Numerical Differentiation and Integration

Transcription

1 Numerl Deretto d Itegrto

2 Overvew Numerl Deretto Newto-Cotes Itegrto Formuls Trpezodl rule Smpso s Rules Guss Qudrture Cheyshev s ormul

3 Numerl Deretto Forwrd te dvded deree Bkwrd te dvded deree Ceter te dvded deree All sed o the Tylor Seres ' h h ''!...

4 Forwrd Fte Deree '' ' '...! '' ' O h h h O h h h h

5 Forwrd Dvded Deree ' O h Oh () ( +,y + ) (, y )

6 Bkwrd Deree Appromto o the Frst Dervtve Epd the Tylor seres kwrds '' h h '...! '' h h '...! ' h h The error s stll O(h)

7 Cetered Deree Appromto o the Frst Dervtve Sutrt kwrd deree ppromto rom orwrd Tylor seres epso ' 6 ''' ' '...! '' ' O h h h h h h

8 () ' h O h (,y ) ( +,y + ) ( -,y - )

9 () () true dervtve orwrd te dvded deree ppro. () () kwrd te dvded deree ppro. etered te dvded deree ppro.

10 Newto-Cotes Itegrto Commo umerl tegrto sheme Bsed o the strtegy o replg omplted uto or tulted dt wth some ppromtg uto tht s esy to tegrte I d d...

11 Newto-Cotes Itegrto Commo umerl tegrto sheme Bsed o the strtegy o replg omplted uto or tulted dt wth some ppromtg uto tht s esy to tegrte I d d... () s th order polyoml

12 () () The ppromto o tegrl y the re uder - rst order polyoml - seod order polyoml

13 () () The ppromto o tegrl y the re uder - rst order polyoml - seod order polyoml We lso ppromted the tegrl y usg seres o polyomls ppled pee wse.

14 () A ppromto o tegrl y the re uder strght le segmets.

15 () A ppromto o tegrl y the re uder strght le segmets.

16 Trpezodl Rule Frst o the Newto-Cotes losed tegrto ormuls Correspods to the se where the polyoml s rst order I d d A strght le e represeted s:

17 Trpezodl Rule I d d d Itegrte ths equto. Results the trpezodl rule. I Multple Applto o the Trpezodl Rule I ( ) d ( ) d ( ) d I h h h I

18 Multple Applto o the Trpezodl Rule I } } wdth verge heght The verge heght represets weghted verge o the uto vlues Note tht the teror pots re gve twe the weght o the two ed pots

19 Smpso s / Rule Correspods to the se where the uto s seod order polyoml I d d Desgte d s d, d estmte () s seod order Lgrge polyoml I d d... d

20 Smpso s / Rule Ater tegrto d lger mpulto, we get the ollowg equtos I h 6 } } wdth verge heght Multple Applto o Smpso s / Rule I d d d I j j ) ( ) ( ) (,,5..,,6..

21 Smpso s /8 Rule Correspods to the se where the uto s thrd order polyoml 8 h I d d I

22 Guss Qudrture Rule Prevously, the Trpezodl Rule ws developed y the method o udetermed oeets. The result o tht developmet s summrzed elow. ( ) d ( ) ( ) ( ) ( ) The two-pot Guss Qudrture Rule s eteso o the Trpezodl Rule ppromto where the rgumets o the uto re ot predetermed s d ut s ukows d. I the two-pot Guss Qudrture Rule, the tegrl s ppromted s I ( )d ( ) ( )

23 Bss o the Guss Qudrture Rule The our ukows,, d re oud y ssumg tht the ormul gves et results or tegrtg geerl thrd order polyoml,. ) ( Hee d )d (

24 It ollows tht )d ( Equtg Equtos the two prevous two epressos yeld

25 Se the ostts,,, re rtrry The our smulteous oler Equtos hve oly oe eptle soluto, ) ( d Hee Two-Pot Guss Qudrture Rule

26 6 Hgher Pot Guss Qudrture Formuls ) ( ) ( ) ( ) ( d s lled the three-pot Guss Qudrture Rule. The oeets,, d, d the utol rgumets,, d re lulted y ssumg the ormul gves et epressos or d 5 5 Geerl -pot rules would ppromte the tegrl ) ( ) ( ) ( )d ( tegrtg th order polyoml

27 Argumets d Weghg Ftors or -pot Guss Qudrture Formuls I hdooks, oeets d rgumets gve or -pot Guss Qudrture Rule re gve or tegrls g( )d s show Tle. g( ) Tle : Weghtg tors d uto rgumets used Guss Qudrture Formuls. Pots Weghtg Ftors =. =. = = = = = = = Futo Argumets = = = =. = = = =.998 =.866 7

28 Argumets d Weghg Ftors or -pot Guss Qudrture Formuls Tle (ot.) : Weghtg tors d uto rgumets used Guss Qudrture Formuls. Pots Weghtg Ftors 5 = = = = = =.79 = = = = =.79 Futo Argumets = = =. = = = = = = = =

29 Argumets d Weghg Ftors or -pot Guss Qudrture Formuls So the tle s gve or g( )d tegrls, how does oe solve ( )d? The swer les tht y tegrl wth lmts o, e overted to tegrl wth lmts, Let mt I, the t I the t, Suh tht: 9 m

30 Argumets d Weghg Ftors or -pot Guss Qudrture Formuls The Hee t dt d Susttutg our vlues o, d d to the tegrl gves us ) ( dt t d

31 Cheyshev's Formul Cheyshev s Qudrture -pot rules would ppromte the tegrl wth ll equl weghts. ( ) d w[ ( ) ( ) ( )] Tle : Weghtg tors d uto rgumets used Guss Qudrture Formuls. Pots Futo Argumets = = = =. = = = = = Pots Futo Argumets 5 = = =. = = = = = = = =

32 THANK YOU Sumtted y: Rkesh Kumr, (Deptt. O Mthemts), P.G.G.C.G, Se-, Chdgrh.