f(x) is a function of x and it is defined on the set R of real numbers. If then f(x) is continuous at x=x 0, where x 0 R.

Transcription

1 MATHEMATICAL PRELIMINARIES Limit Cotiuity Coverget squece Series Dieretible uctios Itegrble uctios Summtio deiitio o itegrl Me vlue theorem Me vlue theorem or itegrls Tylor's theorem Computer represettio o umbers Error lysis d sources o errors Limit () is uctio o d it is deied o the set R o rel umbers. I lim L () the () is sid to hve limit L t =. I = + h, the equtio () c be writte s h L lim h where h is icremet d R. Cotiuity Assume tht () is deied o R o rel umbers. I lim I() is cotiuous t ll poits the () is cotiuous t =, where R. S, the () is cotiuous o the set S o rel umbers. Coverget sequece is iiite sequece d is geerl term o the sequece. I hs limit L, such tht lim L the sequece is coverget sequece. Emple 6 cos is squece. L, thereore squece hs limit L= d it is coverget. lim Error sequece is deied s. Limit o error sequece is lim.

2 () () Series Deiitio : Let be sequece. The is iiite series d clled th prtil sum. I LimS Lim S is L, the the iiite series is coverget d the squece S coverges to limit L, or series is sid to be coverget series i the th prtil sum hs limit L. I series dose't coverge, it is sid to be diverget series. Emple is give. th prtil sum is S i i i i i i L LimS Lim Dieretible uctios Assume tht () is deied o itervl cotiig. I Lim

3 the () is dieretible t d ( ) is clled the derivtive o () t. I icremetl orm or = + h lim h h h I uctio () is dieretible t every poit i R (R represets the set o rel umbers) the () is dieretible o R. Emple Assume tht () = d =., h=. () = d ( ) = (.) =. h lim lim. h h Itegrble uctios C[,b] deotes the set o ll cotiuous uctios o the closed itervl [,b]. b the it is sid tht () is b Assume tht C,b I d F Fb F itegrble d F() is itegrl o () o the closed itervl [,b]. Summtio Deiitio o Itegrl : Assume tht C,b d the itervl [,b] is subdivided ito itervls which re [, ],[, ],[, ],...,[ -, ], where = d =b. Select rbitrry poit t i the itervl [ -, ], =,,,..., d by itroducig the dierece. The b d lim t is clled summtio deiitio o itegrl d t is clled Riem sum o () o the itervl [,b]. Note tht Emple b t d ()= is give o the itervl [,] d d 8 7.

4 Assume tht the itervl [,] is subdivided ito 5 prts d t is te s d., the t d Me Vlue Theorem. Assume tht C[,b] is the set o ll uctios cotiuous o the itervl [,b] d dieretible, the there is umber c, bsuch tht, b c rom igure b below. 4. Me Vlue Theorem or Itegrls C,b or b, the there eists umber c : <c<b such tht b b d c

5 Tylor's Theorem Assume tht () C,b &,b. Where C + is the set o uctios which re + times dieretible d cotiuous. Tylor s series epsio o the uctio () bout is give s!!!! or d! There eists umber,b such tht!!! P Where R () is the remider term. Emple P R ()=cos(), =. Write () s Tylor's series epsio or irst terms or irst ozero 6 terms (=) ()=., ' ()=-si(), ' ()=. '' ()=-cos(), '' ()=-. ''' ()=si(), ''' ()=. (IV) ()=cos(), (IV) ()=. (V) ()=-si(), (V) ()=. (VI) ()=-cos(), (VI) ()=-. (VII) ()=si(), (VII) ()=. (VIII) ()=cos(), (VIII) ()=. (IX) ()=-si(), (IX) ()=. (X) ()=-cos(), (VII) ()= ! 4! 6! 8!!

6 I =(,y) ( hs idepedet vribles) Epsio o (,y) bout P (,y ), y, y!!, y, y y y, y, y y y, y y y y Or i short cut orm!!!! y y y, y y y, y Computer Represettio o Numbers Numericl clcultios re crried out by computers. Computer is the hrdwre o computig. Compilers such s Bsic, C, Fortr etc. d other computer codes (progrms) writte i proper lguges re the sotwre o computig. It is importt to ow tht how the computers store the umbers d me mthemticl clcultios by umbers i umericl lysis. Most o the computers hve iteger mode d lotig-poit mode or represetig umbers. Iteger mode is used or iteger umbers while the lotig-poit orm is used to represet rel umbers. I geerl, computers use the biry umber system while hum beigs use the deciml umber system. There re some computers tht use bse 8 (Octhedrl) or bse 6 (Hedeciml) which re the simple vrits o the biry system. Numbers typed o the computer by hum beigs d the umbers show o the scree re i the deciml umber system; becuse hum beigs c esily commuicte with the computer by usig the deciml umber system. Computers covert our deciml umbers to biry umbers, do rithmetic with biry umbers d illy show the results i deciml umbers. Emple () =* +* +* +* = () where subscripts d std or bses o biry d deciml systems, respectively.

