Romberg's method
numerical integration method
In numerical analysis, Romberg's method is used to estimate the definite integral
∫
a
b
f
(
x
)
d
x
{\displaystyle \int _{a}^{b}f(x)\,dx}
by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangular array. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spaced points. The integrand must have continuous derivatives, though fairly good results may be obtained if only a few derivatives exist. If it is possible to evaluate the integrand at unequally spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate.
The method is named after Werner Romberg, who published the method in 1955.
Method
Using
h
n
=
(
b
−
a
)
2
n
+
1
{\textstyle h_{n}={\frac {(b-a)}{2^{n+1}}}}
, the method can be inductively defined by
R
(
0
,
0
)
=
h
0
(
f
(
a
)
+
f
(
b
)
)
R
(
n
,
0
)
=
1
2
R
(
n
−
1
,
0
)
+
2
h
n
∑
k
=
1
2
n
−
1
f
(
a
+
(
2
k
−
1
)
h
n
−
1
)
R
(
n
,
m
)
=
R
(
n
,
m
−
1
)
+
1
4
m
−
1
(
R
(
n
,
m
−
1
)
−
R
(
n
−
1
,
m
−
1
)
)
=
1
4
m
−
1
(
4
m
R
(
n
,
m
−
1
)
−
R
(
n
−
1
,
m
−
1
)
)
{\displaystyle {\begin{aligned}R(0,0)&=h_{0}(f(a)+f(b))\\R(n,0)&={\tfrac {1}{2}}R(n{-}1,\,0)+2h_{n}\sum _{k=1}^{2^{n-1}}f(a+(2k-1)h_{n-1})\\R(n,m)&=R(n,\,m{-}1)+{\tfrac {1}{4^{m}-1}}(R(n,\,m{-}1)-R(n{-}1,\,m{-}1))\\&={\frac {1}{4^{m}-1}}(4^{m}R(n,\,m{-}1)-R(n{-}1,\,m{-}1))\end{aligned}}}
where
n
≥
m
{\displaystyle n\geq m}
and
m
≥
1
{\displaystyle m\geq 1\,}
. Observe that the first two equations correspond to the first column, and these are the formulas associated to the compound trapezoidal rule. In big O notation, the error for R(n, m) is:
O
(
h
n
2
m
+
2
)
.
{\displaystyle O{\left(h_{n}^{2m+2}\right)}.}
The zeroth extrapolation, R(n, 0), is equivalent to the trapezoidal rule with 2n + 1 points; the first extrapolation, R(n, 1), is equivalent to Simpson's rule with 2n + 1 points.
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