CHAPTER 4a. ROOTS OF EQUATIONS

Transcription

1 CHAPTER 4. ROOTS OF EQUATIONS A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering by Dr. Ibrhim A. Asskk Spring 00 ENCE 03 - Computtion Methods in Civil Engineering II Deprtment o Civil nd Environmentl Engineering University o Mrylnd, College Prk Introduction Qudrtic Formul In high school, students usully lern how to use the qudrtic ormul b ± x b 4c to solve qudrtic eqution o the type () x x + bx + c 0 () () Slide No.

2 Introduction Qudrtic Formul The vlues clculted with Eq. re clled the roots o Eq.. They represent the vlues o x tht mke Eq. equl to zero. Thus, the root (roots) o n eqution cn be deined s the vlue (vlues) o x tht mkes the eqution equl to zero. Slide No. Introduction Root o Eqution Deinition The root (roots) o n eqution is deined s the vlue (vlues) o x tht mke the eqution () x 0 equl to zero. The roots o n eqution sometimes re clled the zeros o the eqution. Slide No. 3

3 Introduction Anlyticl Solution The roots ( two roots) or qudrtic eqution re sid to be ound nlyticlly. The re ound through the use o the qudrtic ormul. For third-order polynomil, the roots (three roots) cn lso be ound nlyticlly. However, no generl solution exists or other higher-order polynomils. Slide No. 4 Introduction Numericl Solution The solution o mny scientiic nd engineering problems requires inding the roots o equtions tht re complex nd nonliner in nture. For exmple, the unction (x) e -x x cnnot be solved nlyticlly. In such instnces, the only lterntive is n pproximtion by numericl methods. Slide No. 5 3

4 Introduction Numericl Solution To obtin the roots o (x) e -x x 0, some type o itertive numericl method must be employed. This is, in generl, requires lrge number o clcultions, prticulrly, i the roots re to be determined to high degree precision. Thus the problem is well suited to numericl nlysis. Slide No. 6 Introduction Types o Equtions There re typiclly two types o equtions tht relte to roots inding: Algebric Trnscendentl Equtions Polynomils re simple clss o lgebric unctions tht represented generlly by n n () x + x + x +... x + 0 n Slide No. 7 4

5 Introduction Types o Equtions Some speciic exmples o lgebric (polynomils) equtions re nd () x 3.47x + 8.5x 3 6 () x 4x x + 6x Slide No. 8 Introduction Types o Equtions A trnscendentl unction is one tht is non- lgebric. These include trigonometric, exponentil, logrithmic, nd others. Speciic exmples re x () x sin() x 0 nd () x ln( x ) Slide No. 9 5

6 Introduction Roots o Equtions Depending on the type, n eqution cn hve one, two, or more roots. Furthermore, the roots o equtions cn be either rel or complex. Exmple complex roots re x +i nd x -i o the ollowing qudrtic eqution: x x b ± x b 4c ± 4 4()(5) ± () 6 ± i Slide No. 0 Introduction Roots o Equtions No Roots (x) x Slide No. 6

7 Introduction (x) Roots o Equtions One Roots Root x Slide No. Introduction (x) Roots o Equtions Two Roots Roots x Slide No. 3 7

8 Introduction Roots o Equtions Three Roots (x) Roots x Slide No. 4 Introduction Roots o Equtions Although there re situtions where complex roots o nonpolynomils re o interest, such cses re less common thn or polynomils. The stndrd methods or locting roots typiclly ll into two somewht relted but primrily distinct clsses o problems: Slide No. 5 8

9 Introduction Roots o Equtions. The determintion o the rel roots o lgebric nd trnscendentl equtions. These techniques re usully designed to determine the vlue o single rel root on the bsis o its pproximte loction.. The determintion o ll rel nd complex roots o polynomils. These methods re speciiclly designed or polynomils. The systemticlly determine ll roots o the polynomils rther thn determining single rel root given n pproximte loction. Slide No. 6 Eigenvlue Anlysis Engineering Applictions A lrge number o engineering problems require the determintion o set o vlues clled eigenvlues or chrcteristic vlues. The electricl engineer, or exmple, uses eigenvlue nlysis in the solution o twoterminl networks nd in the optimiztion o djustments o control system. Slide No. 7 9

