Question

5. Let f a, b R be a 4 times continuously differentiable function. For n even, consider < tn = b, a to < t< an uniform partition of [a, b] with b- a , i = 0,1,.. , n - 1 h t Let T denote the composite Trapezoidal rule associated with the above partition which approx imates eliminate the term containing h2 in the asymptotic expansion. Interprete the result which you obtain as an appropriate numerical quadrature rule f(x)d. Consider the first step of Romberg integration based on T and Ta so as to (10

Answer 1

Richardson Extrapolation is mainly a simple method for boosting the accuracy of certain numerical procedures. This one does the elimination of errors. Though it can only remove errors of the form:

Derivation of the General Form of Richardson Extrapolation

This part will show how the error is being removed through Richardson Extrapolation.

Assumptions:

1. We have an approximate means of computing some quantity G
2. The result depends on a parameter ‘h’ such that the approximation by g(h) is given by: G = g(h) + E(h)

Derivation Proper:

Romberg Integration

Now that we have established the pre-requisites, let’s go to the main thing.

Romberg Integration

Romberg intergation combines the Composite Trapezoidal Rule with Richardson Extrapolation.

Below is the overview of the integration process:

$1*kX_ZlniWDtQ3Y0YUN2gkJw.png$

this is exactly what we are doing

This tells us that we need to compute where the two arrows are from to compute where the two arrows are pointing at. The most accurate estimate of the integral is always the last diagonal term of the array. This process is continued until the difference between two successive diagonal terms becomes sufficiently small.

The overview above can be summarized into the formula: