1 Numerical Integration

Transcription

1 cs4: introduction to numericl nlysis /6/0 Lecture 8: Numericl Integrtion Instructor: Professor Amos Ron Scribes: Mrk Cowlishw, Nthnel Fillmore Numericl Integrtion Recll tht lst lecture, we discussed numericl integrtion. Given n intervl [, b] nd function f : [, b], we would like to find the re under the curve over [, b], I: s shown in Figure. I = f(t)dt I b Figure : The Definite Integrl of f(t) over [, b] Recll from the fundmentl theorem of clculus tht we cn find I using the ntiderivtive, function F with F = f I = F (b) F () However, s we discussed lst lecture, this method is nerly useless in numericl integrtion except in very specil cses (such s integrting polynomils). To illustrte, consider the following exmple: Exmple.. Find the numericl vlue of log(.). We recll tht the nturl log is defined using the definite integrl of simple, rtionl function, s shown in Figure : log(x) = log(.) = x. nd thus

2 f(t) = t log(.). Figure : Clculting the Nturl Logrithm with Definite Integrl So, if we cn find method to give numericl pproximtion of definite integrls, we cn use it to find numericl pproximtions of the nturl log. Mny functions don t even hve ntiderivtives expressible in terms of simple functions like cos, exp, etc. One importnt exmple is e Bsic Rules for Numericl Approximtion of Definite Integrls All of the bsic methods for numericl pproximtion tht we will exmine rely on the sme bsic ide:. Approximte f [,b] using some polynomil p. Integrte the polynomil: I rule = p(t)dt f(t)dt Suppose tht we obtin the pproximting polynomil p through interpoltion. Given n + nodes [t 0, t,..., t n ], we cn write p using the Lgrnge representtion: p(t) = f(t 0 ) l 0 (t) + f(t ) l (t) + + f(t n ) l n (t) Integrting this polynomil, we get our pproximtion of f [,b] :

3 I rule = = = p(t)dt () n f(t i ) l i (t)dt () i=0 n f(t i) i=0 l i (t)dt } {{ } w i (3) Where w i is weight pplied to ech function vlue. Note tht, since Lgrnge polynomils do not depend on the function, we cn clculte the weights w i using only the nodes [t 0, t,..., t n ], thus we cn rewrite eqution 3 s simple weighted sum of function vlues: I rule = n f(t i ) w i (4) i=0 Now we will discuss four simple rules tht follow this generl frmework.. Rectngle Rule In the rectngle rule, we pproximte f [,b] using single interpoltion point [] (the left endpoint of the intervl). Our polynomil interpolnt will thus be constnt polynomil p(t) = f(), s shown in Figure 3 nd we cn clculte its re I R using: I R = f() (b ) (5) b Figure 3: The Rectngle Rule for Approximting Definite Integrl Approximting the solution to exmple. using the rectngle rule yields: 3

4 I =. I R = f() (. ) = 0. = 0. We cn summrize the rectngle rule by the nottion {} This nottion mens tht the rectngle rule pproximtes the integrl f(t) dt by evluting f t, finding the polynomil which interpoltes the point (, f()), nd integrting this polynomil.. Midpoint Rule In the midpoint rule, we gin pproximte f [,b] using single interpoltion point, but this time we use the midpoint +b. Our polynomil interpolnt will gin be constnt polynomil, this time p(t) = f ( ) +b, s shown in Figure 4 nd we cn clculte its re IM using: ( ) + b I M = f (b ) (6) b Figure 4: The Midpoint Rule for Approximting Definite Integrl Approximting the solution to exmple. using the midpoint rule yields: 4

5 I =. I M = f(.) (. ) =. 0. = We cn summrize the midpoint rule by the nottion { } + b This nottion mens tht the midpoint rule pproximtes the integrl f(t) dt by evluting f t, finding the polynomil which interpoltes this point, nd integrting this polynomil. +b.3 Trpezoid Rule In the trpezoid rule, we pproximte f [,b] using the endpoints of the intervl [, b] s our interpoltion. The polynomil interpolnt will thus be liner polynomil (with degree t most ) pssing through f() nd f(b), s shown in Figure 5 nd we cn clculte its re I T simply, using I T = f() + f(b) (b ) (7) = f() (b ) + f(b) (b ) (8) Note tht the two weights w 0, w in this cse re ech / (b ). The sum of the weights will lwys equl the width of the intervl (b ). b Figure 5: The Trpezoid Rule for Approximting Definite Integrl Approximting the solution to exmple. using the trpezoid rule yields: 5

