Numerical Integration

Transcription

1 Numericl tegrtio Newto-Cotes Numericl tegrtio Scheme Replce complicted uctio or tulted dt with some pproimtig uctio tht is esy to itegrte d d 3-7 Roerto Muscedere The itegrtio o some uctios c e very esy or very diicult. some cses it my ot e possile to itegrte uctio usig lyticl mes. Or it is possily we ever ever see the uctios themselves. We just get dt poits d hve to work rom there. these situtios we c use umericl itegrtio. We will egi y lookig t the Newto-Cotes Numericl tegrtio schemes. This scheme pproimtes deiite itegrl,, rom to, to polyomil uctio. Geerlly, the higher-order the polyomil, the etter the pproimtio o the itegrl will e tegrtio

2 Result o itegrtio is ouded y d the is etwee = d = C e pproimted umericlly y dividig the regio ito smll segmets ech o width The re o ech segmet c e pproimted y trpezoid pproimtig ech segmet o the curve y stright lie Trpezoidl Rule 3-7 Roerto Muscedere We will irst look t the trpezoidl rule. The ide ehid this pproch is to id the trpezoidl re etwee the curve d the -is i eve segmets d to sum them up. the width o ech o these segmets is resoly smll compred to the compleity o the curve, we should get good pproimtio tegrtio

3 Trpezoidl Rule Trpezoidl Rule is the irst o the Newto- Cotes closed itegrtio ormuls = is stright lie d d 3-7 Roerto Muscedere 3 We will irst look t how the trpezoidl rule is oud. We pply it i the Newto-Cotes umericl itegrtio scheme with irst-order polyomil. A irst-order polyomil is simply lie. So sed o the iormtio o our curve, we c crete lie segmet tht will itersect the curve t poits d. This uctio o the lie,, hs the orm o y=m+. We c see the slope, m, is the dierece i the y poits o d divided y the poits o d. We c lso see the y itercept,, is set to tegrtio 3

4 tegrtio Roerto Muscedere Trpezoidl Rule Trpezoidl Rule The re uder this stright lie is estimte o the itegrl o etwee the limits o d d we plced this lie equtio ito the itegrl d perorm itegrtio d some lger, we id the re uder the curve.

5 tegrtio Roerto Muscedere Trpezoidl Rule Trpezoidl Rule Sme result i we clculted trpezoid grphiclly We could lso do the sme process grphiclly. we clculte the re o the trigle o top, - * - /, plus the re o the rectgle o the ottom, - *, d sum them up, we get the sme swer.

6 Trpezoidl Rule: Emple Numericlly itegrte: =, =.8 Alyticl swer = Roerto Muscedere 6 Here is emple. Numericlly itegrte the ove uctio etwee d.8. We c do this lyticlly, d i we do we get swer o out tegrtio 6

7 Trpezoidl Rule: Emple Sigiict dierece rom lyticl solutio o Roerto Muscedere 7 We use the sigle-segmet trpezoidl rule d oti.78. This is ot very good pproimtio. But why is tht? tegrtio 7

8 Trpezoidl Rule: Emple Why so d? 3-7 Roerto Muscedere 8 Bsiclly this is wht is hppeig. We re oly clcultig very rough trpezoid sed o the curve d the limits o the itegrl. the limits re chose to e t low poits while there re high poits i etwee, we will get poor pproimtios. The solutio to this is to use multiple segmets with the trpezoidl rule tegrtio 8

9 tegrtio Roerto Muscedere Trpezoidl Rule Trpezoidl Rule Add series o itervls d d d h We do this y epdig our origil itegrl to iclude multiple segmets. We replce d with d s we c hve more s ow.

10 tegrtio 3-7 Roerto Muscedere Trpezoidl Rule Trpezoidl Rule Sustitutig the trpezoidl rule or ech itegrl yields: h h h we sustitute i our sigle segmet trpezoidl rule, we oti the ove pproimtio. This c e geerlized though.

11 tegrtio 3-7 Roerto Muscedere Trpezoidl Rule Trpezoidl Rule Groupig the terms: i i i i h we group the terms together, we c come up with etter uctio.

