Computational Methods CMSC/AMSC/MAPL 460. Quadrature: Integration

Transcription

1 Computatioal Metods CMSC/AMSC/MAPL 6 Quadrature: Itegratio Ramai Duraiswami, Dept. o Computer Siee Some material adapted rom te olie slides o Eri Sadt ad Diae O Leary

2 Numerial Itegratio Idea is to do itegral i small parts, like te way you irst leared itegratio - a summatio Numerial metods just try to make it aster ad more aurate

3 Basi Numerial Itegratio Weigted sum o utio values d i i i a -

4 Numerial Itegratio Caraterized y were te utio is evaluated Newto-Cotes Closed Formulae -- Use ot ed poits Trapezoidal Rule : Liear Simpso s /-Rule : Quadrati Simpso s /8-Rule : Cui Boole s Rule : Fourt-order Newto-Cotes Ope Formulae -- Use oly iterior poits midpoit rule

5 Trapezoid Rule Straigt-lie approimatio d i i i a L

6 Eample:Trapezoid Rule Evaluate te itegral e d Eat solutio itegratio y parts` e d e Trapezoidal Rule I e e e d e %

7 Simpso s /-Rule Approimate te utio y a paraola d i i i a L

8 Simpso s /8-Rule Approimate y a ui polyomial 8 d i i i a L

9 Eample: Simpso s Rules Evaluate te itegral Simpso s /-Rule e d I e d Simpso s /8-Rule 8 e e % I e d 8 / % 56.96

10 Midpoit Rule Newto-Cotes Ope Formula a d a a a m a a m

11 Two-poit Newto-Cotes Ope Formula Approimate y a straigt lie a d a a 8

12 Tree-poit Newto-Cotes Ope Formula Approimate y a paraola a a d 5 7 a

13 Better Numerial Itegratio Composite itegratio Composite Trapezoidal Rule Composite Simpso s Rule Riardso Etrapolatio et lass Romerg itegratio et lass

14 Cost Struture o quadrature programs Typially require several alls to utio routie tat is eig itegrated So ost is i terms o utio alls Auray As a umer As te igest order polyomial tat is itegrated eatly

15 Error aalysis All ormulas tus ar ave orm Q= i=m i i Deie residue or error utio as R=I-Q We aot alulate I i geeral doig so would require us to kow te rigt aswer Istead we a ompute our error o a partiular lass o utios tat are easy to itegrate te polyomials Also tese orm a asis i a utio spae

16 Error Trapezoidal rule is eat or ostat ad liear utios Wat aout oters? Let T deote trapezoidal rule result. Te or some \i [a,] I-T=-[-a /] Similarly Simpso rule is eat or quadratis ad as error proportioal to tird derivative We eed utio evaluatios or a d order error or a rd order ad so o

17 Error aalysis

18 Formula or trapezoidal rule I t ad its st two derivatives are otiuous o [a,], te

19 Error trapezoidal rule

20 How to redue error? I -a is large error is iger Use omposite rules.

21 Apply trapezoid rule to multiple segmets over itegratio limits Two segmets Tree segmets Four segmets May segmets

22 Composite Trapezoid Rule d d d d i a a

23 Composite Trapezoid Rule Evaluate te itegral, I %, I..75%, I % 8,.5 I % 6,.5 I % I e d

24 Composite Trapezoid Rule wit Uequal Segmets Evaluate te itegral =, =, =.5, =.5 I e d I.5 d.5.5 d.5 d e e e e e.5 d 7 8.5e e %.5

25 Composite Simpso s Rule Pieewise Quadrati approimatios a

26 d d d d i i i- a Composite Simpso s Rule Multiple appliatios o Simpso s rule

27 Composite Simpso s Rule Evaluate te itegral =, = I e d I 8 e e % =, = I e e e 8.7% 6 e 8

28 Composite Simpso s Rule wit Uequal Segmets Evaluate te itegral =.5, =.5 I e d I d d % e e e e e

29 Error o omposite trapezoid Error dereases y a ator o -a

30 Adaptive itegratio Oter ator i te error is te seod derivative Idea, keepig te error ied, redue te size o te iterval were seod derivative is ig I we used te worst part o te domai to determie step size we would waste resoures o te easy parts Idea o adaptive use dieret or dieret parts