10 Eigenvlue Anlysis Engineering Applictions A structurl engineer uses eigenvlue nlysis in the design o structure to resist ground motion due n erthquke. The chemicl engineer uses eigenvlue nlysis in the design o rector systems. The eronuticl engineer pplies eigenvlue nlyses in nlyzing the lutter o n irplne wing. Slide No. 8 Eigenvlue Anlysis Wht re Egienvlues? Eigenvlues or chrcteristic vlues re vlues, usully denoted s, or which the ollowibg mtrix system hs nonzero (I.e., nontrivil) solution X. [ I ] A n n mtrix A X 0 (3) I n n identity digonl mtrix prmeter clled eigen vlue Slide No. 9 0

11 Eigenvlue Anlysis Eigenvlues Chrcteristic Eqution The chrcteristic equtions o the eigenvlues cn be obtined by expnding the ollowing expression: A I 0 (4) into polynomil, nd then set this polynomil equl to zero. Solving this eqution or the roots s gives the eigenvlues. Slide No. 0 Eigenvlue Anlysis Exmple : 3 3 mtrix: Find the eigenvlues o the ollowing mtrix mtrix: A Slide No.

12 Eigenvlue Anlysis Exmple (cont d): 3 3 mtrix: A I Slide No. Eigenvlue Anlysis or Exmple (cont d): 3 3 mtrix: ( )( [ )( ) ] [ ( ) ] + [ ( ) ] or A I b + b + b where b 0, b, nd b re unctions o the elements ij o A. The solution (roots) o the chrcteristic eqution provides the three eigenvlues Slide No. 3

13 3 Slide No. 4 Eigenvlue Anlysis Exmple : Mtrix Find the eigenvlues o 3 4 A I A Slide No. 5 Eigenvlue Anlysis Exmple (cont d): Mtrix ( ) ( )( ) ( ) det I A The chrcteristic eqution o A is det(a - ), or Solving or, we get -, nd 5 Hence, the eigenvlues o A re - nd 5.

14 Eigenvlue Anlysis Exmple 3: Mtrix Find the eigenvlues o A A I Slide No. 6 Eigenvlue Anlysis Exmple 3 (cont d): Mtrix det ( A I ) ( )( ) ( ) ( ) + 3 The chrcteristic eqution o A is det(a - ), or Solving or, we get + i,nd i Hence, the eigenvlues o A re + i,nd i Note: Even i the elements o A re rel, the eigenvlues my be complex. Slide No. 7 4

15 Methods or Finding the Roots o Equtions. Grphicl Methods. Direct-Serch Method 3. Bisection Method 4. Newton-Rphson Itertion 5. Secnt Method 6. Polynomil Reduction nd Synthetic Division Slide No. 8 Grphicl Methods One method to obtin n pproximte solution is to plot the unction nd determine where it crosses the x xis. This point, which represents the x vlue or which (x) 0, is the root. Although grphicl methods re useul or obtining rough estimtes o roots, they re limited due to their lck o precision. Slide No. 9 5

16 Grphicl Methods Exmple: Flling Prchutist Problem Using the grphicl pproch to determine the drg coeicient or prchutist o mss m 68. kg to hve velocity o 40 m/s ter ree lling or t 0 s. Note tht the ccelertion due to grvity 9.8 m/s. ( ) [ ] c / e m t gm v( t) (4) c Slide No. 30 Grphicl Methods Exmple: Flling Prchutist Problem Eqution 4 cn be rewritten s or or gm ( c) c () c 9.8 c ( c / m) t [ e ] v 0 ( 68.) ( c / 68.)( 0) ( e ) c () c ( e ) c Slide No. 3 6

17 Grphicl Methods Exmple (cont d): Flling Prchutist Problem Vrious vlues o c cn be substituted into the right-hnd side o c () c ( e ) c 40 To check pproximtely which one will mke the unction (x) 0. The ollowing tble nd plot show the results. Slide No. 3 Grphicl Methods Exmple (cont d): Flling Prchutist Problem c () c ( e ) c 40 c (c) c (c ) Slide No. 33 7

18 Grphicl Methods Exmple (cont d): Flling Prchutist Problem (c ) Root Root 4.75 c Slide No. 34 Grphicl Methods Exmple: Eigenvlues The ollowing chrcteristic eqution resulted rom the mtrix A: ( ) A Estimte the eigenvlues (roots) by grphicl pproch. Slide No. 35 8

19 Grphicl Methods Exmple (cont d): Eigenvlues () () () () ( ) Slide No. 36 Grphicl Methods Exmple (cont d): Eigenvlues 0. Root () Root Root ( ) , 0.7, nd 3.9 Slide No. 37 9