7.2 Composite Trapezoidal and Simpson s Rule

Transcription

1 364 CHAP. 7 NUMERICAL INTEGRATION 7. Composite Trpezoidl nd Simpson s Rule An intuitive method of finding the re under the curve y = f (x) over [, b] is by pproximting tht re with series of trpezoids tht lie bove the intervls {[x k, x k+1 ]}. Theorem 7. (Composite Trpezoidl Rule). Suppose tht the intervl [, b] is subdivided into M subintervls [x k, x k+1 ] of width h = (b )/M by using the eqully spced nodes x k = + kh,fork = 0, 1,..., M. The composite trpezoidl rule for M subintervls cn be expressed in ny of three equivlent wys: (1) T ( f, h) = h or ( f (x k 1 ) + f (x k )) (1b) T ( f, h) = h ( f 0 + f 1 + f + f f M + f M 1 + f M ) or (1c) T ( f, h) = h M 1 ( f () + f (b)) + h f (x k ). This is n pproximtion to the integrl of f (x) over [, b], nd we write () f (x) dx T ( f, h). Proof. Apply the trpezoidl rule over ech subintervl [x k 1, x k ] (see Figure 7.6). Use the dditive property of the integrl for subintervls: (3) f (x) dx = xk x k 1 f (x) dx h ( f (x k 1) + f (x k )). Since h/ is constnt, the distributive lw of ddition cn be pplied to obtin (1). Formul (1b) is the expnded version of (1). Formul (1c) shows how to group ll the intermedite terms in (1b) tht re multiplied by. Approximting f (x) = + sin( x) with piecewise liner polynomils results in plces where the pproximtion is close nd plces where it is not. To chieve ccurcy, the composite trpezoidl rule must be pplied with mny subintervls. In the next exmple we hve chosen to integrte this function numericlly over the intervl [1, 6]. Investigtion of the integrl over [0, 1] is left s n exercise.

2 SEC. 7. COMPOSITE TRAPEZOIDAL AND SIMPSON S RULE y y = f(x) x Figure 7.6 Approximting the re under the curve y = + sin( x) with the composite trpezoidl rule. Exmple 7.5. Consider f (x) = + sin( x). Use the composite trpezoidl rule with 11 smple points to compute n pproximtion to the integrl of f (x) tken over [1, 6]. To generte 11 smple points, we use M = 10 nd h = (6 1)/10 = 1/. Using formul (1c), the computtion is T ( f, 1 ) = 1/ ( f (1) + f (6)) + 1 ( f ( 3 ) + f () + f ( 5 ) + f (3) + f ( 7 ) + f (4) + f ( 9 11 ) + f (5) + f ( )) = 1 ( ) ( ) = 1 4 ( ) + 1 ( ) = =

3 SEC. 7. COMPOSITE TRAPEZOIDAL AND SIMPSON S RULE 367 Error Anlysis The significnce of the next two results is to understnd tht the error terms E T ( f, h) nd E S ( f, h) for the composite trpezoidl rule nd composite Simpson rule re of the order O(h ) nd O(h 4 ), respectively. This shows tht the error for Simpson s rule converges to zero fster thn the error for the trpezoidl rule s the step size h decreses to zero. In cses where the derivtives of f (x) re known, the formuls E T ( f, h) = (b ) f () (c)h 1 nd E S ( f, h) = (b ) f (4) (c)h 4 cn be used to estimte the number of subintervls required to chieve specified ccurcy. Corollry 7. (Trpezoidl Rule: Error Anlysis). Suppose tht [, b] is subdivided into M subintervls [x k, x k+1 ] of width h = (b )/M. The composite trpezoidl rule (7) T ( f, h) = h M 1 ( f () + f (b)) + h is n pproximtion to the integrl f (x k ) 180 (8) f (x) dx = T ( f, h) + E T ( f, h).

