Chapter Bisection Method of Solving a Nonlinear Equation

Transcription

1 Chpter 00 Bisection Method o Solving Nonliner Eqtion Ater reding this chpter, yo shold be ble to: 1 ollow the lgorith o the bisection ethod o solving nonliner eqtion, se the bisection ethod to solve eples o inding roots o nonliner eqtion, nd enerte the dvntges nd disdvntges o the bisection ethod Wht is the bisection ethod nd wht is it bsed on? One o the irst nericl ethods developed to ind the root o nonliner eqtion ( ) 0 ws the bisection ethod (lso clled binry-serch ethod) he ethod is bsed on the ollowing theore heore An eqtion ( ) 0, where () is rel continos nction, hs t lest one root between nd i ( ) ( ) < 0 (See Figre 1) Note tht i ( ) ( ) > 0, there y or y not be ny root between nd (Figres nd ) I ( ) ( ) < 0, then there y be ore thn one root between nd (Figre 4) So the theore only grntees one root between nd Bisection ethod Since the ethod is bsed on inding the root between two points, the ethod lls nder the ctegory o brcketing ethods Since the root is brcketed between two points, nd, one cn ind the idpoint, between nd his gives s two new intervls 1 nd, nd nd 001

2 00 Chpter 00 () l Figre 1 At lest one root eists between the two points i the nction is rel, continos, nd chnges sign () l Figre I the nction () does not chnge sign between the two points, roots o the eqtion ( ) 0 y still eist between the two points

3 Bisection Method 00 () () l l Figre I the nction () does not chnge sign between two points, there y not be ny roots or the eqtion ( ) 0 between the two points () l Figre 4 I the nction () chnges sign between the two points, ore thn one root or the eqtion ( ) 0 y eist between the two points Is the root now between nd or between nd? Well, one cn ind the sign o ( ) ( ), nd i ( ) ( ) < 0 then the new brcket is between nd, otherwise, it is between nd So, yo cn see tht yo re literlly hlving the intervl As one repets this process, the width o the intervl [, ] becoes sller nd sller, nd yo cn zero in to the root o the eqtion ( ) 0 he lgorith or the bisection ethod is given s ollows

4 004 Chpter 00 Algorith or the bisection ethod he steps to pply the bisection ethod to ind the root o the eqtion ( ) 0 re 1 Choose nd s two gesses or the root sch tht ( ) ( ) < 0, or in other words, () chnges sign between nd Estite the root,, o the eqtion ( ) 0 s the id-point between nd s + Now check the ollowing ) I ( ) ( ) < 0, then the root lies between nd ; then nd b) I ( ) ( ) > 0, then the root lies between nd ; then nd c) I ( ) ( ) 0 ; then the root is Stop the lgorith i this is tre 4 Find the new estite o the root + Find the bsolte reltive pproite error s where - new old new 0 new estited root ro present itertion old estited root ro previos itertion 5 Copre the bsolte reltive pproite error with the pre-speciied reltive error tolernce s I > s, then go to Step, else stop the lgorith Note one shold lso check whether the nber o itertions is ore thn the i nber o itertions llowed I so, one needs to terinte the lgorith nd notiy the ser bot it

5 Bisection Method 005 Eple 1 A trnnion hs to be cooled beore it is shrink itted into steel hb Figre 5 rnnion to be slid throgh the hb ter contrcting he eqtion tht gives the tepertre to which the trnnion hs to be cooled to obtin the desired contrction is given by ( ) Use the bisection ethod o inding roots o eqtions to ind the tepertre to which the trnnion hs to be cooled Condct three itertions to estite the root o the bove eqtion Find the bsolte reltive pproite error t the end o ech itertion nd the nber o signiicnt digits t lest correct t the end o ech itertion Soltion Fro the designer s records or the previos bridge, the tepertre to which the trnnion ws cooled ws 8 F Hence ssing the tepertre to be between 0 F nd 150 F, we hve 150 F, 0 F,, Check i the nction chnges sign between, nd, 150 Hence ( ) ( ), ( ), ( 0) ( 150) ( 150) ( 0) ( 0) ( 150) ( 0)

