Numerical Integration Problems

Transcription

1 Integrtion COS 33

2 Nuericl Integrtion Proles Bsic D nuericl integrtion Given ility to evlute or ny, ind Gol: est ccurcy wit ewest sples Clssic prole even nlytic unctions not necessrily integrle in closed or Oter proles uture lectures: Multi-diensionl integrtion Ordinry dierentil equtions Prtil dierentil equtions G = d e t dt

3 Qudrture Sple t set o points Approite y unction Integrte unction Alterntives: it single unction vs. ultiple piecewise Even vs. uneven spcing

4 Trpezoidl Rule Approite unction y trpezoid

5 Trpezoidl Rule d

6 Etended Trpezoidl Rule d d n Divide into segents o widt, piecewise trpezoidl pproition

7 Trpezoidl Rule Error Anlysis How ccurte is tis pproition? Strt wit Tylor series or round idpoint E d

8 Trpezoidl Rule Error Anlysis Epnd LHS: Epnd RHS: d [ ] E E =

9 Trpezoidl Rule Error Anlysis So, E = In generl, error or single segent proportionl to 3 Error or sudividing entire intervl proportionl to Cuic locl ccurcy, qudrtic glol ccurcy Ect or liner unctions Note tt only even-power ters in error:, 4, etc.

10 Deterining Step Size Cnge in integrl wen reducing step size is resonle guess or ccurcy or trpezoidl rule, esy to go ro / witout wsting previous sples

11 Sipson s Rule Approite integrl y prol troug tree points d O Better ccurcy or se # o evlutions Glol error O 4, ect or cuic! unctions Higer-order polynoils Newton-Cotes: Glol error O k or k odd, O k or k even

12 Ricrdson Etrpoltion Better wy o getting iger ccurcy or given # o sples Suppose we ve evluted integrl or step size nd step size / using trpezoidl rule: Ten = = 4 / 4 β α β α 4 3 / 3 4 O =

13 Ricrdson Etrpoltion Tis trets te pproition s unction o nd etrpoltes te result to =0 Cn repet: / 4 / / O O O O /3 4/3 /5 6/5 /63 64/63

14 Open Metods Trpezoidl rule won t work i unction undeined t one o te points were evluting Most oten: unction ininite t one endpoint 0 d Open etods only evlute unction on te open intervl i.e., not t endpoints

15 Midpoint Rule Approite unction y rectngle evluted t idpoint

16 Etended Midpoint Rule d 3 d Divide into segents o widt :

17 Midpoint Rule Error Anlysis ollowing siilr nlysis to trpezoidl rule, ind tt locl ccurcy is cuic, qudrtic glol ccurcy Surprisingly, leding-order constnt is ½ s ig! Better tn trpezoidl rule wit ewer sples orul suitle or dptive etods nd Ricrdson etrpoltion, ut cn t lve intervls witout wsting sples

18 Etended / Adptive Midpoint Rule Cn cut intervl into tirds:

19 Liits t Ininity Usul trick: cnge o vriles d = / t / t dt Works wit, se sign, one o te ininite Oterwise, split into ultiple pieces Also requires to decrese ster tn / Else need dierent cnge o vriles, i possile!

20 Oter Qudrture Rules Nonunior spling: copleity vs. ccurcy Clensw-Curtis: Ceysev polynoils Cnge o vriles: =cos θ Sple t etre o polynoils T-sed lgorit to ind weigts Gussin qudrture Optiize spling loctions to get igest possile ccurcy: O n or n spling points

21 Discontinuities All te ove error nlyses ssued nice continuous, dierentile unctions In te presence o discontinuity, ll etods revert to ccurcy proportionl to In generl, i te k-t order derivtive is discontinuous, cn do no etter tn O k Loclly-dptive etods: do not sudivide ll intervls eqully, ocus on tose wit lrge error estited ro cnge wit single sudivision