<s>
In	O
numerical	B-Algorithm
integration	I-Algorithm
,	O
Simpson	B-Algorithm
's	I-Algorithm
rules	I-Algorithm
are	O
several	O
approximations	O
for	O
definite	B-Algorithm
integrals	I-Algorithm
,	O
named	O
after	O
Thomas	O
Simpson	O
(	O
1710	O
–	O
1761	O
)	O
.	O
</s>
<s>
Simpson	O
's	O
1/3	O
and	O
3/8	O
rules	O
are	O
two	O
special	O
cases	O
of	O
closed	O
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
formulas	I-Algorithm
.	O
</s>
<s>
In	O
naval	O
architecture	O
and	O
ship	O
stability	O
estimation	O
,	O
there	O
also	O
exists	O
Simpson	O
's	O
third	O
rule	O
,	O
which	O
has	O
no	O
special	O
importance	O
in	O
general	O
numerical	O
analysis	O
,	O
see	O
Simpson	B-Algorithm
's	I-Algorithm
rules	I-Algorithm
(	O
ship	O
stability	O
)	O
.	O
</s>
<s>
Simpson	O
's	O
1/3	O
rule	O
,	O
also	O
simply	O
called	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
,	O
is	O
a	O
method	O
for	O
numerical	B-Algorithm
integration	I-Algorithm
proposed	O
by	O
Thomas	O
Simpson	O
.	O
</s>
<s>
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
gains	O
an	O
extra	O
order	O
because	O
the	O
points	O
at	O
which	O
the	O
integrand	O
is	O
evaluated	O
are	O
distributed	O
symmetrically	O
in	O
the	O
interval	O
.	O
</s>
<s>
Since	O
the	O
error	O
term	O
is	O
proportional	O
to	O
the	O
fourth	O
derivative	O
of	O
at	O
,	O
this	O
shows	O
that	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
provides	O
exact	O
results	O
for	O
any	O
polynomial	O
of	O
degree	O
three	O
or	O
less	O
,	O
since	O
the	O
fourth	O
derivative	O
of	O
such	O
a	O
polynomial	O
is	O
zero	O
at	O
all	O
points	O
.	O
</s>
<s>
Because	O
of	O
the	O
factor	O
,	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
is	O
also	O
referred	O
to	O
as	O
"	O
Simpson	O
's	O
1/3	O
rule	O
"	O
(	O
see	O
below	O
for	O
generalization	O
)	O
.	O
</s>
<s>
This	O
weighted	O
average	O
is	O
exactly	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
Using	O
another	O
approximation	O
(	O
for	O
example	O
,	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
with	O
twice	O
as	O
many	O
points	O
)	O
,	O
it	O
is	O
possible	O
to	O
take	O
a	O
suitable	O
weighted	O
average	O
and	O
eliminate	O
another	O
error	O
term	O
.	O
</s>
<s>
This	O
is	O
Romberg	B-Algorithm
's	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
This	O
yields	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
If	O
the	O
interval	O
of	O
integration	O
is	O
in	O
some	O
sense	O
"	O
small	O
"	O
,	O
then	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
with	O
subintervals	O
will	O
provide	O
an	O
adequate	O
approximation	O
to	O
the	O
exact	O
integral	O
.	O
</s>
<s>
For	O
such	O
a	O
function	O
,	O
a	O
smooth	O
quadratic	O
interpolant	O
like	O
the	O
one	O
used	O
in	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
will	O
give	O
good	O
results	O
.	O
</s>
<s>
In	O
these	O
cases	O
,	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
may	O
give	O
very	O
poor	O
results	O
.	O
</s>
<s>
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
is	O
then	O
applied	O
to	O
each	O
subinterval	O
,	O
with	O
the	O
results	O
being	O
summed	O
to	O
produce	O
an	O
approximation	O
for	O
the	O
integral	O
over	O
the	O
entire	O
interval	O
.	O
</s>
<s>
This	O
sort	O
of	O
approach	O
is	O
termed	O
the	O
composite	O
Simpson	O
's	O
1/3	O
rule	O
,	O
or	O
just	O
composite	B-Algorithm
Simpson	I-Algorithm
's	I-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
This	O
composite	O
rule	O
with	O
corresponds	O
with	O
the	O
regular	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
of	O
the	O
preceding	O
section	O
.	O
</s>
<s>
This	O
leads	O
to	O
the	O
adaptive	B-Algorithm
Simpson	I-Algorithm
's	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Simpson	O
's	O
3/8	O
rule	O
,	O
also	O
called	O
Simpson	O
's	O
second	O
rule	O
,	O
is	O
another	O
method	O
for	O
numerical	B-Algorithm
integration	I-Algorithm
proposed	O
by	O
Thomas	O
Simpson	O
.	O
</s>
<s>
A	O
further	O
generalization	O
of	O
this	O
concept	O
for	O
interpolation	O
with	O
arbitrary-degree	O
polynomials	O
are	O
the	O
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
formulas	I-Algorithm
.	O
</s>
<s>
In	O
the	O
task	O
of	O
estimation	O
of	O
full	O
area	O
of	O
narrow	O
peak-like	O
functions	O
,	O
Simpson	B-Algorithm
's	I-Algorithm
rules	I-Algorithm
are	O
much	O
less	O
efficient	O
than	O
trapezoidal	B-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
Namely	O
,	O
composite	O
Simpson	O
's	O
1/3	O
rule	O
requires	O
1.8	O
times	O
more	O
points	O
to	O
achieve	O
the	O
same	O
accuracy	O
as	O
trapezoidal	B-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
Integration	O
by	O
Simpson	O
's	O
1/3	O
rule	O
can	O
be	O
represented	O
as	O
a	O
weighted	O
average	O
with	O
2/3	O
of	O
the	O
value	O
coming	O
from	O
integration	O
by	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
with	O
step	O
h	O
and	O
1/3	O
of	O
the	O
value	O
coming	O
from	O
integration	O
by	O
the	O
rectangle	O
rule	O
with	O
step	O
2h	O
.	O
</s>
<s>
These	O
rules	O
are	O
very	O
much	O
similar	O
to	O
the	O
alternative	O
extended	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
These	O
two	O
rules	O
can	O
be	O
associated	O
with	O
Euler	B-Algorithm
–	I-Algorithm
MacLaurin	I-Algorithm
formula	I-Algorithm
with	O
the	O
first	O
derivative	O
term	O
and	O
named	O
First	O
order	O
Euler	O
–	O
MacLaurin	O
integration	O
rules	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
generate	O
higher	O
order	O
Euler	O
–	O
Maclaurin	O
rules	O
by	O
adding	O
a	O
difference	O
of	O
3rd	O
,	O
5th	O
,	O
and	O
so	O
on	O
derivatives	O
with	O
coefficients	O
,	O
as	O
defined	O
by	O
Euler	B-Algorithm
–	I-Algorithm
MacLaurin	I-Algorithm
formula	I-Algorithm
.	O
</s>
<s>
Simpson	B-Algorithm
rule	I-Algorithm
for	O
irregularly	O
spaced	O
data	O
.	O
</s>