7 Flotig-poit orm o umber: As it is ow tht scietiic ottio is the stdrd wy o writig umbers d it hs oly oe digit beore the deciml poit ollowed by rest o the digits with the pproprite power o te. Emple Geerl orm Scietiic ottio * -5 56, 5.6* 5 Assume tht R represets rel umber d its lotig-poit orm is R r r q or R.d d d d 4 d () where q is clled mtiss deotes the bse o the umber the system beig used; the umber r is clled epoet or chrcteristic ; d, d,...,d re the digits betwee d di d d should be dieret rom zero. Emple I deciml umber system R=±(.d d...d )* r I biry umber system R=±(.d...d )* r The epoetil i equtio () stisies m< r<m with m d M beig itegers. m d M c te dieret vlues i dieret computers. Geerlly, m=-m or m=-m±. Becuse o limittio o m d M, there is limittio o the smllest d the lrgest umber tht vry with computers d compilers. I the limittio is violted, the overlow problems c occur. For emple, o IBM PC (Bsic), the rge is pproimtely rom.9* -9 to.7* 8 while o the sme mchie with QuicC, the rge is rom.4* -45 to.4* 8. The mtiss q, cotis digits i the bse system. This iite umber o digits, i mtiss is clled precisio (e.g double precisio or sigle precisio). vries with the computer. I the umber o digits i umber is greter th, the the umber o digits i mtiss o lotig-poit orm o umber must be decresed i some wy (It c be doe by roudig o or choppig o which itroduces error to the umber).

8 Emple Cosider computer with =6 digits d compute 7 True represettio o umbers 6 digit represettio o umbers As it is see rom the tble, becuse o the limittio o digits error is itroduced. The computed umber is pproimte umber. It is correct oly or 4 digits ter the deciml poit. This c be see clerly rom the true swer. O computer, we do't hve cotiuous umbers. The smllest positive umber er zero o IBM PC (Bsic) is.9* -9. There re o umbers betwee zero d.9* -9 s well s betwee zero d -.9* -9. The et smllest positive iteger tht c be represeted o IBM PC (Bsic) is.9* * -45. The smllest umber greter th is +.9* -7 o IBM PC with QuicC d Bsic. The itervl betwee d et umber greter th is.9* -7, d it is clled mchie epsilo. The itervl betwee y rel umber R d the et rel umber pproimtely equls to mchie epsilo*r. How the umber stored i the computer: Bit is croym o biry digit d correspods to o d o positios o electric switch. 8 bits is clled byte. Computer ottio o umber is ±(.d d...d )* r where d i is biry bit tht is or. Thereore vlue o is limited by hrdwre desig d compiler. It could be 8, 6,, 64, 8, etc. For emple, i computer hs bits cpcity, the mog the bits, bit could be or the sig, 8 bits or the epoet, d bits or the mtiss. Alloctio o bits o IBM PC (QuicC d Bsic) Mtiss d epoet deped o computer brd d compiler. (See the ollowig tble, Microsot Bsic)

9 Sigle Precisio Double Precisio Mtiss Epoet Mtiss Epoet IBM PC, AT, XT IBM CDC 76 d Cyber V ii Cry XMP Tble: umber o bits or lotig poit umbers Redig Assigmet: Biry umbers i irst chpter till error lysis rom the Mthews' boo, secod editio. Error Alysis d Sources o Errors Error : The error i the result o clcultio or i umber is the dierece betwee the true vlue d the pproimte vlue o the umber. Emple is the pproimte vlue o the umber. is the true vlue o the umber. The, Error = E =-, the = +E. But, it is better to deie the reltive error. Reltive error, ER where E R True vlue Approimte Reltive Error, E vlue Error, E R From the bove tble, it c be misledig to decide bout where the error is lrge or smll by looig t E. This is obvious i the third row o bove tble. Numbers re smll d error, E loos smll. But the reltive error is %5 which is high. The, the Reltive Error, E R re better mesure i umericl lysis. But, i geerl, ect vlue is ot ow. I this is the cse pproimte reltive error deied below is itroduced d used:

