<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
Romberg	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
used	O
to	O
estimate	O
the	O
definite	O
integral	O
by	O
applying	O
Richardson	B-Algorithm
extrapolation	I-Algorithm
repeatedly	O
on	O
the	O
trapezium	B-Algorithm
rule	I-Algorithm
or	O
the	O
rectangle	O
rule	O
(	O
midpoint	O
rule	O
)	O
.	O
</s>
<s>
Romberg	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
a	O
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
formula	I-Algorithm
–	O
it	O
evaluates	O
the	O
integrand	O
at	O
equally	O
spaced	O
points	O
.	O
</s>
<s>
If	O
it	O
is	O
possible	O
to	O
evaluate	O
the	O
integrand	O
at	O
unequally	O
spaced	O
points	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
generally	O
more	O
accurate	O
.	O
</s>
<s>
The	O
zeroeth	O
extrapolation	O
,	O
,	O
is	O
equivalent	O
to	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
with	O
points	O
;	O
the	O
first	O
extrapolation	O
,	O
,	O
is	O
equivalent	O
to	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
with	O
points	O
.	O
</s>
<s>
The	O
second	O
extrapolation	O
,	O
,	O
is	O
equivalent	O
to	O
Boole	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
with	O
points	O
.	O
</s>
<s>
The	O
further	O
extrapolations	O
differ	O
from	O
Newton-Cotes	B-Algorithm
formulas	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
further	O
Romberg	O
extrapolations	O
expand	O
on	O
Boole	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
in	O
very	O
slight	O
ways	O
,	O
modifying	O
weights	O
into	O
ratios	O
similar	O
as	O
in	O
Boole	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
In	O
contrast	O
,	O
further	O
Newton-Cotes	B-Algorithm
methods	O
produce	O
increasingly	O
differing	O
weights	O
,	O
eventually	O
leading	O
to	O
large	O
positive	O
and	O
negative	O
weights	O
.	O
</s>
<s>
This	O
is	O
indicative	O
of	O
how	O
large	O
degree	O
interpolating	O
polynomial	O
Newton-Cotes	B-Algorithm
methods	O
fail	O
to	O
converge	O
for	O
many	O
integrals	O
,	O
while	O
Romberg	B-Algorithm
integration	I-Algorithm
is	O
more	O
stable	O
.	O
</s>
<s>
By	O
labelling	O
our	O
approximations	O
as	O
instead	O
of	O
,	O
we	O
can	O
perform	O
Richardson	B-Algorithm
extrapolation	I-Algorithm
with	O
the	O
error	O
formula	O
defined	O
below	O
:	O
</s>
<s>
To	O
estimate	O
the	O
area	O
under	O
a	O
curve	O
the	O
trapezoid	B-Algorithm
rule	I-Algorithm
is	O
applied	O
first	O
to	O
one-piece	O
,	O
then	O
two	O
,	O
then	O
four	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
After	O
trapezoid	B-Algorithm
rule	I-Algorithm
estimates	O
are	O
obtained	O
,	O
Richardson	B-Algorithm
extrapolation	I-Algorithm
is	O
applied	O
.	O
</s>
<s>
obtained	O
by	O
the	O
trapezium	B-Algorithm
rule	I-Algorithm
in	O
the	O
first	O
column	O
of	O
the	O
triangular	O
array	O
.	O
</s>
<s>
Here	O
is	O
an	O
example	O
of	O
a	O
computer	O
implementation	O
of	O
the	O
Romberg	B-Algorithm
method	I-Algorithm
(	O
in	O
the	O
C	B-Language
programming	I-Language
language	I-Language
)	O
:	O
</s>
<s>
Here	O
is	O
an	O
example	O
of	O
a	O
computer	O
implementation	O
of	O
the	O
Romberg	B-Algorithm
method	I-Algorithm
in	O
the	O
Javascript	B-Language
programming	I-Language
language	I-Language
.	O
</s>