6 006 Chpter 00 ( ) ( ) ( 150) ( 0) ( 190 )( 1890 ) 0,, < So there is t lest one root between, nd, tht is between 150 nd 0 Itertion 1 he estite o the root is, +, ( 0) ( ) ( ) ( 15) ( 15) ( 15) ( ) ( ) ( ) ( 15) ( 190 )( 56 ) 0, < Hence the root is brcketed between, nd, tht is, between 150 nd 15 So, the lower nd pper liits o the new brcket re, 150,, 15 At this point, the bsolte reltive pproite error hve previos pproition Itertion he estite o the root is, +, ( ) ( ) ( ) ( 175) ( 175) cnnot be clclted, s we do not ( 175) ( ) ( ) ( ) ( 15) ( 576 )( 56 ) 0, < Hence, the root is brcketed between nd,, tht is, between 15 nd 17 5 So the lower nd pper liits o the new brcket re, 17 5,, 15 he bsolte reltive pproite error t the end o Itertion is

7 Bisection Method 007 new old new ( 15) % None o the signiicnt digits re t lest correct in the estited root o 175 s the bsolte reltive pproite error is greter tht 5 % Itertion he estite o the root is, +, ( 15) 115 ( ) ( 115) ( 115) ( 115) ( 115) (, ) ( ) ( 15) ( 115) ( 56 )( 1540 ) < 0 Hence, the root is brcketed between, nd, tht is, between 15 nd 11 5 So the lower nd pper liits o the new brcket re, 11 5,, 15 he bsolte reltive pproite error t the ends o Itertion is new old 0 new 115 ( 175) % he nber o signiicnt digits t lest correct is 1 Seven ore itertions were condcted nd these itertions re shown in the ble 1 below

8 008 Chpter 00 ble 1 Root o ( ) 0 Itertion s nction o nber o itertions or bisection ethod,, , % ( ) th At the end o the itertion, 0079% Hence, the nber o signiicnt digits t lest correct is given by the lrgest vle o or which log ( ) log( ) 101 So he nber o signiicnt digits t lest correct in the estited root o is Advntges o bisection ethod ) he bisection ethod is lwys convergent Since the ethod brckets the root, the ethod is grnteed to converge b) As itertions re condcted, the intervl gets hlved So one cn grntee the error in the soltion o the eqtion Drwbcks o bisection ethod ) he convergence o the bisection ethod is slow s it is siply bsed on hlving the intervl b) I one o the initil gesses is closer to the root, it will tke lrger nber o itertions to rech the root c) I nction () is sch tht it jst toches the -is (Figre 6) sch s ( ) 0 it will be nble to ind the lower gess,, nd pper gess,, sch tht

9 Bisection Method 009 ( ) ( ) < 0 d) For nctions () where there is singlrity 1 nd it reverses sign t the singlrity, the bisection ethod y converge on the singlrity (Figre 7) An eple incldes 1 ( ) where, re vlid initil gesses which stisy ( ) ( ) < 0 However, the nction is not continos nd the theore tht root eists is lso not pplicble () Figre 6 he eqtion ( ) 0 hs single root t 0 tht cnnot be brcketed 1 A singlrity in nction is deined s point where the nction becoes ininite For eple, or nction sch s 1 /, the point o singlrity is 0 s it becoes ininite

10 00 Chpter 00 () Figre 7 he eqtion ( ) 0 1 hs no root bt chnges sign NONLINEAR EQUAIONS opic Bisection Method-More Eples Sry etbook notes on the bisection ethod o inding roots o nonliner eqtions, inclding convergence nd pitlls Mjor Mechnicl Engineering Athors Atr Kw Dte Noveber 15, 01 Web Site