Numerical Methods. Lecture 5. Numerical integration. dr hab. inż. Katarzyna Zakrzewska, prof. AGH. Numerical Methods lecture 5 1

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1 Numeril Methods Leture 5. Numeril itegrtio dr h. iż. Ktrzy Zkrzewsk, pro. AGH Numeril Methods leture 5

2 Outlie Trpezoidl rule Multi-segmet trpezoidl rule Rihrdso etrpoltio Romerg's method Simpso's rule Gussi method Numeril Methods leture 5

3 Numeril itegrtio - ide d Y The itegrl e pproimted y: d S i i Δ i X i i i Numeril Methods leture 5

4 Newto Cotes methods Newto Cotes itegrtio elogs to lss o methods with ied odes: utio is iterpolted y polyomil e.g. Lgrge polyomil where: 0... The, the itegrl o e pproimted s itegrl o the iterpolted utio d d Numeril Methods leture 5

5 5 Trpezoidl rule The trpezoidl rule ssumes:, thus: 0 d d d 0 But wht is 0 d? 0 Now i oe hooses,, i, s the two poits to pproimte y stright lie rom to. t ollows tht: Numeril Methods leture 5

6 Trpezoidl rule d d re o trpezoid Y X Numeril Methods leture 5 6

7 Trpezoidl rule Emple : Veloity vt o roket rom t 8 s to t 0 s is give y: 0000 v t 000 l 9. 8t t Use the sigle segmet trpezoidl rule to id the diste overed y the roket rom t 8 s to t 0 s Fid the true reltive error. Numeril Methods leture 5 7

8 Trpezoidl rule 0000 t 000 l 9. 8t t l l s 0 s m / m / s s m Numeril Methods leture 5 8

9 Trpezoidl rule The true vlue Δ l 9.8t dt t 0 06m The reltive error: t % 06 Numeril Methods leture 5 9

10 Multi-segmet trpezoidl rule The true error usig Y sigle segmet trpezoidl rule ws lrge. We divide the itervl rom to ito smller segmets o equl legth h d pply the trpezoidl rule over eh segmet: h or X d h h h d d d h h h h d Numeril Methods leture 5 0

11 d ih i h h h h h h d d... d d d... h h h h h ] [... h h Multi-segmet trpezoidl rule Numeril Methods leture 5

12 Emple : Multi-segmet trpezoidl rule Veloity vt o roket rom t 8 s to t 0 s is give y: 0000 v t 000 l 9. 8t t Use the omple segmet trpezoidl rule to id the diste overed rom t 8 s to t 0 s or Fid the true reltive error. Numeril Methods leture 5

13 Multi-segmet trpezoidl rule 8 s 0 s ih i 0 8 h s i ih 0 [ 8 9 0] [ ] 66 m Numeril Methods leture 5

14 Multi-segmet trpezoidl rule true vlue: l 9. 8t dt t 06m The reltive error: 0666 t 00.85% 06 Numeril Methods leture 5

15 Multi-segmet trpezoidl rule E t % % t Numeril Methods leture 5 5

16 Estimtio o error The reltive error or simple trpezoidl rule is E t " ζ, < ζ < The reltive error i the multi-segmet trpezoidl rule is sum o errors or eh segmet. The reltive error withi the irst segmet is give y: [ h ] E " ζ, < ζ < h " ζ h Numeril Methods leture 5 6

17 Estimtio o error By logy: [ ih i h ] Ei " ζi, i h < ζi < ih h " ζ i or -th segmet : [ { h} ] E " ζ, h < ζ < h " ζ Numeril Methods leture 5 7

18 Estimtio o error The totl error i the omple trpezoidl rule is sum o the errors or sigle segmet: E t E i i Formul: " ζ h i " ζi i i yields pproimte verge vlue o the seod derivtive i the rge o E t " ζ i α i < < Numeril Methods leture 5 8

19 Estimtio o error Tle elow presets the results or the itegrl l 9. 8t dt t s utio o the umer o segmets. Whe twie ireses, the solute error E t dereses our times! Vlue E % % t t Numeril Methods leture 5 9

