O
perfectly	O
fit	O
four	O
data	B-Application
points	I-Application
,	O
and	O
so	O
on	O
.	O
</s>
<s>
The	O
terminology	O
,	O
in	O
this	O
context	O
,	O
represents	O
a	O
high	O
level	O
of	O
precision	B-Architecture
required	O
to	O
account	O
for	O
an	O
effect	O
which	O
is	O
inferred	O
to	O
be	O
very	O
small	O
when	O
compared	O
to	O
the	O
overall	O
subject	O
matter	O
.	O
</s>
<s>
The	O
higher	O
the	O
order	O
,	O
the	O
more	O
precision	B-Architecture
is	O
required	O
to	O
measure	O
the	O
effect	O
,	O
and	O
therefore	O
the	O
smallness	O
of	O
the	O
effect	O
in	O
comparison	O
to	O
the	O
overall	O
measurement	O
.	O
</s>
