Richardson Extrapolation Method

Use case

It is an Error correction technique that improves the result of the an Integration by improving the integral itself. (As opposed to doing so through iterations)

Derivation

An integral $I$ can be written as

$$I=I(h)+E(h)$$

Where $I(h)=$ Numerical approximation, $E(h)=$ the truncation error.

Suppose we make to estimates step sizes $h_1$ and $h_2$ and we get the same values for the error.

$$I\left(h_1\right)+E\left(h_1\right)=I\left(h_2\right)+E\left(h_2\right)\text{\hspace{1em}(2)}$$

For the Trapezoid Rule, the error is $E\cong -\frac{b-a}{12}{h}^2{\bar{f}}^{\prime \prime }$

Assume $f^{\prime \prime }$ is constant regardless of step size. Then the ratio of the errors is given by.

$$E\left(h_1\right)\cong E\left(h_2\right){\left(\frac{h_1}{h_2}\right)}^2$$

Substitution in eqn. $(2)$