1 Error Analysis of Simple Rules for Numerical Integration

Transcription

1 cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion Lst time we discussed pproximting the definite integrl If) = ft)dt The generl pproch introduced lst time ws to interpolte function f using some polynomil pt), choosing interpoltion points ccording to some rule r nd compute the integrl of the polynomil, pt)dt s the pproximtion. Let X = {t 0, t 1,, t n } e the interpolting node set. We cn write the integrl of pt) using Lgrnge polynomils: I r f) = = = = pt)dt ft 0 )l 0 t) + ft 1 )l 1 t) + + ft n )l n t)) dt n ft i )l i t)dt n ft i ) l i t)dt } {{ } w i Since the Lgrnge polynomils l i t) depend only on the interpoltion points nd not the corresponding function vlues, we cn rewrite this pproximtion s simple weighted sum of function vlues: I r f) = n ft i )w i Lst time we presented four rules tht used this scheme to pproximte definite integrl: Rectngle Rule The rectngle rule uses node set X = {}, the left endpoint of the intervl [, ] to interpolte f [,] using constnt polynomil pt) = f)). The corresponding estimte of the definite integrl is given y: I R = f) ) 1

2 Midpoint Rule The midpoint rule uses node set X = { } +, the midpoint of the intervl [, ] to interpolte f [,] using constnt polynomil pt) = f + )). The corresponding estimte of the definite integrl is given y: Trpezoid Rule ) + I M = f ) The trpezoid rule uses node set X = {, }, the left nd right endpoints of the intervl [, ] to interpolte f [,] using polynomil of degree t most 1 pt) = f) t t + f) ). The corresponding estimte of the definite integrl is given y: Simpson s Rule I T = f) + f)) Simpson s rule uses node set X = {, +, }, the left endpoint, midpoint, nd right endpoint of the intervl [, ] to interpolte f [,] using polynomil of degree t most pt) = t )t ) t )t m) + fm) + f), where m is the midpoint of [, ]). The f) t )t m) ) m) m )m ) ) m) corresponding estimte of the definite integrl is given y: ) ) + I S = f) + 4f + f) 6 In lst lecture s exmple, we estimted ln1.) using the four rules nd otined the following results: 1.1 Error Anlysis I R = 0. I M = I T = I S = Recll tht lst time we showed tht the error of pproximting definite integrl using polynomil interpoltion over T = {t 0, t 1,..., t n } is given y: E r f) = = [ft) pt)] dt f n+1)) c) n + 1)! n t t i ) dt }{{} ωt) We split the error nlysis into two cses:

3 Cse 1: ωt) [,] is lwys nonnegtive, or lwys non-positive In this cse, we cn clculte the error s: E r f) = f n+1) c) n + 1)! ωt)dt The Rectngle nd Trpezoid rule fit this cse, nd lst time we showed tht the error for ech cn e written s: E R f) = f c) ) E T f) = f c) )3 1 Cse : ωt)dt = 0 It is esy to see tht the midpoint rule flls into this cse, since: ω M t)dt = t + ) dt [t + )/] = = 0 nd Simpson s rule ehves similrly. An interesting property of rules tht fll into cse is tht dding nother interpoltion point does not chnge the integrl of the polynomil interpolnt. This is esy to see, since ωt) is the next Newton polynomil nd since its integrl is 0, the weight of the corresponding function vlue w n+1 will e Error Anlysis of Midpoint Rule Since the midpoint rule fits into cse of our error nlysis, tht is: ωt)dt = [t + )/] = 0 s shown in Figure 1, we cn dd n interpoltion point without ffecting the re of the interpolted polynomil, leving the error unchnged. We cn therefore do our error nlysis of the midpoint rule with ny single point dded - since dding ny point in [, ] does not ffect the re, we simply doule the midpoint, so tht X = { + )/, + )/}. We cn now exmine the vlue of the next Newton polynomil, ωt) for the modified rule: ωt) = t + ) t + ) 3

