2.3 Error Analysis
2.3 Error Analysis
Corollary 2. 2(Trapezoidal Rule: Error Analysis). Suppose that [a is subdivided into M subintervals [ ack, kl of width h=(6-a)/M. The composite trapezoidal rule (b)=(f()+f6)+b∑f(ak is an approximation to the integral f( dar=T(, h)+Er(, (2.30) Furthermore, if fEC a, b, there exists a value c with a c< b so that the error term Er(, h)h nas th ne form Er(, h)= (b-a)f2(c) 12 O(h
Corollary 2.3(Simpson,s Rule: Error Analysis). Suppose that a, b is subdivided into 2M subintervals [ ok, k+1 of equal width h=(b-a)/(2M) The composite Simpson rule S(b)=(fa)+f6)+3∑f(2)+∑f(a2-1)(236 is an approximation to the integral f( d r= s(, h)+Es(f, h) (2.37) Furthermore, if f E C4(a, 61, there exists a value c with a <c<b so that the) error term Es(f, h)ha as the form Es(,h)= a)r(4 ()h O(h (2.38) 180
Example 2. 7. Consider f(a)=2+sin(23). 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(2v a)over 1,6 T(, h) Er(, h)=O(h2) 10 0.58.19385457 0.01037540 200.25818604926 0.00257006 400.125818412019 0.00064098 800.06258.18363936 0.00016015 1600.031258.18351924 0.00004003
Example 2.8. Consider f(a)=2 +sin (2v). 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) over [1, 6 s(, h) Es(f, h)=O(h4 0.5 8.18301549 0.00046371 100.258.18344750 0.00003171 200.1258.18347717 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 error Er(, h) for the composite trapezoidal rule is less than 5 x 10- for the ap- proximation S2 d /a T(, h)
Example 2. 10. Find the number M and the step size h so that the er- ror Es(, h) for the composite Simpson rule is less than 5 x 10-g for the approximation 52 dx/aa S(, 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 M is the number of subintervals %Output -s is the tra pezoidal rule sum h=(b-a)/M: 0: for k=1: (M-1) x=a+h*k s=s+feva(f,×) en s=h*(feval(f, a)+feval(, b)/2+h*s