12 Trpezoidl Rule: Emple Numericlly itegrte: =, =.8, = Alyticl swer = Roerto Muscedere Lets try tht emple gi, this time usig segmets tegrtio

13 Trpezoidl Rule: Emple Sigiictly etter th with oe segmet o the trpezoidl rule Roerto Muscedere 3 Our pproimtio is much etter ow. t still is t tht gret, ut it is cosiderly etter th with oe segmet. We c chieve etter pproimtio usig lrger tegrtio 3

14 tegrtio Code p. clss tegrtio { privte: doule, ; // limits o itegrtio doule *doule ; //uctio to e itegrted pulic: tegrtiodoule *Fdoule, doule, doule { =F; =; =;} doule Trpezoidlit ; }; 3-7 Roerto Muscedere 4 t is ovious tht we will hve to perorm lot o clcultios to oti good umericl pproimtio o itegrl. So we should write some code to do this. Agi to mke our code simpler we will use uctio to crete our dt poits. We cll our clss tegrtio. t hs some privte memers,,, d poiter to uctio. We hve costructor which tkes ll the pssed vlues, the uctio,, d d stores them i to the privte memers. We lso declre the Trpezoidl uctio to clculte the re uder the curve tegrtio 4

15 tegrtio Code p. doule tegrtio::trpezoidlit { doule h=-/; doule sum=.; doule =; doule ; orit i=; i<=-; i++ { +=h; sum+=; } =++.*sum*h/.; } retur ; 3-7 Roerto Muscedere 5 The Trpezoidl uctio is reltively smll. We irst clculte h sed o, d. We set the sum o the summtio prt o the uctio to zero. We will e usig loop to id the sum o the elemets through -. Our iitil is set to. We ow go though ech elemet, ut strtig t d goig to -. Our loop irst dds h to so tht our correspods to the proper i. We the dd the result o the uctio with rgumets to the sum. We do this util elemet -. We the use the sum log with the vlues o d to id the re uder the curve. We retur this vlue tegrtio 5

16 Trpezoidl Emple - Code doule FUNdoule // User supplied uctio { retur. + 5* - ** + 675*** - 9**** + 4*****; } it mi { tegrtio FUN,,.8; it i; ori=;i<=6;i++ cout << "=" << i << ", =" <<.Trpezoidli << edl; } retur ; 3-7 Roerto Muscedere 6 We declre our uctio, the sme s the emples eore, ut you c chge this or y uctio. Our mi code is simple. We crete the oject rom the clss tegrtio d pss the uctio to it d the limits d.8 log with 8 segmets. We the output the result o the Trpezoidl pproimtio or chgig rom to tegrtio 6

17 Trpezoidl Emple - Output =, =.78 =, =.688 =3, = =4, =.4848 =5, = =6, =.577 =7, = =8, =.68 =9, =.699 =, =.654 =, =.6945 =, =.68 =3, =.654 =4, =.675 =5, =.698 =6, = Roerto Muscedere 7 The result, or 8 segmets, is.68. we use 6 segmets, the result is We c see tht usig doule the segmets hs oly give us little it more ccurcy tegrtio 7

18 Simpso s s /3 Rule Simpso s Rule is the secod o the Newto- Cotes closed itegrtio ormuls = usig Lgrge terpoltig Polyomils d i i j ji d i j j 3-7 Roerto Muscedere 8 We c improve our umericl itegrtio methods y usig d order polyomil. This method is kow s the Simpso s /3 rule. The reso it is clled /3 will ecome ppret soo. this cse, we will use Lgrge terpoltig polyomil. The uctio is show ove, ut or =, the result is ot too comple tegrtio 8

19 tegrtio Roerto Muscedere Simpso Simpso s /3 Rule s /3 Rule Ater itegrtio d lgeric mipultio: 4 3 h h we epd out the uctio or =, we get the ove epressio. Ater itegrtio d lgeric mipultio we c sigiictly reduce the uctio. We c ow see where the /3 cme rom s it is the ctor pplied to h.