20 Rihrdso s etrpoltio d Romerg s method o itegrtio Rihrdso s etrpoltio d Romerg s method o itegrtio ostitute etesio o the trpezoidl method d give etter pproimtio o the itegrl y reduig the true error. Numeril Methods leture 5 0

21 Rihrdso etrpoltio The true error otied whe usig the multi-segmet trpezoidl rule with segmets to pproimte itegrl is give y: E t where: C is pproimte ostt o proportiolity C Sie: E TV t true vlue pproimte vlue usig -segmets Numeril Methods leture 5

22 Rihrdso etrpoltio t e show tht: C TV the umer o segmets is douled rom to : C C TV TV We get: TV Numeril Methods leture 5

23 Rihrdso etrpoltio Emple : The veloity vt o roket rom t 8 s to t 0 is give y: 0000 v t 000 l 9. 8t t Use Rihrdso etrpoltio rule to id the diste overed or Fid the reltive true error Numeril Methods leture 5

24 Tle o results or 8 segmets trpezoidl rule E t % % t Numeril Methods leture 5

25 Rihrdso etrpoltio 66m m TV TV or 66 06m Numeril Methods leture 5 5

26 Rihrdso etrpoltio The true vlue: l 9. 8t dt t 06 m The solute true error: Et 0606 m Numeril Methods leture 5 6

27 Rihrdso etrpoltio The reltive error: 0606 t % 06 Compriso o dieret methods:. 8 m Trpezoidl rule % m % t Trpezoidl rule Rihrdso etrpoltio t Rihrdso etrpoltio Numeril Methods leture 5 7

28 Romerg's method Romerg s method uses the sme ptter s Rihrdso etrpoltio. However, Romerg used reursive lgorithm or the etrpoltio s ollows: TV The true vlue TV is repled y the result o the Rihrdso etrpoltio R Note lso tht the sig is repled y the sig R Numeril Methods leture 5 8

29 Romerg's method TV Ch Esimted true vlue is give y: R where: Ch is the vlue o the error o pproimtio Aother vlue o itegrl otied while doulig the umer o segmets rom to : R Estimted true vlue is give y: TV R R 5 R R R R Numeril Methods leture 5 9

30 Romerg's method A geerl epressio or Romerg itegrtio e writte s: k, j k, j k, j k, j, k k The ide k represets the order o etrpoltio k represets the vlues otied rom the regulr trpezoidl rule k represets the vlues otied usig the true error estimte s Oh The vlue o itegrl with or j is more urte th the vlue o the itegrl or j ide Numeril Methods leture 5 0

31 Romerg's method Emple : Veloity vt o roket rom t 8 s to t 0 s is give y: 0000 v t 000 l 9. 8t t Use Romerg s method to id the diste overed. Use,,, d 8 Fid the solute true error d the reltive pproimte error Numeril Methods leture 5

32 Tle o results or 8 segmets trpezoidl rule E t % % t Numeril Methods leture 5

33 Romerg's method From this tle, the iitil vlues rom the trpezoidl rule re: ,,, 07, To get the irst order etrpoltio vlues:,,,, Numeril Methods leture 5

34 Romerg's method Similrly,,,,,,,,, Numeril Methods leture 5

35 Romerg's method For the seod order etrpoltio: Similrly,,,, 06 06, 06 06, 5, ,, Numeril Methods leture 5 5

36 Romerg's method For the third order etrpoltio,,, 6, m Numeril Methods leture 5 6

37 Romerg's method Appr. Appr. Appr. -segmet segmet segmet segmet 07 mproved estimtes o the vlue o itegrl usig Romerg itegrtio Numeril Methods leture 5 7

38 Simpso's rule The trpezoidl rule ws sed o pproimtig the itegrd y irst order polyomil, d the itegrtig the polyomil over itervl rom to. Simpso s rule ssumes tht the itegrd e pproimted y seod order polyomil. d d where: 0 Numeril Methods leture 5 8

39 Simpso's rule A prol pssig through three poits :,, Y,,, X Numeril Methods leture 5 9

40 0 Coeiiets 0,, re: Simpso's rule Numeril Methods leture 5 0

41 Simpso's rule Sie: d 0 d 0 0 Numeril Methods leture 5

42 Simpso's rule d 6 h t ollows tht: h d it is lled Simpso s / rule Numeril Methods leture 5

43 d d d 0... Multi-segmet Simpso's rule 0 d d Numeril Methods leture d i i h i,,...,...

