lecture 22: Newton Cotes quadrature

Transcription

1 lecture : Newto Cotes qudrture f () 3. Newto Cotes qudrture You ecoutered the most bsic method for pproimtig itegrl whe you lered clculus: the Riem itegrl is motivted by pproimtig the re uder curve by the re of rectgles tht touch tht curve, which gives rough estimte tht becomes icresigly ccurte s the width of those rectgles shriks. This mouts to pproimtig the fuctio f by piecewise costt iterpolt, d the computig the ect itegrl of the iterpolt. Whe oly oe rectgle is used to pproimte the etire itegrl, we hve the most simple Newto Cotes formul; see Figure 3.. Newto Cotes formuls re iterpoltory qudrture rules where the qudrture otes,..., re uiformly spced over [, b], b j = j Give the lessos we lered bout polyomil iterpoltio t uiformly spced poits i Sectio., you should rightly be suspicious of pplyig this ide with lrge (i.e., high degree iterpolts). A more relible wy to icrese ccurcy follows the led of bsic Riem sums: prtitio [, b] ito smller subitervls, d use low-degree iterpolts to pproimte the itegrl o ech of these smller domis. Such methods re clled composite qudrture rules. I some cses, the fuctio f my be firly regulr over most of the domi [, b], but the hve some smll regio of rpid growth or oscilltio. Moder dptive qudrture rules re composite rules o which the subitervls of [, b] vry i size, depedig o estimtes of how rpidly f is chgig i give prt of the domi. Such methods seek to blce the competig gols of highly ccurte pproimte itegrls d s few evlutios of f s possible.we shll ot dwell much o these sophisticted qudrture procedures here, but rther strt by uderstdig some methods you were probbly itroduced to i your first clculus clss.. Figure 3.: Estimtes of R f () d, show i gry: the first pproimtes f by costt iterpolt; the secod, composite rule, uses piecewise costt iterpolt. You probbly hve ecoutered this secod pproimtio s Riem sum. 3.. The trpezoid rule The trpezoid rule is simple improvemet over pproimtig the itegrl by the re of sigle rectgle. A lier iterpolt to f c be costructed, requirig evlutio of f t the itervl ed poits = d = b. Usig the iterpoltory qudrture methodology

2 3 described i the lst sectio, we write b p () = f () + f (b) b b, d compute its itegrl s p () d = f () b + f (b) b b d I summry, Trpezoid rule: = f () b = f () d + f (b) f () d b b + f (b). f ()+ f (b). d (trpezoid) (ect) The procedure behid the trpezoid rule is illustrted i Figure 3. where the re pproimtig the itegrl is colored gry. To derive error boud for the trpezoid rule, simply itegrte the fudmetl iterpoltio error formul i Theorem.3. Tht gve, for ech [, b], some [, b] such tht f () p () = f ()( )( b) Figure 3.: Trpezoid rule estimte of R f () d, show i gry. Note tht will vry with, which we emphsize by writig (). Itegrte this formul to obti f () d p () d = f (())( )( b) d = f (h) ( )( b) d = f (h)( 3 b + b b3 ) = f (h)(b ) 3 for some h [, b]. The secod step follows from the me vlue theorem for itegrls. I forthcomig lecture we shll develop much more geerl theory, bsed o the Peo kerel, from which we c derive this error The me vlue theorem for itegrls sttes tht if h, g C[, b] d h does ot chge sig o [, b], the there eists some h [, b] such tht R b g(t)h(t) dt = g(h) R b h(t) dt. The requiremet tht h ot chge sig is essetil. For emple, if g(t) =h(t) =t the R g(t)h(t) dt = R t dt = /3, yet R h(t) dt = R t dt =, so for ll h [, ], g(h) R h(t) dt = = R g(t)h(t) dt = /3.

3 boud, plus bouds for more complicted schemes, too. For ow, we summrize the boud i the followig Theorem. Theorem 3.. Let f C [, b]. The error i the trpezoid rule is f () d b f ()+ f (b) = f (h)(b ) 3 for some h [, b]. This boud hs iterestig feture: if we re itegrtig over the smll itervl, b = h, the the error i the trpezoid rule pproimtio is O(h 3 ) s h!, while the error i the lier iterpolt upo which this qudrture rule is bsed is oly O(h ) (from Theorem.3). Emple 3. ( f () =e (cos + si )). Here we demostrte the differece betwee the error for lier iterpoltio of fuctio, f () =e (cos + si ), betwee two poits, = d = h, d the trpezoid rule pplied to the sme itervl. The theory revels tht lier iterpoltio will hve O(h ) error s h!, while the trpezoid rule hs O(h 3 ) error, s cofirmed i Figure iterpoltio error O(h ) trpezoid error Figure 3.3: Error of lier iterpoltio d trpezoid rule pproimtio for f () =e (cos + si ) for [, h] s h! O(h 3 ) h Simpso s rule To improve the ccurcy of the trpezoid rule, icremet the degree of the iterpoltig polyomil. This will icrese the umber of evlutios of f (ofte very costly), but hopefully will sigifictly

