We consider the problem of computing approximations to gener

We consider the problem of computing approximations to general integrals where a b and f a, b H R is an integrable function. 1. Write a Matlab program that implements the version of the composite trapezoidal rule pre- sented in class. Note that this version is designed to use as few function evaluations as possible and that it produces a conservative error estimate Ei along with the final approximation T to I. The inputs for your program should be The integration limits a and b A Matlab function to compute f(ar) for any r E a, b An error tolerance tol; A safeguard jmax to limit the iteration index j The output for your program should be: The final approximation Ti to I

Solution

1-a = x0 < x1 < · · · < xn = b

The composite trapezoidal rule is obtained by applying the trapezoidal rule in each subinterval [xi1, xi ], i = 1, . . . , n, i.e., Z b a f(x)dx = Xn i=1 Z xi xi1 f(x)dx 1 2 Xn i=1 (xi xi1)[f(xi1) + f(xi)],

A particular case is when these points are uniformly spaced, i.e., when all intervals have an equal length. For example, if xi = a + ih, where h = b a n

then Z b a f(x)dx h 2 \" f(a) + 2Xn1 i=1 f(a + ih) + f(b) # = h Xn i=0 00f(a + ih). (6.13) The notation of a sum with two primes, P00, means that we sum over all the terms with the exception of the first and last terms that are being divided by 2. We can also compute the error term as a function of the distance between neighboring points, h. We know from (6.11) that in every subinterval the quadrature error is h 3 12 f 00(x).

Hence, the overall error is obtained by summing over n such terms: Xn i=1 h 3 12 f 00(i) = h 3n 12 \" 1 n Xn i=1 f 00(i) # . Here, we use the notation i to denote an intermediate point that belongs to the i th interval. Let M = 1 n Xn i=1 f 00(i). Clearly min x[a,b] f 00(x) 6 M 6 max x[a,b] f 00(x) If we assume that f 00(x) is continuous in [a, b] (which we anyhow do in order for the interpolation error formula to be valid) then there exists a point [a, b] such that f 00() = M. Hence (recalling that (b a)/n = h, we have E = (b a)h 2 12 f 00(), [a, b]. (6.14) This means that the composite trapezoidal rule is second-order accurate.

Then use this function as follows: