CHAPTER 4d. ROOTS OF EQUATIONS
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1 CHAPTER 4d. ROOTS OF EQUATIONS A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng by Dr. Ibrahm A. Assakka Sprng 00 ENCE 03 - Computaton Methods n Cvl Engneerng II Department o Cvl and Envronmental Engneerng Unversty o Maryland, College Park A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Although the bsecton method wll always converge on the root, the rate o convergence s very slow. A aster method or convergng on a sngle root o a uncton s the Newton- Raphson method. Perhaps t s the most wdely used method o all locatng ormulas. Assakka Slde No. 03
2 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Dervaton o ( Lne tangent to the curve at pont slope ( ( slope tanθ + ( ( Root θ Assakka Slde No. 04 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Dervaton o Graphcal Dervaton From the prevous gure, d Slope ( d or or + + ( ( ( ( ( ( Assakka Slde No
3 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Dervaton o Dervaton usng Taylor Seres Recall Taylor seres epanson, h! ( ( ( ( ( ( 3 ( ( n ( 0 + h 0 + h ( 0 + Rn+ 3 h 3! n h n! I we let 0 + h +h + and termnate the seres at ts lnear term, then or ( ( ( ( + h + ( ( ( + ( ( Assakka Slde No. 06 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Dervaton o Dervaton usng Taylor Seres ( h 0 X Assakka Slde No. 07 3
4 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Dervaton o Dervaton usng Taylor Seres ( + X Assakka Slde No. 08 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Dervaton o Note that snce the root o the uncton relatng ( and s the value o when ( + 0 at the ntersecton, hence, or or ( 0 ( + ( ( + ( ( - ( + + ( ( + Assakka Slde No. 09 4
5 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Newton-Raphson Iteraton where + ( ( value o the root at teraton + a revsed value o the root at teraton + ( value o the uncton at teraton ( dervatve o ( evaluated at teraton Assakka Slde No. 0 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample Use the Newton-Raphson teraton method to estmate the root o the ollowng uncton employng an ntal guess o 0 0: ( e Let s nd the dervatve o the uncton rst, ( ( d d e Assakka Slde No. 5
6 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample (cont d The ntal guess s 0 0, hence, 0: (0 e -(0-0 (0 -e -( ( ( + ( 0 ( 0 ( ( e ( d d e Assakka Slde No. A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample (cont d Now 0.5, hence, (0 e -(0.5 ( (0 -e -( ( + ( ( ( ( ( e ( d d e Assakka Slde No. 3 6
7 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample (cont d Now , hence, (0 e -( ( (0 -e -( ( + 3 ( ( ( ( ( e ( d d e Assakka Slde No. 4 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample (cont d Now , hence, d ( 3 d (0 e -(0.567 ( (0 -e -( ( ( ( 3 ( ( e ( e Assakka Slde No. 5 7
8 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample (cont d Thus, the approach rapdly converges on the true root o to our sgncant dgts. ( ' ( Percent ε r E E E E-3 Hence, the root s Assakka Slde No. 6 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample The ollowng polynomal has a root wthn the nterval : I a tolerance o 0.00 (0.% s requred, nd ths root usng both the bsecton and Newton-Raphson methods. Compare the rate o convergence on the root between the two methods. 3 ( Assakka Slde No. 7 8
9 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample (cont d s m Bsecton Method: 3.75, ( s ( 3.75 ( 3.75 ( ( ( m ( ( ( ( ( e ( 5 ( 5 ( 5 0( ( s ( m < 0 (negatve ( ( > 0 (postve m s e e 5.00 e 3 ( Assakka Slde No A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample (cont d Bsecton Method: s m ( s ( ( m ( ( ( e s e e 3 ( ( ( < 0 (negatve s m Assakka Slde No. 9 9
10 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample (cont d Bsecton Method: Iteraton s m e ( s ( m ( e ( s ( m ( m ( e error ε d error ε d Assakka Slde No. 0 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample (cont d Newton-Raphson Iteraton: The ntal guess s , hence, 0: (3.75 ( (3.75 0( (3.75 3(3.75 ( ( ( ( 0 ( ( 0 8 ( 3 0 Assakka Slde No. 0
11 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample (cont d Newton-Raphson Iteraton: Now we have 4.066, hence, : ( ( ( ( ( ( ( 0 8 ( Assakka Slde No. A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample (cont d ( ' ( ε d Percent ε r E E The rate o convergence wth Newton-Raphson teraton s much aster than the bsecton method. N-R method converges to the eact root n 3 teratons. Assakka Slde No. 3