4 5 decrese the error. Ideed it does by eve greter mrgi th we might epect. Simpso s rule itegrtes the qudrtic iterpolt p P to f t the uiformly spced poits =, =( + b)/, = b. Usig the iterpoltory qudrture formultio of the lst sectio, p () d = w f ()+w f ( ( + b)) + w f (c), where w = w = w = d = b (b ) d = 3 d = b. I summry: Simpso s rule: f () d b f ()+ f ( ( + b)) + f (b) (Simpso) (ect) Simpso s rule ejoys remrkble feture: though it oly pproimtes f by qudrtic, it itegrtes y cubic polyomil ectly! Oe c verify this by directly pplyig Simpso s rule to geeric cubic polyomil. Write f () = 3 + q(), where q P. Let I( f )= R b f () d d let I ( f ) deote the Simpso s rule pproimtio. The, by lierity of the itegrl, I( f )=I( 3 )+I(q) Figure 3.: Simpso s rule estimte of R f () d, show i gry. d, by lierity of Simpso s rule, I ( f )=I ( 3 )+I (q). Sice Simpso s rule is iterpoltory qudrture rule bsed o qudrtic polyomils, its degree of ectess must be t lest (Theorem 3.), i.e., it ectly itegrtes q: I (q) =I(q). Thus I( f ) I ( f )= I( 3 ) I ( 3 ).

5 So Simpso s rule will be ect for ll cubics if it is ect for 3.A simple computtio gives I ( 3 )= b + b b 3 = b b + 3b + 3b 3 = b = I( 3 ), cofirmig tht Simpso s rule is ect for 3, d hece for ll cubics. For ow we simply stte error boud for Simpso s rule, which we will prove i future lecture. I fct, Newto Cotes formuls bsed o pproimtig f by eve-degree polyomil lwys ectly itegrte polyomils oe degree higher. Theorem 3.3. Let f C [, b]. The error i the Simpso s rule is b f () d f ()+f(( + b)/)+ f (b) = 9 f () (h)(b ) 5 for some h [, b]. This error formul cptures the fct tht Simpso s rule is ect for cubics, sice it fetures the fourth derivtive f () (h), two derivtives greter th f (h) i the trpezoid rule boud, eve though the degree of the iterpolt hs oly icresed by oe. Perhps it is helpful to visulize the ectess of Simpso s rule for cubics. Figure 3.5 shows f () = 3 (blue) d its qudrtic iterpolt (red). O the left, the re uder f is colored gry: its re is the itegrl we seek. O the right, the re uder the iterpolt is colored gry. Accoutig re below the is s egtive, both itegrls give ideticl vlue eve though the fuctios re quite differet. It is remrkble tht this is the cse for ll cubics. Typiclly oe does ot see Newto Cotes rules bsed o polyomils of degree higher th two (i.e., Simpso s rule). Becuse it c be fu to see umericl myhem, we give emple to emphsize why high-degree Newto Cotes rules c be bd ide. Recll tht Ruge s fuctio f () =/( + ) gve ice emple for which the polyomil iterpolt t uiformly spced poits over [ 5, 5] fils to coverge uiformly to f. This fct suggests tht Newto Cotes qudrture will lso fil to coverge s the degree of the iterpolt grows. The ect vlue of the itegrl we seek is Itegrtig the cubic iterpolt t four uiformly spced poits is clled Simpso s three-eighths rule. Z d = t (5) = Just s the iterpolt t uiformly spced poits diverges, so too does the Newto Cotes itegrl. Figure 3. illustrtes this divergece, d shows tht itegrtig the iterpolt t Chebyshev

6 re uder f () = re uder qudrtic iterpolt Figure 3.5: Simpso s rule pplied to f () = 3 o [, 3/]. The res uder f () (blue) d its qudrtic iterpolt (red) re the sme, eve though the fuctios re quite differet poits, clled Cleshw Curtis qudrture, does ideed coverge. Sectio 3. describes this ltter qudrture i more detil. Before discussig it, we describe wy to mke Newto Cotes rules more robust: itegrte low-degree polyomils over subitervls of [, b]. p() d 3 uiform poits Figure 3.: Itegrtig iterpolts p t + uiformly spced poits (red) d t Chebyshev poits (blue) for Ruge s fuctio, f () =/( + ) over [ 5, 5]. Z 5 5 Z 5 f () d Chebyshev poits Composite rules As ltertive to itegrtig high-degree polyomil, oe c pursue simpler pproch tht is ofte very effective: Brek the itervl [, b] ito subitervls, the pply stdrd Newto Cotes rule (e.g., trpezoid or Simpso) o ech subitervl. Applyig the