31 Adaptive itegratio I geeral we do ot ave a grap to tell us were tigs are ad. Need a utio wi estimates te error loally Idea: use two ormulae: oe more aurate, ad oe less aurate i ea iterval ad estimate te error Dieree gives a estimate o te error loally Were error is larger we eed to do sometig

32 I loal error estimate is less ta tolerae i a partiular regio we a stop dividig it. Oterwise split te iterval i two piees, ad repeat te proedure Ea su-iterval tolerae requiremet eeds to e al tat o te parets Upo overgee ea suiterval aieves suess. Some suitervals eeded lots o poits, oters ew Add up all su iterval aswers ad report to allig program

33 Adaptive metods Allow us to aieve a give tolerae at a give ost

34 Riardso Etrapolatio Wikipedia ttp://e.wikipedia.org/wiki/riardso_etrapolatio

35 Riardso Etrapolatio For trapezoidal rule k t level o etrapolatio B 6B 5 C B A B A B A A A A A A A A d A a C C / D k k

36 Trapezoid k k k k k O O O 6 O 8 O I, I, I, I, I, / I, I, I, I, / I, I, I, /8 I, I, /6 I, I j, I j, Romerg Itegratio Aelerated Trapezoid Rule I j,k k I j,k k I j,k ; k,,, 6I j, I j, 5 6I j, I j, 6 56I j, I j, 55

37 Romerg Itegratio Aelerated Trapezoid Rule I e d k k k k k O O O 6 O 8 O %.57%.5%.68%.5%

38 Gaussia Quadratures Newto-Cotes Formulae use evely-spaed utioal values Did ot use te leiility we ave to selet te quadrature poits I at a quadrature poit as several degrees o reedom. I= i=m i i A ormula wit m utio evaluatios requires speiiatio o m umers i ad i Gaussia Quadratures selet ot tese weigts ad loatios so tat a iger order polyomial a e itegrated alteratively te error is proportioal to a iger derivatives Prie: utioal values must ow e evaluated at ouiormly distriuted poits to aieve iger auray Weigts are o loger simple umers Usually derived or a iterval su as [-,] Oter itervals [a,] determied y mappig to [-,]

39 Gaussia quadrature More ormally Gaussia quadrature is deied wit a weigt utio = i=m i t i Here te weigt utio is Speial ame: Gauss-Legedre quadrature Ca deie quadratures or oter wt as log as

40 Gauss-Legedre Quadrature o [-, ] Two utio evaluatios: Coose,,, su tat te metod yields eat itegral or =,,, d i i i d : -

41 Fidig quadrature odes ad weigts Oe way is troug te teory o ortogoal polyomials. Here we will do it via rute ore Set up equatios y requirig tat te m poits guaratee tat a polyomial o degree m- is itegrated eatly. I geeral proess is o-liear ivolves a polyomial utio ivolvig te ukow poit ad its produt wit ukow weigt Ca e solved y usig a multidimesioal oliear solver Alteratively a sometimes e doe step y step

42 Gauss-Legedre Quadrature o [-, ] Eat itegral or =,,, Four equatios or our ukows d : d d d d d I

43 Error I we approimate a utio wit a Gaussia quadrature ormula we ause a error proportioal to t derivative

44 Gaussia Quadrature o [-, ] : d - Coose,,,,, su tat te metod yields eat itegral or =,,,,, 5

45 Gaussia Quadrature o [-, ] d d d d d d 5 / 5 / 9 5 / 9 8 / 9 5 /

46 Gaussia Quadrature o [-, ] Eat itegral or =,,,,, d I

47 Gaussia Quadrature o [a, ] Coordiate trasormatio rom [a,] to [-,] t t a a d g d a a a dt t t a t a a t

48 Eample: Gaussia Quadrature Evaluate Coordiate trasormatio Two-poit ormula.% e e d I dt te I t d dt ; d d e dt te I a a t t

49 Eample: Gaussia Quadrature Tree-poit ormula I d e e % Four-poit ormula e I d %

50 Gauss-Loatto: Oter rules requirig ed poits e iluded i te ormula Gauss-Radau Require oe ed poit e i te ormula

51 Higer dimesios Ca take similar approa it polyomials ad evaluate However, as dimesioality ireases umer o poits eeded ireases epoetially i dimesio Very ig dimesios: oly pratial way is Mote- Carlo itegratio Evaluates itegrals proailistially I tis ase epeted value is te omputed itegral Error is te variae o te estimate.