6 I =. I T f() + f(.) = (. ) = ( + ) 0.. = We cn summrize the trpezoid rule by the nottion {, b} This nottion mens tht the trpezoid rule pproximtes the integrl f(t) dt by evluting f t nd b, finding the polynomil which interpoltes these two points, nd integrting this polynomil..4 Simpson s Rule In Simpson s rule, we pproximte f [,b] using three equidistnt interpoltion points [, +b, b], corresponding to the endpoints nd midpoint of the intervl. The resulting polynomil interpolnt will thus hve degree t most, s shown in Figure 6 nd cn be written s: p(t) = f() (t m)(t b) ( m)( b) + f(m) (t )(t b) (t m)(t ) + f(b) (m )(m b) (b m)(b ) where m is the midpoint +b. Integrting the Lgrnge polynomils over [, b] gives the following formul for the re I S : I S = b [ ( ) ] + b f() + 4f + f(b) (9) 6 note gin tht the weights b 6, 4 b 6, b 6 dd up to the width of the intervl (b ). Approximting the solution to exmple. using Simpson s rule yields: I =. I S =. [f() + 4 f(.) + f(.)] 6 = 0. [ ]. = We cn summrize Simpson s rule by the nottion {, + b }, b 6

7 b Figure 6: Simpson s Rule for Approximting Definite Integrl This nottion mens tht Simpson s rule pproximtes the integrl f(t) dt by evluting f t, +b, nd b, finding the polynomil which interpoltes these three points, nd integrting this polynomil. 3 Error Anlysis Since we used polynomil interpoltion s the bsis for ech of the bsic rules for pproximting the definite integrl, we cn use the error formul for polynomil interpoltion to clculte the error. For ny of these rules, we cn clculte the error E rule s: E rule = I I rule = = f(t)dt [f(t) p(t)] dt p(t)dt We cn use the error formul for polynomil interpoltion over n+ points to express f(t) p(t) f(t) p(t) = f n+ (c) (n + )! (t x 0)(t x )... (t x n ) where c is point tht exists somewhere on the intervl. It is importnt to note tht this is not n pproximtion of the error, but n exct error formul. Substituting into the error formul for pproximting definite integrls yields: E rule = = f n+ (c) (n + )! f n+ (c) (n + )! (t x 0)(t x )... (t x n )dt (t x 0 )(t x )... (t x n ) }{{} ω(t) 7

8 We cnnot clculte this error exctly (since we don t know c), but we will consider two cses for ω(t). These cses re not exhustive, but they do cover ll four simple rules (nd, in fct, ll interesting rules of this kind). Cse Cse ω(t) does not chnge sign on [, b]. Tht is, either ω [,b] 0 or ω [,b] 0 3. Cse ω(t)dt = 0 Which of the rules hve ω(t) not chnging sign over [, b]? Rectngle Rule For the rectngle rule, ω(t) = t, which is lwys positive on the intervl, so the rectngle rule flls into cse. Midpoint Rule For the midpoint rule, ω(t) = t ( + b)/, which is negtive to the left of the midpoint nd positive to the right of the midpoint, so the midpoint rule does not fll into cse. Trpezoid Rule For the trpezoid rule, ω(t) = (t )(t b). On (, b), this vlue is lwys negtive, since t is lwys positive nd t b is lwys negtive. Thus, the trpezoid rule flls into cse. Simpson s Rule In Simpson s rule, ω(t) = (t )[t ( + b)/](t b). Since (t )(t b) is lwys negtive, nd t ( + b)/ flips signs on either side of the midpoint, Simpson s rule does not fll into cse. 3.. Error Anlysis of Rectngle Rule For rules tht mtch cse, we hve E rule = f (n+) (c) (n + )! ω(t)dt 8

9 for the rectngle rule, this becomes: E R = f (c) = f (c) (t )dt (b ) Note tht the width of the intervl (b ) is rised to the second power in this error. As we will see in lter lectures, we would like this power to be s lrge s possible, so tht the error will shrink rpidly s the width of the intervl decreses. It is importnt to note tht the errors we obtin using this method re exct error expressions. A negtive error (which will depend both on the sign of the error expression nd the sign of the function s derivtive) indictes tht the method is overestimting the error, so tht we could djust our estimte using the error expression. Looking t exmple. we see tht our error is given by: E R = f (c) (b ) = c (0.) = 0.0 c This indictes tht our pproximtion tht I 0. ws n overestimte by t most Error Anlysis of Trpezoid Rule For the trpezoid rule, we hve error: E T = f (c) = f (c) (t )(t b)dt ) (b )3 ( 6 = f (c) (b )3 Looking t exmple. we see tht our error is given by: E T = f (c) (b )3 = 6 c 3 (0.)3 = c 3 This indictes tht our pproximtion tht I ws n overestimte by t most

10 3. Cse Cse demnds tht ω(t)dt = 0 (0) It is esy to see tht this condition holds for both the midpoint rule nd Simpson s rule. An interesting feture of rules tht mtch cse is tht dding point to the polynomil interpoltion results in no difference in the estimte of the re of the interpolnt over [, b]. We know this is true, since eqution 0 tells us tht the definite integrl of the next Newton polynomil over [, b] is zero. This mens tht we cn clculte the error for ny rule mtching cse by dding n dditionl point to the interpoltion. We will discuss this in the next lecture. 0