10 e i i i where i d i re the successive pproimte vlues o umber. Sigiict Digits The umber is pproimte to with "m" sigiict digits i "m" is the lrgest positive iteger or which.5 m Or digits i umber which re ow to be correct or digits we re sure bout such s (uderlied digits re sigiict) 7,855 ive sigiict digits d writte s.7855*.75 three sigiict digits d writte s.75* - 7 our sigiict digits d writte s.7* 6 Emple.644 E.64 R.5 is pproimte to with sigiict digits (two deciml-plces ccurcy) Sources o Errors Mthemticl modelig o physicl problem Roudig o d choppig o errors Tructio errors Ucertity i physicl dt Progrmmig errors (Bluders) Mthemticl Modelig o Physicl Problem I geerl, our im is to solve physicl problem by usig mthemtics. First, mthemticl model should be derived. It should be ever orgotte tht the derivtio o mthemticl equtios or epressios to simulte physicl evet is ideliztio process which itroduces some ssumptios. It is lmost impossible to simulte physicl evet by ll mes. Mthemticl modelig or simultio o relity by the mthemticl equtios is pproimtio to physicl world d it is vlid uder some coditios. Roudig o d Choppig o Errors

11 We lered tht umber i the computer c be represeted by the limited umber o bits. Emple To represet umbers.646 d.745 i three digits, they should be rouded o or cut o. True Represettio o Numbers Three digits Represettio o Numbers (cut o or choppig o) (rouded o) * whe choppig o *.5* whe rouded o Mimum possible error whe cut o is -m, m is the # o digits tht re cut o, d mimum possible error whe rouded o is.5* -m, m is the # o digits tht re rouded o. I bove emple, - is the mimum possible error i choppig o d.5* - is the mimum possible error i roudig o. I cse o computers, there is o umber betwee d +, where is mchie epsilo, the umbers betwee d + re rouded to or +. I i the umber + is betwee somewhere i or the the umber + is cut o to or i the the umber is rouded o to +. Thus is the mimum possible error. I geerl, whe the rel umber R is cut o, error will be R; whe it is rouded o, error will be * R. Thus, it is better to ow the mchie epsilo o the computer beig used. Followig c progrm ids the mchie epsilo o computer. # iclude <stdio.h> mi() { lot ep, g=, eps; do { g=g/ ep=g*.98+ ep=ep- prit("g=%6.8e ep=%6.8e\\",eps); }while (ep>); prit ("\ Mchie epsilo=%8.8e\\",eps) } Or i ortr

12 Mchie epsilo or ew computers implicit rel (-h,o-z) g=. i= cotiue g=g/ ep=g*.98+. ep=ep-. i=i+ c write(*,*) 'i=',i,' g=',g,' ep=',ep i(ep.gt.) the eps=ep goto edi write(*,*) 'mchie epsilo=', eps write(*,*) i stop ed Precisio IBM PC * IBM 7 W II Cry XMD Sigle.9* * -7.9* -7.55* -5 Double.77* -7.* -6.77* -7.6* -9 * QuicC d Bsic Amout o roud-o error is serious d occurs i ollowig situtios: Whe very smll umber is dded to or subtrcted rom very lrge umber; umber is subtrcted rom other tht is very close; umber hs iiite digits such s (...) Eercises ) Add to thousd times ) Te ()=e d compute lim h e () h h e e h strt by tig i= d the,,... i h=.* -i, plot the ecessry grphs, observe the behvior d me your commets.

13 Tructio Error It occurs whe very comple uctio or physicl evet is represeted by simple uctios, such s polyomils. It ivolves the pproimtio o iiite or cotiuous process by iite or discrete oe. Emple () e d Tylor's epsio o () is give s () e 4! 6! 8 4! 5!...!... I () is represeted by iite terms o Tylor's series epsio i y mthemticl opertio, such s itegrtio, the error itroduced will be tructio error. I 4 6 P6 ()!! is chose s tructed Tylor's series epsio o () the itegrtio o () c be pproimted s ollows e d 4! 6 d! So the tructio error is cused by tructig series. Cosider series S i i The pproimte vlue o S is S Ad the remider R... Suppose tht the series is tructed ter i= S S i i So S S d R e error due to tructio. Emple: Ect itegrtio o () e or 4 digits is e d.46. I irst 4 term cosidered i Tylor Series epsio:

14 4 6 d.457 I irst 5 term cosidered i Tylor Series epsio: d.46 I irst 6 term cosidered i Tylor Series epsio: d.46