4 0 + )/ Figure 1: ωt) in the Midpoint rule over [, ] Clerly, ωt) [,] 0, so tht this new rule cn e nlyzed using cse 1, this yields: E M f) = f c) = f c) = f c) = f c) t + ) dt ) t ) 3 3 ) 3 4 = f c) )3 4 Note tht this error is constnt fctor of two smller thn the error for the trpezoid rule. 1.3 Error Anlysis of Simpson s Rule Since Simpson s rule lso fits into cse of our error nlysis, tht is: ωt)dt = 0 s shown in Figure, we cn dd n interpoltion point without ffecting the re of the interpolted polynomil, leving the error unchnged. We cn therefore do our error nlysis of Simpson s rule with ny single point dded - since dding ny point in [, ] does not ffect the re, we simply doule the midpoint, so tht our node set X = {, + )/, + )/, }. We cn now exmine the vlue of the next Newton polynomil, ωt) for the modified rule: ) 3 ωt) = t ) t + ) t ) 4

5 0 + )/ Figure : ωt) in Simpson s rule over [, ] Clerly, ωt) [,] 0, so tht this new rule cn e nlyzed using cse 1, this yields: Composite Rules E M f) = f 4) c) 4 t ) = f 4) c) )5 880 t + ) t )dt Notice tht the error formul for ech of the simple rules depends on high power of the size of the intervl, so tht smll intervl mkes for smller error. This motivtes the following generl ide for creting composite rules for numericl integrtion. Step 1 Step Step 3 Prtition the intervl [, ] into N suintervls, equidistnt y defult, with width h = N Apply simple pproximtion rule r to ech suintervl [x i, x i+1 ] nd use the re I r s the pproximtion of the integrl for tht suintervl: xi+1 x i ft)dt I r [xi,x i+1 ]f) Note tht in this ppliction, the ppernce of ech of the piecewise polynomils is unimportnt, we re only interested in their pproximtion of the definite integrl. 5

6 Add up the pproximtion of the re over ech suintervl to otin the pproximtion over the entire intervl [, ]: I [,] f) n 1 I r [xi,x i+1 ]f) Exmple.1. To illustrte, consider pplying the composite rectngle rule to n intervl [, ], s shown in Figure 4. In ech suintervl, the left endpoint gets weight h. Thus every point except the lst one in our prtition hs weight 1; the lst point hs weight 0. This yields the following estimte of the definite integrl: I CR = h f) + h f + h) + h f + h) + + h f h) h h h x 0 x 1 x... x n weight h/ h h h/ 0 Figure 3: Function Vlue Weights in the Composite Rectngle Rule Exmple.. AS nother illustrtion, consider pplying the composite trpezoid rule to n intervl [, ], s shown in Figure 4. In ech suintervl, the endpoints get weight h/. Since ech of the interior points is included in two suintervls, this yields the following estimte of the definite integrl: I CT = h [f) + f + h) + f + h) + + f h) + f)] Exmple.3. Now, consider pplying composite Simpson s rule to n intervl [, ] s shown in Figure 5. For ech suintervl [x i, x i+1 ], the endpoints get weight 1/6 nd the midpoint gets weight 4/6. Since ech interior endpoint ll nodes except nd ) is counted twice, this yields the following estimte of the definite integrl: I CS = h [ f) + 4 f + h 6 ) + f + h) + 4 f + 3h ) + f + h) f h) + 4 f h ] ) + f) 6

7 h h h x 0 x 1 x... x n weight h/ h h h/ Figure 4: Function Vlue Weights in the Composite Trpezoid Rule h h h x 0 x 1 x... x n weight h/6 h/6 h/6 h/6 4h/6 4h/6 Figure 5: Function Vlue Weights in Composite Simpson s Rule.1 Error Anlysis for Composite Simpson s Rule In composite rule, we re mking use of the fct tht definite integrl over n intervl [, ] is simply the sum of the definite integrls of the suintervls. ft)dt = n 1 xi+1 x i ft)dt To get n expression for the error of composite rule, we tke the sum of the errors over ech suintervl, noting tht over nd underestimtes my cncel out: E CS f) = n 1 E s [xi,x i+1 ] = 1 n 1 h5 880 N f 4) c i ) N = f 4) c) h 4 N h) }{{} ) = f 4) c) 880 h4 ) Note tht ) is n verge of the vlues of fc i ). This is generl formul for the error of 7

8 composite rule. In generl, simple rule r with error of the form: E r = f n) c) ) n+1 k will produce composite rule Cr with error of the form: E Cr = f n) c) h n ) k We will discuss this further in the next lecture. 8