Numerical Integration Formulas

Transcription

1 Numerical Itegratio Formulas Berli Che Departmet o Computer Sciece & Iormatio Egieerig Natioal Taiwa Normal Uiversity Reerece: 1. Applied Numerical Methods with MATLAB or Egieers, Chapter 19 & Teachig material

2 Chapter Objectives (1/) Recogizig that Newto-Cotes itegratio ormulas are based o the strategy o replacig a complicated uctio or tabulated data with a polyomial that is easy to itegrate Kowig how to implemet the ollowig sigle applicatio Newto-Cotes ormulas: Trapezoidal rule Simpso s 1/3 rule Simpso s 3/8 rule Kowig how to implemet the ollowig composite Newto-Cotes ormulas: Trapezoidal rule Simpso s 1/3 rule NM Berli Che

3 Chapter Objectives (/) Recogizig that eve-segmet-odd-poit ormulas like Simpso s 1/3 rule achieve higher tha epected accuracy Kowig how to use the trapezoidal rule to itegrate uequally spaced data Uderstadig the dierece betwee ope ad closed itegratio ormulas NM Berli Che 3

4 Itegratio Itegratio: I b d a is the total value, or summatio, o () d over the rage rom a to b: NM Berli Che 4

5 Newto-Cotes Formulas The Newto-Cotes ormulas are the most commo umerical itegratio schemes Geerally, they are based o replacig a complicated uctio or tabulated data with a polyomial that is easy to itegrate: I b d d a b a where () is a th order iterpolatig polyomial NM Berli Che 5

6 Newto-Cotes Eamples The itegratig uctio ca be polyomials or ay order - or eample, (a) straight lies or (b) parabolas The itegral ca be approimated i oe step or i a series o steps to improve accuracy NM Berli Che 6

7 The Trapezoidal Rule The trapezoidal rule is the irst o the Newto-Cotes closed itegratio ormulas; it uses a straight-lie approimatio or the uctio: b I d I a b (a) b a ad a b a I b a a b NM Berli Che 7

8 Error o the Trapezoidal Rule A estimate or the local trucatio error o a sigle applicatio o the trapezoidal rule is: E t 1 1 b a 3 where is somewhere betwee a ad b This ormula idicates that the error is depedet upo the curvature o the actual uctio as well as the distace betwee the poits Error ca thus be reduced by breakig the curve ito parts NM Berli Che 8

9 Trapezoidal Rule: A Eample Eample 19.1 NM Berli Che 9

10 Composite Trapezoidal Rule Assumig +1 data poits are evely spaced, there will be itervals over which to itegrate The total itegral ca be calculated by itegratig each subiterval ad the addig them together: 1 I d d d d 0 I I h 1 0 i i NM Berli Che 10

11 Composite Trapezoidal Rule: A Eample Eample 19. NM Berli Che 11

12 MATLAB Program NM Berli Che 1

13 Simpso s Rules Oe drawback o the trapezoidal rule is that the error is related to the secod derivative o the uctio More complicated approimatio ormulas ca improve the accuracy or curves - these iclude usig (a) d ad (b) 3rd order polyomials The ormulas that result rom takig the itegrals uder these polyomials are called Simpso s rules NM Berli Che 13

14 Simpso s 1/3 Rule Simpso s 1/3 rule correspods to usig secod-order polyomials. Usig the Lagrage orm or a quadratic it o three poits: Itegratio over the three poits simpliies to: I I h 3 0 b a 4 0 d h b a NM Berli Che 14

15 Error o Simpso s 1/3 Rule A estimate or the local trucatio error o a sigle applicatio o Simpso s 1/3 rule is: E t b a 5 where agai is somewhere betwee a ad b This ormula idicates that the error is depedet upo the ourth-derivative o the actual uctio as well as the distace betwee the poits Note that the error is depedet o the ith power o the step size (rather tha the third or the trapezoidal rule) Error ca thus be reduced by breakig the curve ito parts NM Berli Che 15

16 Simpso s 1/3 Rule: A Eample Eample 19.3 NM Berli Che 16

17 Composite Simpso s 1/3 Rule Simpso s 1/3 rule ca be used o a set o subitervals i much the same way the trapezoidal rule was, ecept there must be a odd umber o poits Because o the heavy weightig o the iteral poits, the ormula is a little more complicated tha or the trapezoidal rule: NM Berli Che 17 a b I h I h h h I d d d d I j j i i i i j j i i i i eve, 1 odd, 1 0 eve, 1 odd, a b h

18 Composite Simpso s 1/3 Rule: A Eample Eample 19.4 NM Berli Che 18

19 Simpso s 3/8 rule correspods to usig thirdorder polyomials to it our poits. Itegratio over the our poits simpliies to: I I 3h 8 Simpso s 3/8 Rule b a 3 h d 0 3 b a Simpso s 3/8 rule is geerally used i cocert with Simpso s 1/3 rule whe the umber o segmets is odd NM Berli Che 19 3

20 Simpso s 3/8 Rule: A Eample (1/) Eample 19.5 NM Berli Che 0

21 Simpso s 3/8 Rule: A Eample (/) NM Berli Che 1

22 Higher-Order Formulas Higher-order Newto-Cotes ormulas may also be used - i geeral, the higher the order o the polyomial used, the higher the derivative o the uctio i the error estimate ad the higher the power o the step size As i Simpso s 1/3 ad 3/8 rule, the eve-segmet-oddpoit ormulas have trucatio errors that are the same order as ormulas addig oe more poit. For this reaso, the eve-segmet-odd-poit ormulas are usually the methods o preerece NM Berli Che

23 Itegratio with Uequal Segmets Previous ormulas were simpliied based o equispaced data poits - though this is ot always the case The trapezoidal rule may be used with data cotaiig uequal segmets: 1 I d d d d 0 0 I NM Berli Che 3

24 Itegratio Code or Uequal Segmets NM Berli Che 4

25 MATLAB Fuctios MATLAB has built-i uctios to evaluate itegrals based o the trapezoidal rule z = trapz(y) z = trapz(, y) produces the itegral o y with respect to. I is omitted, the program assumes h=1 z = cumtrapz(y) z = cumtrapz(, y) produces the cumulative itegral o y with respect to. I is omitted, the program assumes h=1 NM Berli Che 5

26 Multiple Itegrals Multiple itegrals ca be determied umerically by irst itegratig i oe dimesio, the a secod, ad so o or all dimesios o the problem T (, y) y y 7 NM Berli Che 6