# The Romberg Method

## Contents

# The Romberg Method#

The Romberg method generates a triangular array of approximations \(R_{i,j}, j \leq i\) of an integral \(I = \int_a^b f(x)\; dx\) with the end-of-row values \(R_{0,0}, R_{1,1}, \dots R_{n,n}\) being the main, successively better (usually) approximations.

It starts with the trapezoid rule \(R_{0,0} := T_1 = \frac{f(a) + f(b)}{2}(b-a)\); the basic trapezoid rule.

Then using \(T_{2n} = \frac{T_n + M_n}{2}\), one defines

Finally, Richardson extrapolation leads to

and with further extrapolations to the more general formula

## Exercise#

Implement this, using these three formulas plus the function for the composite midpoint rule in section Summation and Integration

One natural data structure is a 2D array with unused entries above the main diagonal. However, you might consider how to store this triangular collection of data as a list of lists, succesively of lengths 1, 2 and so on up to \(n\).