12 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Ptalls o the Nonconvergence Nonconvergence can occur the ntal estmate s selected such that the dervatve o the uncton equals zero. In such case, ( would be zero and ( / ( would go to nnty. + ( ( ( 0 Assakka Slde No. 4 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Ptalls o the Nonconvergence ( Zero Slope, ( 0 0 Assakka Slde No. 5
13 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Ptalls o the ( Nonconvergence Nonconvergence can also occurs ( / ( equals -( + / ( + as shown 0 0 Assakka Slde No. 6 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Ptalls o the Ecessve Iteraton A large number o teratons wll be requred the value o ( s much larger than (. In ths case, ( / ( s small, whch leads to a smaller adjustment at each teraton. Ths stuaton can occur, or eample, when the root o a polynomal s near zero. ( + small number ( Assakka Slde No. 7 3
14 Secant Method A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng A potental problem n utlzng Newton- Raphson method s the evaluaton o the dervatve. Although ths s not true or polynomals and many other unctons, there are certan unctons whose dervatves may be etremely dcult or nconvenent to evaluate. Assakka Slde No. 8 Secant Method A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng The secant method s smlar to the Newton-Raphson method wth the derence that the dervatve ( s numercally evaluated, rather computed analytcally. Assakka Slde No. 9 4
15 Secant Method ( - ( A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Development o the Secant Method ( ( ( Root + + ( + X - + Assakka Slde No. 30 Secant Method A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Development o the Secant Method Usng the geometrc smlartes o two trangles o the prevous gure, Hence ( ( or ( [ ] ( ( Assakka Slde No. 3 5
16 Secant Method A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng The Secant Method A new estmate o the root can be obtaned usng values o the uncton ( and ( - at two other estmates and - o the root, and applyng the ollowng teratve procedure: + ( [ ] ( ( Assakka Slde No. 3 Secant Method A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample Use the secant method to estmate the root o the ollowng uncton: ( e Start wth ntal estmates o - 0 and. Assakka Slde No. 33 6
17 Secant Method A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample (cont d Frst teraton, : 0 0 (0 ( 0 e ( ( e ( [ 0 ] ( ( 0 ( [ ] ( Assakka Slde No. 34 Secant Method 3 A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample (cont d Second teraton, :, 0.670, ( 0.63 ( ( [ ] ( ( [ 0.670] ( Assakka Slde No. 35 7
18 Secant Method A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample (cont d Thrd teraton, 3: 0.670, ( , ( ( 3[ 3] ( ( 3 ( [ ] Hence, the root s to 4 sgncant dgts Assakka Slde No. 36 Polynomal Reducton A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Ater one root o a polynomal has been ound, the process can be repeated usng a new estmate. However, proper consderaton s not gven to the selecton o the new ntal estmate o the second root, then applcaton o some method mght result n the same root beng ound. Assakka Slde No. 37 8
19 Polynomal Reducton A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Denton Polynomal reducton states that the polynomal ( equals zero and root s the root o (, then there s a reduced polynomal * ( such that ( * ( 0, where * ( ( I ( s a polynomal o order n, the reduced polynomal s o order n. Assakka Slde No. 38 Polynomal Reducton A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample Usng Newton-Raphson teraton, a root o 4 was ound or the ollowng polynomal: Reduce ths polynomal *********** ****** 8 8*** 0 error Assakka Slde No. 39 9
20 Polynomal Reducton A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng Eample The reduced polynomal can be used to nd addtonal roots or the orgnal polynomal Any other method then can be used to nd a root o the reduced polynomal, and the polynomal can be reduced agan usng polynomal reducton untl all o the roots are ound. Assakka Slde No. 40 0
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