44 Simpso's rule d 0 h 6... h... 6 h 6... h 6 Numeril Methods leture 5

45 Simpso's rule h d 0 {... } [...] {... } }]... h odd i i 0 i i i i eve odd i i 0 i i i i eve Numeril Methods leture 5 5

46 Estimtio o errors i Simpso's rule Approimte vlues o the itegrl, usig Simpso's rule with multiple segmets Approimte vlues E t Є t % 0.007% % 0.000% % Numeril Methods leture 5 6

47 Simpso's rule errors Error or oe segmet 5 E t ζ, < ζ < 880 Error or the multi-segmet E E 5 h 5 0 ζ ζ, h ζ ζ, < 0 < ζ < < ζ E i i i ζ i 5 h 90 ζ i, i < ζi < i Numeril Methods leture 5 7

48 Simpso's rule errors True error E i t E i i ζ i 5 h 90 i ζ i ζ i i 90 E t i ζ the verge vlue o the derivtive i Numeril Methods leture 5 8

49 Guss-Qudrture Method Gussi itegrl is give y: d ostt oeiiets Poits d, whih deie the vlue o the itegrd re ot ied s eore, ut there re priori distriuted rdomly withi <,>. Numeril Methods leture 5 9

50 50 Guss-Qudrture Method. 0 d d There re our ukows,,,. These re oud y ssumig tht the ormul gives et results or itegrtig geerl third order polyomil: Numeril Methods leture 5

51 5 Guss-Qudrture Method 0 0 d 0 d d d 0 0 Hee: The ormul would the give: Numeril Methods leture 5

52 5 Guss-Qudrture Method 0 0 Numeril Methods leture 5

53 5 Guss-Qudrture Method This gives us our equtios s ollows: we id tht the ove our simulteous olier equtios hve oly oe eptle solutio Numeril Methods leture 5

54 5 Guss-Qudrture Method d Hee: d Geerl -poit rules would pproimte the itegrl: Numeril Methods leture 5

55 Guss-Qudrture Method The oeiiets d rgumets or -poit Guss method re give withi the rge o <-,> : g d i i g i Coeiiets Futio rgumets Numeril Methods leture 5 55

56 Guss-Qudrture Method Coeiiets Futio rgumets Numeril Methods leture 5 56

57 So i the tle is give or: Guss-Qudrture Method g d d? The swer lies i tht y itegrl with limits o [, ] e overted ito itegrl with limits: [,] itegrls, how does oe solve Let, mt i, t ir t, t ollows tht: m Numeril Methods leture 5 57

58 58 Hee: t dt d Sustitutig our vlues o d d ito the itegrl gives us: dt t d Guss-Qudrture Method Numeril Methods leture 5

59 Emple 5: Guss-Qudrture Method Use two-poit Guss qudrture rule to pproimte the diste overed y roket rom t 8 s to t 0 s i the veloity is give y: 0000 v t 000 l 9. 8t t Use the Guss-Qudrture Mmethod to id the diste overed rom t 8 s to t 0 s Fid the true error. Numeril Methods leture 5 59

60 Guss-Qudrture Method First, hge the limits o itegrtio rom rom [8,0] to [-,] 0 t dt d 8 The weightig tors d utio rgumet vlues re : 9 d Numeril Methods leture 5 60

61 Guss-Qudrture Method The ormul is: 9 d m Numeril Methods leture 5 6

62 Guss-Qudrture Method Sie: l l Numeril Methods leture 5 6

63 Guss-Qudrture Method The solute true error: E t m The reltive error: t t 00% % Numeril Methods leture 5 6