4 368 CHAP. 7 NUMERICAL INTEGRATION Furthermore, if f C [, b], there exists vlue c with < c < b so tht the error term E T ( f, h) hs the form (9) E T ( f, h) = (b ) f () (c)h = O(h ). 1 Proof. We first determine the error term when the rule is pplied over [x 0, x 1 ]. Integrting the Lgrnge polynomil P 1 (x) nd its reminder yields (x x 0 )(x x 1 ) f () (c(x)) (10) f (x) dx = P 1 (x) dx + dx. x 0 x 0 x 0! The term (x x 0 )(x x 1 ) does not chnge sign on [x 0, x 1 ], nd f () (c(x)) is continuous. Hence the second men vlue theorem for integrls implies tht there exists vlue c 1 so tht (11) f (x) dx = h x 0 ( f 0 + f 1 ) + f () (x x 0 )(x x 1 ) (c 1 ) dx. x 0! Use the chnge of vrible x = x 0 + ht in the integrl on the right side of (11): f (x) dx = h x 0 ( f 0 + f 1 ) + f () (c 1 ) h(t 0)h(t 1)hdt (1) = h ( f 0 + f 1 ) + f () (c 1 )h 3 0 (t t) dt = h ( f 0 + f 1 ) f () (c 1 )h 3. 1 Now we re redy to dd up the error terms for ll of the intervls [x k, x k+1 ]: (13) f (x) dx = = xk x k 1 f (x) dx h ( f (x k 1) + f (x k )) h3 1 f () (c k ). The first sum is the composite trpezoidl rule T ( f, h). In the second term, one fctor of h is replced with its equivlent h = (b )/M, nd the result is ( ) b (b )h 1 f (x) dx = T ( f, h) f () (c k ). 1 M The term in prentheses cn be recognized s n verge of vlues for the second derivtive nd hence is replced by f () (c). Therefore, we hve estblished tht f (x) dx = T ( f, h) (b ) f () (c)h, 1 nd the proof of Corollry 7. is complete.

5 SEC. 7. COMPOSITE TRAPEZOIDAL AND SIMPSON S RULE 369 Exmple 7.7. Consider f (x) = + sin( x). Investigte the error when the composite trpezoidl rule is used over [1, 6] nd the number of subintervls is 10, 0, 40, 80, nd 160. Tble 7. shows the pproximtions T ( f, h). The ntiderivtive of f (x) is nd the true vlue of the definite integrl is F(x) = x x cos( x) + sin( x), 6 1 f (x) dx = F(x) x=6 = x=1 This vlue ws used to compute the vlues E T ( f, h) = T ( f, h) in Tble 7.. It is importnt to observe tht when h is reduced by fctor of 1 the successive errors E T ( f, h) re diminished by pproximtely 1 4. This confirms tht the order is O(h ).

6 370 CHAP. 7 NUMERICAL INTEGRATION Tble 7. Composite Trpezoidl Rule for f (x) = + sin( x) over [1, 6] M h T ( f, h) E T ( f, h) = O(h ) Exmple 7.9. Find the number M nd the step size h so tht the error E T ( f, h) for the composite trpezoidl rule is less thn for the pproximtion 7 dx/x T ( f, h). The integrnd is f (x) = 1/x nd its first two derivtives re f (x) = 1/x nd f () (x) = /x 3. The mximum vlue of f () (x) tken over [, 7] occurs t the endpoint x =, nd thus we hve the bound f () (c) f () () = 1 4,for c 7. This is used with formul (9) to obtin (17) E T ( f, h) = (b ) f () (c)h 1 (7 ) 1 4 h 1 = 5h 48. The step size h nd number M stisfy the reltion h = 5/M, nd this is used in (17) to get the reltion (18) E T ( f, h) 15 48M Now rewrite (18) so tht it is esier to solve for M: 5 (19) M. Solving (19), we find tht M. Since M must be n integer, we choose M =,8, nd the corresponding step size is h = 5/,8 = When the composite trpezoidl rule is implemented with this mny function evlutions, there is

7 SEC. 7. COMPOSITE TRAPEZOIDAL AND SIMPSON S RULE 371 possibility tht the rounded-off function evlutions will produce significnt mount of error. When the computtion ws performed, the result ws ( ) 5 T f, = ,,8 which compres fvorbly with the true vlue 7 dx/x = ln(x) x=7 x= = The error is smller thn predicted becuse the bound 1 4 for f () (c) ws used. Experimenttion shows tht it tkes bout 10,001 function evlutions to chieve the desired ccurcy of , nd when the clcultion is performed with M = 10,000, the result is T ( f, 5 10,000 ) = The composite trpezoidl rule usully requires lrge number of function evlutions to chieve n ccurte nswer. This is contrsted in the next exmple with Simpson s rule, which will require significntly fewer evlutions.

8 Numericl Methods Using Mtlb, 4 th Edition, 004 John H. Mthews nd Kurtis K. Fink ISBN: Prentice-Hll Inc. Upper Sddle River, New Jersey, USA