CHAPTER 4c. ROOTS OF EQUATIONS

Transcription

1 CHAPTER c. ROOTS OF EQUATIONS A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring by Dr. Ibrahim A. Aakka Spring 00 ENCE 03 - Computation Mthod in Civil Enginring II Dpartmnt o Civil and Environmntal Enginring Univrity o Maryland, Collg Park A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Th biction mthod or intrval-halving i an tnion o th dirct-arch mthod. It i ud in ca whr it i known that only on root occur within a givn intrval o. For th am lvl o prciion, thi mthod rquir wr calculation than th dirct arch mthod. Aakka Slid No. 75

2 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Undrlying Concpt Th biction (intrval-halving) mthod i on o th implt tchniqu or dtrmining root o a unction. Th bai or thi mthod can b aily illutratd by conidring th ollowing unction: y () Aakka Slid No. 76 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Undrlying Concpt On objctiv i to ind an valu or which y i zro. Uing thi mthod, th unction can b valuatd at two valu, ay and uch that ( ) ( ) 0 < Aakka Slid No. 77

3 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Undrlying Concpt Th implication i that on o th valu i ngativ and th othr i poitiv. Alo, th unction mut b continuou or Th condition can b aily atiid by ktching th unction a hown in th ollowing igur: Aakka Slid No. 78 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Undrlying Concpt y y () 3 Aakka Slid No. 79 3

4 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Undrlying Concpt Looking at th igur, it i clar that th unction i ngativ at and poitiv at, and i continuou or. Thror, th root mut btwn and and a nw approimation to th root can b calculatd a + 3 Aakka Slid No. 80 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Undrlying Concpt Clarly, 3 and can b ud to comput yt anothr valu. Thi proc i continud until () 0 or th dird accuracy i achivd. It i to b notd that at ach itration, th nw valu and on o th two prviou valu ar ud o that continuity and unctional product ar atiid. Aakka Slid No. 8

5 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Eampl : Uing graphical mthod, th ollowing unction wa ound to hav a ral root btwn and 3: () Approimat th root. Aakka Slid No. 8 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Eampl (cont d): () ( ) Aakka Slid No. 83 5

6 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Eampl (cont d): Evaluating th unction at th initial valu: : 3: Obviouly,()(3) ()(-) < 0 and th root ha a valu btwn and 3. Thror, a nw valu i approimatd by : ( 3 ) ( ) 6 ( ) ( ) Aakka Slid No. 8 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Eampl (cont d): It i vidnt that th root i btwn 3 and, which mut now b ud to comput a nw valu. Thi i bcau ( )( 3 ) < 0. Procding with th nt iv itration, giv +.5 : ( ) (.5) ( ) ( ) ( ) (.5) ( )( 0.875) < Aakka Slid No. 85 6

7 5 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Eampl (cont d): + ( ) (.5) ( ) ( 5 ) ( ) (.5) ( )(.875) > 0 ( ) ( ) ( ) (.5) ( 0.875)(.6063) : : 5 < ( ) (.375) Aakka Slid No. 86 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Eampl (cont d): ( 5 ) ( 6) (.5) (.375) (.6063)( ) ( ) ( ) (.375) (.5) ( )( 0.875) < : > 0 ( ) (.375) ( ) ( ) (.375) (.375) ( )( ) < Aakka Slid No. 87 7

8 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Eampl (cont d): : 7 ( ) (.065) It i vidnt that th unctional valu ar approaching zro a th numbr o itration i incrad. Atr i itration th approimatd root o.065 compar avorably with th act valu o Aakka Slid No. 88 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Gnral Procdur. Sktch th unction undr conidration.. Etablih th tarting point and th nd point o th intrval uch that ( )( ) < 0 3. From th tarting point and th nd point, locat th midpoint m at th cntr o th intrval a ollow: m + Aakka Slid No. 89 8

9 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Gnral Procdur y ( ) ( )( m ) > 0 ( ) ( ) < 0 m ( ) m ( ) Root m Aakka Slid No. 90 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Gnral Procdur. At th tarting point, midpoint m, and th nd point, valuat th unction rulting rom ( ), ( m ), and ( ), rpctivly. 5. Comput th product o th unction valuatd at th nd o th two intrval, that i, ( )( m ) and ( m )( ). Th root li in th intrval or which th product i ngativ, and m i ud a an timat. Aakka Slid No. 9 9

10 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Gnral Procdur 5. Chck or convrgnc a ollow: a. I th convrgnc critrion (tolranc) i atiid, thn u m a th inal timat o th root. b. I th tolranc ha not bn mt, pciy th nd o hal-intrval in which th root i locatd a th tarting and nding point or a nw intrval and go to tp 3. Aakka Slid No. 9 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Error Analyi and Convrgnc Critrion To nur clour o th itration loop, a convrgnc critrion i ndd to trminat th itrativ procdur or inding th root o a unction. Th convrgnc critrion ud in tp 6 o th biction mthod can b prd in trm o ithr th abolut valu o th dirnc prcnt rlativ rror. Aakka Slid No. 93 0

11 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Error Analyi and Convrgnc Critrion ε ε d r i+ i+ i+ i i 00 Whr ε d abolut dirnc, ε r prcnt rlativ rror, i th midpoint in th prviou root-arch itration, and i+ i th midpoint in a nw root-arch itration. Aakka Slid No. 9 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Error Analyi and Convrgnc Critrion Th tru accuracy o th olution at any itration can b computd i th th tru olution (root t ) i known. Th tru rror ε t in th i th itration i givn by ε t t i t Aakka Slid No. 95

12 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Eampl : Th ollowing polynomial ha a root within th intrval : 3 () I a tolranc o 0.0 (%) i rquird, ind thi root uing biction mthod. Aakka Slid No. 96 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Eampl : 3.75, i m () ( ) ( 3.75) ( 3.75) ( 3.75) 0( 3.75) 8 3 ( m ) (.375) (.375) (.375) 0(.375) 3 ( ) () 5 () 5 () 5 0() ( ) ( m ) < 0 (ngativ) ( ) ( ) > 0 (poitiv) m Aakka Slid No. 97

13 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Eampl : i m ( ) ( 3.75) 6.88 ( m ) (.063).98 ( ) (.375) () ( ) ( ) < 0 (ngativ) m Aakka Slid No. 98 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Eampl : rror ε 3.75 i 3 m ε r + i+ i+ i+ i i ( ) ( 3.75) 6.88 ( m ) ( 3.907) ( ) (.063). 93 d %.063 ( ) ( ) < 0 (ngativ) m Aakka Slid No. 99 3

14 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Eampl : Itration i m ( ) ( m ) ( ) ( ) ( m ) ( m ) ( ) rror ε d rror ε d Aakka Slid No. 00 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Eampl : () ( ) Aakka Slid No. 0

15 A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring Diadvantag o Although th biction (intrval halving) mthod will alway convrg on th root, th rat o convrgnc i vry low. A atr mthod or convrging on a ingl root o a unction i th Nwton-Raphon itration Mthod. Aakka Slid No. 0 5