Question 4) Romberg Integration:
Calculate the integral 2 [ sin(1) dx
with the help of Romberg integration: The trapezoid rule is used here with different interval widths hi and the result is extrapolated to an interval width of 0.
As the step size, take h; = 2^(+1) T, j=0, 1,2
Then we get:
pio = Ihi [(f) . Pi+lk-1(0) Pik-1(0) (ii) pik (0) = Pi+lk-1(0) + 22k
For the efficient calculation of Ihi(") we use the relation: 1(1) = 1/4(f) + 2^(h/3)(f)^2
between midpoint rule // (0) (f) and trapezoidal rule /,( (f):
Since the results of the extrapolation have an increasing order of error with an increasing number of support points, the comparison of two successive results (the best results with a given number of support points are again found as po) an a posteriori error estimate. Calculate the integral with an error less than 10^-4