<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
the	O
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
formulas	I-Algorithm
,	O
also	O
called	O
the	O
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
quadrature	I-Algorithm
rules	I-Algorithm
or	O
simply	O
Newton	O
–	O
Cotes	O
rules	O
,	O
are	O
a	O
group	O
of	O
formulas	O
for	O
numerical	B-Algorithm
integration	I-Algorithm
(	O
also	O
called	O
quadrature	O
)	O
based	O
on	O
evaluating	O
the	O
integrand	O
at	O
equally	O
spaced	O
points	O
.	O
</s>
<s>
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
formulas	I-Algorithm
can	O
be	O
useful	O
if	O
the	O
value	O
of	O
the	O
integrand	O
at	O
equally	O
spaced	O
points	O
is	O
given	O
.	O
</s>
<s>
If	O
it	O
is	O
possible	O
to	O
change	O
the	O
points	O
at	O
which	O
the	O
integrand	O
is	O
evaluated	O
,	O
then	O
other	O
methods	O
such	O
as	O
Gaussian	B-Algorithm
quadrature	I-Algorithm
and	O
Clenshaw	B-Algorithm
–	I-Algorithm
Curtis	I-Algorithm
quadrature	I-Algorithm
are	O
probably	O
more	O
suitable	O
.	O
</s>
<s>
There	O
are	O
two	O
classes	O
of	O
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
quadrature	I-Algorithm
:	O
they	O
are	O
called	O
"	O
closed	O
"	O
when	O
and	O
,	O
i.e.	O
</s>
<s>
A	O
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
formula	I-Algorithm
of	O
any	O
degree	O
can	O
be	O
constructed	O
.	O
</s>
<s>
Methods	O
such	O
as	O
Gaussian	B-Algorithm
quadrature	I-Algorithm
and	O
Clenshaw	B-Algorithm
–	I-Algorithm
Curtis	I-Algorithm
quadrature	I-Algorithm
with	O
unequally	O
spaced	O
points	O
(	O
clustered	O
at	O
the	O
endpoints	O
of	O
the	O
integration	O
interval	O
)	O
are	O
stable	O
and	O
much	O
more	O
accurate	O
,	O
and	O
are	O
normally	O
preferred	O
to	O
Newton	O
–	O
Cotes	O
.	O
</s>
<s>
Alternatively	O
,	O
stable	O
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
formulas	I-Algorithm
can	O
be	O
constructed	O
using	O
least-squares	O
approximation	O
instead	O
of	O
interpolation	B-Algorithm
.	O
</s>
<s>
This	O
allows	O
building	O
numerically	B-General_Concept
stable	O
formulas	O
even	O
for	O
high	O
degrees	O
.	O
</s>
<s>
This	O
table	O
lists	O
some	O
of	O
the	O
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
formulas	I-Algorithm
of	O
the	O
closed	O
type	O
.	O
</s>
<s>
Boole	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
is	O
sometimes	O
mistakenly	O
called	O
Bode	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
,	O
as	O
a	O
result	O
of	O
the	O
propagation	O
of	O
a	O
typographical	O
error	O
in	O
Abramowitz	O
and	O
Stegun	O
,	O
an	O
early	O
reference	O
book	O
.	O
</s>
<s>
This	O
table	O
lists	O
some	O
of	O
the	O
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
formulas	I-Algorithm
of	O
the	O
open	O
type	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
one	O
usually	O
performs	O
numerical	B-Algorithm
integration	I-Algorithm
by	O
splitting	O
into	O
smaller	O
subintervals	O
,	O
applying	O
a	O
Newton	O
–	O
Cotes	O
rule	O
on	O
each	O
subinterval	O
,	O
and	O
adding	O
up	O
the	O
results	O
.	O
</s>
<s>
See	O
Numerical	B-Algorithm
integration	I-Algorithm
.	O
</s>
