Lecture 20: Numerical Integration III


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1 cs4: introduction to numericl nlysis /8/0 Lecture 0: Numericl Integrtion III Instructor: Professor Amos Ron Scribes: Mrk Cowlishw, Yunpeng Li, Nthnel Fillmore For the lst few lectures we hve discussed numericl pproximtions of the definite integrl If) = ft)dt Our generl pproch is to perform polynomil interpoltion of the function over [, b], then integrte the polynomil. We showed tht, if we use the Lgrnge representtion, the pproximtion of the definite integrl cn be written s simple weighted sum of the function vlues. If) n ft i ) w i i=0 Where wi) is the integrl of the Lgrnge polynomil for t i : w i = l i t)dt Note tht the weight w i is distinct from the nth Newton polynomil ωt), which comes from the error formul for pproximtion of definite integrls: ft)dt pt)dt = f n+) c) n + )! n t t j ) dt j=0 } {{ } ωt) It is lso importnt to note tht the quntity f n+) c), cnnot simply be tken outside the integrl, s c my be different vlue for every t. For this reson, we split up the error nlysis into cses. The error nlysis for cse, in which we put restrictions on ωt) [,b] llows us to clculte the error s ft)dt pt)dt = f n+) c) n + )! n t t j )dt cse ) However, the nlysis tht llows us to express the error this wy is beyond the scope of this lecture. j=0
2 Composite Rules If we exmine the error formul for ny of the simple rules, for exmple, the error for Simpson s rule: E S f) = f 4) c) b )5 880 we see tht the error becomes smller s the size of the intervl [, b] shrinks. As we discussed lst time, this leds to the ide of composite rules. In composite rule, we split up the intervl [, b] into equidistnt prtitions, estimte the integrl over ech prtition using simple rule, then dd the estimtes together to rech our finl pproximtion. As we sw lst time, simple rule with n error formul of the form: E rule = #f j) c) b ) j+ where # is some constnt, hs corresponding composite rule with error formul E compositerule = #f j) c) h j b ) where h = b )/N is the size of ech subintervl. To illustrte, here re the error formuls for the composite rules corresponding to the simple rules we hve studied. Rectngle Rule E CR f, h) = f c) h b ) ) Midpoint Rule Trpezoid Rule E CM f, h) = 4 f c) h b ) ) E CT f, h) = f c) h b ) ) Simpson s Rule E CS f, h) = 880 f 4) c) h 4 b ) 4) Note tht, in ech cse, high power of h, corresponding to n lgorithm tht quickly converges to smll error, requires higher fidelity of the function f. Note tht the composite rules re to bsic rules s spline interpoltion is to polynomil interpoltion. Indeed composite rule essentilly fits spline to the given function nd then interpoltes the spline. To illustrte the use of these error formuls, consider the following exmple:
3 Exmple.. Compute the number of function evlutions required to pproximte log) with error 0 8 Solution: Recll tht we cn represent the nturl logrithm of s n integrl. log) = t dt To pproximte this integrl, we should first consider which lgorithm to use. Since /t is infinitely differentible on [, ], we re free to use ny of the rules, so we choose composite Simpson s rule, since it converges t rte of h 4. We know tht f 4) t) = 4/t 5, substituting into eqution yields: h 4 ),[,] c h h N 0 Thus our pproximtion will hve bout 0 subintervls. This will require 0 evlutions t the midpoints nd evlutions t the endpoints, for grnd totl of 6 function evlutions. Extrpoltion If we re given n error formul for numericl method tht is n exct mesure of the error, we cn sometimes use tht formul to derive better numericl methods. This process is clled extrpoltion. For exmple, consider the error formul for the composite midpoint rule. E CM f, h) = 4 f c) b ) h 5) }{{} ) This formul is lmost good enough to use for extrpoltion. However, the constnt ) depends both on the function f nd the size of the intervl h. For extrpoltion, we would like constnt tht depends only on the function f. As it turns out, it is possible to rewrite eqution 5 s follows: E CM f, h) = c f h + c f h) h 4 where c f is constnt dependent only on f, nd c f h) is constnt dependent on both f nd h. We cn rewrite the error formul for the composite trpezoid rule eqution ) similrly: E CT f, h) = k f h + k f h) h 4
4 It lso turns out tht, similr to the stndrd form of the error functions, k f = c f. Now, consider the definition of the error functions for the composite midpoint nd composite trpezoid rules. If) I CM f, h) = E CM f, h) 6) If) I CT f, h) = E CT f, h) 7) If we multiply the top eqution by nd dd it to the bottom eqution, this yields: If) I CT f, h) I CM f, h) = E CT f, h) + E CM f, h) Now, since k f = c f, the first term in ech error formul cncels, leving only some constnt times h 4, thus we hve: If) I CT f, h) I CM f, h) = Oh 4 ) How does this trnslte into new method? We simply estimte the definite integrl by pplying the composite midpoint rule with subintervl size h, multiplying the result by two nd dding this to the result of one ppliction of the trpezoid rule with identicl subintervl size h, then dividing the entire sum by three. This method hs error Oh 4 ), since: If) I CT f, h) I CM f, h) = Oh 4 ) How good is this new method? If we look t the weights pplied to ech interpoltion point, we see tht the composite midpoint rule pplies weights: [ f + h ) + f + h ) + + f b h )] while the trpezoid rule pplies weights: 6 Combining these yields: [ f) + 4 f [ f) + f + h) + f + h) + + fb h) + fb) + h ) + f + h) + 4 f + h ) + f + h) fb h) + 4 f b h ) ] + fb) Close inspection revels tht this new rule is identicl to composite Simpson s rule. Cn we use extrpoltion to produce rules tht re better thn composite Simpson s rule? The nswer is yes, we cn do this by mixing composite Simpson s rule with itself. ] 4
5 As before, we cn write the error formul for composite Simpson s rule s lower order term with constnt independent of h, nd higher order term with constnt dependent on h. E CS f, h) = c f h 4 + c f h) h 6 8) Furthermore, we cn clculte the error formul for composite Simpson s rule with subintervl width twice s lrge: E CS f, h) = c f h) 4 + c f h) h 6 9) Note tht the first term in eqution 9 will be exctly sixteen times the first term in eqution 8, so tht combining the two rules produces error proportionl to the sixth power of h. 6 I CS f, h) I CS f, h) = Oh 6 ) Thus, we cn produce rule with error tht converges s the sixth power of h using: If) 6 I CSf, h) I CS f, h) 5 = Oh 6 ) 5
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