Develop an M-file function to implement adaptive quadrature based on Fig. 20.6. Test the f...

Develop an M-file function to implement adaptive quadrature based on Fig. 20.6. Test the function by using it to determine the integral of the polynomial from Example 20.1. Then use it to solve Prob. 20.20.

Fig. 20.6:

Example 20.1: Richardson Extrapolation

Problem Statement. Use Richardson extrapolation to evaluate the integral of f (x) = 0.2 + 25x − 200x² + 675x³ − 900x⁴ + 400x⁵ from a = 0 to b = 0.8.

Solution. Single and composite applications of the trapezoidal rule can be used to evaluate the integral:

Richardson extrapolation can be used to combine these results to obtain improved estimates of the integral. For example, the estimates for one and two segments can be combined to yield

The error of the improved integral is E_t = 1.640533 − 1.367467 = 0.273067(ε_t = 16.6%), which is superior to the estimates upon which it was based.

In the same manner, the estimates for two and four segments can be combined to give

which represents an error of E_t = 1.640533 − 1.623467 = 0.017067 (ε_t = 1.0%).