7 trpezoid rule o subitervls gives f () d = Z j  j f () d  ( j j ) f ( j + f ( j ). The stdrd implemettio ssumes tht f is evluted t uiformly spced poits betwee d b, j = + jh for j =,..., d h =(b )/, givig the followig fmous formultio: Composite Trpezoid rule: f () d h f ()+  f ( + jh)+ f (b). (Of course, oe c redily djust this rule by prtitioig [, b] ito subitervls of differet sizes.) The error i the composite trpezoid rule c be derived by summig up the error i ech pplictio of the trpezoid rule: f () d h f ()+  f ( + jh)+ f (b) =  = h3  f (h j ) f (h j )( j j ) 3 for h j [ j, j ]. We c simplify these f terms by otig tht ( f (h j )) is the verge of vlues of f evluted t poits i the itervl [, b]. Nturlly, this verge cot eceed the mimum or miimum vlue tht f ssumes o [, b], so there eist poits, [, b] such tht f ( ) pple  f (h j ) pple f ( ). Thus the itermedite vlue theorem gurtees the eistece of some h [, b] such tht f (h) =  f (h j ). We rrive t boud o the error i the composite trpezoid rule. Theorem 3.. Let f C [, b]. The error i the composite trpezoid rule over itervls of uiform width h =(b )/ is f () d for some h [, b]. h f ()+  f ( + jh)+ f (b) = h (b ) f (h).

8 9 This error lysis hs importt cosequece: the error for the composite trpezoid rule is oly O(h ), ot the O(h 3 ) we sw for the usul trpezoid rule (i which cse b = h sice = ). A similr costructio leds to the composite Simpso s rule. We ow must esure tht is eve, sice ech itervl o which we pply the stdrd Simpso s rule hs width h. Simple lgebr leds to the followig formul. Composite Simpso s rule: f () d h 3 / f () + Â f ( +(j )h) + / Â f ( + jh) + f (b). Now use Theorem 3.3 to derive error formul for the composite Simpso s rule, usig the sme pproch s for the composite trpezoid rule. Theorem 3.5. Let f C [, b]. The error i the composite Simpso s rule over / itervls of uiform width h = (b )/ is f () d h 3 / f () + Â f ( +(j )h) + / Â f ( + jh) + f (b) = h (b ) f () (h) for some h [, b]. The illustrtios i Figure 3.7 compre the composite trpezoid d Simpso s rules for the sme umber of fuctio evlutios. Oe c see tht Simpso s rule, i this typicl cse, gives cosiderbly better ccurcy. Reflect for momet. Suppose you re willig to evlute f fied umber of times. How c you get the most bg for your buck? If f is smooth, rule bsed o high-order iterpolt (such s the Cleshw Curtis d Gussi qudrture rules we will preset i few lectures) re likely to give the best result. If f is ot smooth (e.g., with kiks, discotiuous derivtives, etc.), the robust composite rule would be good optio. (A fmous specil cse: If the fuctio f is sufficietly smooth d is periodic with period b, the the trpezoid rule coverges epoetilly.)

9 (comp. trpezoid) (ect) (comp. Simpso) (ect) Adptive Qudrture Figure 3.7: Composite trpezoid rule (left) d composite Simpso s rule (right). If f is cotiuous, we c tti rbitrrily high ccurcy with composite rules by tkig the spcig betwee fuctio evlutios, h, to be sufficietly smll. This might be ecessry to resolve regios of rpid growth or oscilltio i f. If such regios oly mke up smll proportio of the domi [, b], the uiformly reducig h over the etire itervl will be uecessrily epesive. Oe wts to cocetrte fuctio evlutios i the regio where the fuctio is the most orery. Robust qudrture softwre djusts the vlue of h loclly to hdle such regios. To ler more bout such techiques, which re ot foolproof, see W. Gder d W. Gutschi, Adptive qudrture revisited, BIT (). This pper criticizes the routies qud d qud tht were icluded i MATLAB versio 5. I light of this lysis MATLAB improved its softwre, essetilly icorportig the two routies suggested i this pper strtig i versio s the routies qud (dptive Simpso s rule) d qudl ( dptive Guss Lobtto rule).