Romberg Method
Romberg method is a powerful tool which uses the method of extrapolation for the trapezium rule.
Romberg method for the trapezium rule :
Let the integral
be computed by the composite trapezium rule. Let I denote the exact value of the integral and IT denote the value obtained by the composite trapezium rule.
The error, I – IT , in the composite trapezium rule in computing the integral is given by
or
.........................1.1
where c1, c2, c3, ... are independent of h.
To illustrate the extrapolation procedure, first consider two error terms
.......................1.2
Let I be evaluated using two step lengths h and qh, 0 < q < 1. Let these values be denoted by IT (h) and IT(qh). The error equations become
........................1.3
.....................1.4
From (1.3), we obtain
..........................1.5
From (1.4), we obtain
........................1.6
Multiply (1.5) by q2 to obtain
...........................1.7
Eliminating c1q2h2 from (1.6) and (1.7), we obtain
Solving for I, we obtain
Note that the error term on the right hand side is now of order O(h4).