2. 3 Error Analysis
2.3 Error Analysis
Corollary 2.2 (Trapezoidal Rule: Error Analysis). Suppose that [a, b] s subdivided into M subintervals [ak, k+1] of width h (b-a)/M.The composite trapezoidal rule h M-1 T(f, h)= (f(a)+f(b)+h ∑f(xk) (2.29) k=1 is an approximation to the integral b (x)dx T(f, h)+ Er(f, h). (2.30) Furthermore,iff∈c2la,b, there exists value with<c< so that the ,, error term Er(f, h) has the form b-a)f(2)(ch2-o(h2 12 (2.31)
Corollary 2.3(Simpson's Rule: Error Analysis ) Suppose that a, b is subdivided into 2M subintervals [ ak, k+1 of equal width h=(b-a)/(2M The composite Simpson rule S(f,b)=f()+)+∑fa2)+a∑f(a-1)(236) k=1 k=1 is an approximation to the integral A f(aydr=S(f, h)+Es(f, h) (2.37) Furthermore, if f e C a, b], there exists a value c with a c< b so that the error term Es (f, h)has the form b-af(4)(c)h O(h 2.38
Example 2.7. Consider f(a)=2+sin(2v). Investigate the error then the composite trapezoidal rule is used over[1,6 and the number of subintervals is10.20.40.80.and160
Table 2.2. The Composite Trapezoidal rule for f(a)=2+sin(2va)over [1, 6 I h T(, h) Er(f, h)=O(h2) 10 0.58.19385457 0.01037540 200.258.18604926 0.00257006 400.1258.18412019 0.00064098 800.0625818363936 0.00016015 1600.031258.18351924 0.00004003
Example 2.8. Consider f(a)=2+sin(2). Investigate the error when the composite Simpson rule is used over [1,6 and the number of subintervals is10.20.40.80.and160
Table 2.3 The Composite Simpson rule for f(a)=2+sin(2v a) over [1, 6 M s(,h)Es(f, h)=O(h) 5 0.5 8.18301549 0.00046371 100.25818344750 0.00003171 20 ( 0.125818347717 0.00000204 400.06258.18347908 0.00000013 800.031258.18347920 0.00000001
Example 2.9. Find the number M and the step size h so that the e error Er(, h) for the composite trapezoidal rule is less than 5 x 10- for the ap- proximation2d/x≈r(,b
Example 2.10. Find the number M and the step size h so that the er- ror Es(f, h) for the composite Simpson rule is less than 5 x 10- for r the approximation S2 d. c/a a s(f, h)
Function s=traprl(f, a, b, M oInput -f is the integrand input as a string f a and b are upper and lower limits of integration 0 M is the number of subintervals Output -s is the trapezoidal rule sum a 0: for k=1: M-1) X-a +h*k; s=S+feval(f, x) en s=h* (feval(f, a)+feval(f, b)/2+h*s:
