18.5 Applying Richardson Extrapolation to the Numerical Integration Methods

18.5 Applying Richardson Extrapolation to the Numerical Integration Methods#

Note that all of the numerical integration methods we have covered (the midpoint, trapezoidal and Simpson’s rules) have errors of the form:

\[ I(f) - I_n(f) = a h^k \]

given a sufficiently smooth \(f\). Thus, we can apply at least 1 iteration of Richardson extrapolation to these methods.

Note

Remember, for a numerical method of the form:

\[ A = A_0(h) + a_0 h^{k_0} + a_1 h^{k_1} + ... \]

with \(0 < h < 1\) and \(k_0 < k_1 < k_2 < ...\). Richardson extrapolation can be appied recursively to imprive the approximation:

\[ A_{i+1} (h) = \frac{t^{k_i} A_i \left( \tfrac{h}{t} \right) - A_i(h)}{t^{k_i} - 1} \]

with leading error of \(O(h^{k_{i + 1}})\). Error can be estimated as:

\[ E_{i+1}(h) = A_{i+1}(h) - A_i \left( \tfrac{h}{t} \right) \]

We will use a slightly different notation for this:

\[ A_i(h) = I_{n}^{[i]}(f) \]

If we choose \(t=2\), then we effectively double the number of sub-intervals used for the integration method:

\[ A_i\left(\tfrac{h}{t}\right) = A_i\left(\tfrac{h}{2}\right) = I_{2n}^{[i]}(f) \]

Using this, the first iteration of the extrapolated method is:

(26)#\[ I_n^{[1]}(f) = \frac{2^k I_{2n}(f) - I_n}{2^k - 1} \]

and the error can be estimated as:

(27)#\[ I(f) - I_n^{[1]}(f) \approx I_n^{[1]}(f) - I_{2n}(f) \]

To apply this to the methods, note that: