Roots of Polynomials. Ch. 7. Roots of Polynomials. Roots of Polynomials. dy dt. a dt. y = General form:

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1 Roots o Polynomils C. 7 Generl orm: Roots o Polynomils ( ) n n order o te polynomil i constnt coeicients n Roots Rel or Comple. For n n t order polynomil n rel or comple roots. I n is odd At lest rel root 3. I comple roots eist, tey re in comple conjugte pirs λ μi λ μi i Roots o Polynomils Polynomils Represent Mtemticl models o rel systems Result rom crcteristic equtions o n ODE Te roots o te polynomil re Eigenvlues Given Homogeneous ODE: d y dy y () Solution is o te orm: rt y e

2 Roots o Polynomils Polynomils Represent Mtemticl models o rel systems Result rom crcteristic equtions o n ODE Te roots o te polynomil re Eigenvlues Given Homogeneous ODE (I.e. dynmic liner system): d y Solution is o te orm: dy y rt y e () Sustitute into eqution () Roots o Polynomils r rt rt rt e re e r r Crcteristic Eqution Te Roots r s Eigenvlues o te system Eigenvlues tell us importnt inormtion out te system evior. Roots represent importnt Engineering inormtion For te Qudrtic Cse, te eigenvlues tells us: Overdmped rel roots Criticlly Dmped one rel root (te discriminnt is zero) Underdmped comple roots (te discriminnt is negtive) Roots o Polynomils I only rel Roots Eist: Brcketing nd Open Metods will work Bot metods require initil guesses I Roots re Comple Brcketing metods will not work Newton Rpson will work i te lnguge ndles comple numers. (Still susceptile to diverging) Introduce severl New metods tt void tese prolems.

3 Project prol troug 3 points on te unction Mueller s Metod () ( ) ( ) ( ) Root Estimte Need to ind te coe s tt orce te prol troug te 3 points Use te coeicients & qudrtic ormul to ind were te prol intersects te -is Root Estimte y Mueller s Metod Given n Eqution or prol y Rewrite reltive to te point ( ) c ( ) ( ) Sustitute te 3 points (,, ) into te Generl eqution () () (3) ( ) c ( ) ( ) ( ) c ( ) ( ) ( ) c ( ) ( ) c 3 Equtions & Unknowns Mueller s Metod Sustitute te vlue o c into () & () (4) (5) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Solve te Equtions nd unknowns First, Mke te ollowing cnges o vriles to elp us solve te equtions ( ) ( ) ( ) ( ) 3

4 4 Mueller s Metod Net, dd nd sutrct ( ) to te LHS (4) nd dd nd sutrct to te RHS o (4) [ ] [ ] [ ] [ ] ( ) [ ] [ ] ( ) Mueller s Metod Net, dd nd sutrct ( ) to te LHS (4) nd dd nd sutrct to te RHS o (4) [ ] [ ] [ ] [ ] ( ) [ ] [ ] ( ) Mueller s Metod Simpliying: Solve or nd : ( ) c

5 Mueller s Metod () ( ) ( ) ( ) 3 - Root Estimte Now tt we know, nd c return to our eqution or prol: ( ) c And solve or our root estimte 3 using te Qudrtic Formul ( ) c ( ) ( ) Mueller s Metod ( ) c ( ) ( ) Use lterntive orm o te Qudrtic ormul to reduce round-o error Approimte Root (Current Estimte) c 3 ± 4c Previous Root Estimte () ( ) ( ) ( ) 3 - Root Estimte Mueller s Metod-Error & Implementtion Strtegies Reltive Percent Error: ε 3 3 % () ( ) Coose te sign in te denomintor to gree wit te sign o result lrgest denomintor 3 will te root estimte closest to Strtegies or discrding o te s wen moving on to te net itertion. Tke te originl points closest to 3. Replce wit 3 Best wen comple roots re needed ( ) ( ) 3 - Root Estimte c 3 ± 4c 5

6 Mueller s Metod Emple Find te roots o 3 ( ) 5 3 Mueller s Metod - Notes Compred to Newton-Rpson, Mueller s metod only requires unction vlues, NOT derivtives. Will ind comple roots Mueller s metod cn e used to ind comple roots. Mueller s metod ils wen ( ) ( ) ( 3 ) Rte o convergence is sligtly less tn qudrtic. Cn diverge Review o Root Finding Metods Brcketing Metods Grpicl Bisection Flse Position Guesses needed Alwys Converge Slow Convergence Open Metods Successive Itertion Newton Rpson Modiied Newton Rpson Secnt Guess needed Possile Divergence Rpid Convergence 6

7 Review o Root Finding Metods Roots o Polynomils Muller s Metod (cn ndle imginry roots) 3 Guesses needed Possile Divergence Rpid Convergence 7