20 Trpezoidl Rule: Emple Wht does this represet? 3-7 Roerto Muscedere This uctio represets the re etwee two poits, ut y usig curve, ot stright lie s i the Trpezoidl rule. For this grphicl depictio, we c see tht our pproimtio is greter th the uctio give, s the uctio my e hve cosiderly higher order. y cse, this is deiitely etter th usig stright lie tegrtio

21 Simpso s s /3 Rule: Emple Numericlly itegrte: =, =.8 Alyticl swer = Roerto Muscedere We will try the sme emple s eore, ut usig the Simpso s /3 rule tegrtio

22 Simpso s s /3 Rule: Emple Better th with Trpezoidl Rule with two segmets Roerto Muscedere heretly will e whe usig the Simpso s /3 rule. We oti re o.367 which is much etter th the Trpezoidl rule.688 usig =. We c see the pproimtio is etter, ut with ew more clcultios tegrtio

23 Simpso s s /3 Rule Add series o itervls d h 4 d d 3-7 Roerto Muscedere 3 We c improve the Simpso s /3 rule y pplyig it to multi segmets o the uctio. We do this y rekig the itegrl up ito multiple segmets tegrtio 3

24 tegrtio Roerto Muscedere Simpso Simpso s /3 Rule s /3 Rule Sustitutig Simpso s /3 rule or ech itegrl yields: h h h We c ow sustitute i the sigle-segmet Simpso s /3 rule to oti the ove uctio.

25 tegrtio Roerto Muscedere Simpso Simpso s /3 Rule s /3 Rule Groupig the terms: must e eve,4,6,3,5 4 3 j j i i we group the terms, we get simpler uctio. There re two seprte summtios i this uctio. Oe or the odd i s d oe or the eve i s. This uctio is roke up ito two lies so it c e see etter. This is ot mtri.

26 Simpso s s /3 Rule: Emple Numericlly itegrte: =, =.8, = 4 Alyticl swer = Roerto Muscedere 6 Lets try the sme emple gi, this time with = tegrtio 6

27 Simpso s s /3 Rule: Emple Better th with Trpezoidl Rule with eight segmets Roerto Muscedere 7 Pluggig i ll the umers we c see the pproimtio is etter the the trpezoidl rule with =8. ct this Simpso s /3 rule pproimtio is slightly worse th the Trpezoidl with = tegrtio 7

28 tegrtio Code p. clss tegrtio { privte: doule, ; // limits o itegrtio doule *doule ; //uctio to e itegrted pulic: tegrtiodoule *Fdoule, doule, doule { =F; =; =;} doule Trpezoidlit ; doule Simpsos3it ; }; 3-7 Roerto Muscedere 8 We c eted our tegrtio clss to iclude the Simpso s /3 rule. We do this y ddig ew uctio Simpsos tegrtio 8

29 tegrtio Code p.3 doule tegrtio::simpsos3it { doule h=-/; doule sum=.; doule =; doule ; orit i=; i<; i++ { +=h; i i% sum+=4*; else sum+=*; } =++sum*h/3.; } retur ; 3-7 Roerto Muscedere 9 For the Simpso s /3 rule, we slightly modiy the Trpezoidl uctio. For Simpso s we eed to sum the odd d eve i s seprtely. our loop we check to see i i modulo is ot zero, i so, is odd d we dd *4 to the sum. it is eve, we dd * to the sum. Our clcultio o is tht o Simpso s /3 rule. The uctios re ot very dieret rom ech other tegrtio 9

30 Simpso s s /3 Emple - Code doule FUNdoule // User supplied uctio { retur. + 5* - ** + 675*** - 9**** + 4*****; } it mi { tegrtio FUN,,.8; it i; ori=;i<=6;i+= cout << "=" << i << ", trp=" <<.Trpezoidli << ", simp=" <<.Simpsos3i << edl; } retur ; 3-7 Roerto Muscedere 3 We hve the sme uctio s eore, just s our erlier emples. The mi code hs ee eteded to clculte the re usig oth the Trpezoidl rule d Simpso s /3 rule or oly eve vlues o tegrtio 3

31 Simpso s s /3 Emple - Output =, trp=.688, simp= =4, trp=.4848, simp=.6347 =6, trp=.577, simp=.6376 =8, trp=.68, simp= =, trp=.654, simp=.64 =, trp=.68, simp=.643 =4, trp=.675, simp=.644 =6, trp=.6355, simp= Roerto Muscedere 3 We c see the results or = to 6. Oviously the Simpso s /3 rule otis etter pproimtio with lesser clcultios, however, it sometimes otis poor clcultios do to the ct it is pproimtig secod order segmets tegrtio 3