r 1,2 = a 1 (a 2 2 4a 2 a 0 )

Transcription

1 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa CHAPTER. Roots of Equatos, Lear ystems ad Algera ystems.. Roots of Polyomal The polyomal equato s: f( ) = a d Gettg the roots MATLAB s easy whh a e evaluated d usg (roots). The rules of gettg the roots of the polyomal equato are:. For th -order equato, there are real or omple roots. It should e oted that these roots wll eessarly e dstt.. If s odd, there s at least oe real root.. If omple roots est, they est ojugate pars (that s, λ+μ ad λ-μ), where = The roots of the polyomal a e evaluated as: f( ) = The th roots a e foud ased o the order of the polyomal equato. The roots ad the soluto of Ordary dfferetal equato: d y a d + a dy d + a =, r = dy d a r + a r + a = r, = a (a a a ) a The geeral soluto s: y = C e r + C e r I the ase of the omple roots: r = λ μ, where μ ad λ are ostats. y = C e λ+μ + C e λ μ To get the ostat C ad C, the lower ad upper oudares a e employed. I the polyomal equato, there are dfferet methods:. MULL ER s methods Estmato the roots y projetg a straght le to the - as through two futos. Muller s methods tae smlar approah, ut projets a paraola through three pots.

2 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa Where the futos a e wrtte as:,, ad are three pots. Therefore, the futos a e rewrtte as: Fgure. a) eat Method ad ) Muller s Method. eat Method a. Fte dfferees

3 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa A fte dfferee s merely the dfferee etwee two umeral values. Dervatves are appromated y dvded dfferees. or f ( ) f ( ) f ( ) f We may regard ths dvded dfferee as a estmate of ad +. f at or at + or at the mdpot etwee We smply replae f y the dvded dfferee the Newto-Raphso formula: f ( ). f ( ) f ( ) Note the des: +,,. Wth the eat Method, we do t use a futoal form for have to arry alog two values of f, however. f. We do Care must e tae that f ) f ( ) ot e too small, whh would ause a overflow error y the ( omputer. Ths may our f f ) f ( ) due to the fte preso of the mahe. Ths may also ( gve a msleadg result for the overgee test of f ) f ( ). To avod that, we mght use the relatve devato to test for overgee. ( f ( ) f ( ) f ( ). Compare ad otrast Both the Newto-Raphso ad eat Methods loate just oe root at a tme. Newto: requres evaluato of f ad of f at eah step; overges rapdly. eat: requres evaluato oly of f at eah step; overges less rapdly.. Hyrd Methods A hyrd method omes the use oe program of two or more spef methods. For stae, we mght use seto to loate a root roughly, the use the eat Method to ompute the root more presely. For stae, we mght use seto to loate multple roots of a equato, the use Newto- Raphso to refe eah oe.

4 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa B. ystems of Nolear Equatos Cosder a system of olear equatos wth uows. f,,,, ( ) f,,,, ( ) f,,,, ( ). Newto-Raphso a. Matr otato Let s wrte the system of equatos as a matr equato. f f f f The uows form a olum matr also. ompatly as f ( ).. We mght wrte the system of equatos. The Method The Newto-Raphso method for smultaeous equatos volves evaluatg the dervatve matr, F, f whose elemets are defed to e Fj. If the verse F ests, the we a geerate a sequee of appromatos for the roots of futos {f }. F j ( ) f ( )

5 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa At eah step, all the partal dervatves must e evaluated ad the F matr verted. The terato otues utl all the. If the verse matr does ot est, the the method fals. If the umer f of equatos,, s more tha a hadful, the method eomes very umersome ad tme osumg.. Implt Iteratve Methods The Newto-Raphso method s a teratve method the sese that t geerates a sequee of suessve appromatos y repeatg, or teratg, the same formula. However, the term teratve method as ommoly used refers to a partular lass of algorthms whh mght more desrptvely e alled mplt teratve methods. uh algorthms our may umeral otets as we ll see susequet setos of ths ourse. At ths pot, we apply the approah to the system of smultaeous olear equatos. A. Geeral form Let e the soluto matr to the equato f ( ). I.e., f ( ). Now, solve algeraally eah ( ) for. Ths reates a ew set of equatos, F ( ), where f refers to the set of uows {j} eludg. Algeraally, ths loos fuy, eause eah uow s epressed terms of all the other uows, hee the term mplt. Of ourse, what we really mea s F( ). Alteratvely, terms of matr elemets, the equatos tae the form F (,,,,, ).,. Algorthm I a program, the teratve method s mplemeted thusly: ) Choose a tal guess, o ) Compute F( ) o 5

6 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa ) test f ( ) v) f yes, set ad et v) f o, ompute F( ), et.. Covergee We hope that lm. For what odtos wll ths e true? Cosder a rego R the spae of {} suh that umer suh that j j j h for ad suppose that for R there s a postve F ( ). The, t a e show that f les R, the teratve j j method wll overge. What does ths mea, pratally? It meas that f the tal guess,, s lose eough to, the the method wll overge to after some umer,, of teratos. multaeous Lear Equatos. The Prolem o o a. multaeous equatos We wsh to solve a system of lear equatos uows. Where the {j} ad the {} are ostats.. Matr otato The system of equatos a e wrtte as a matr multplato. 6

7 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa B, where, ad B. Whe s small (, say) a dret or oe-step method s used. For larger systems, teratve methods are preferred.. Gaussa Elmato I a oe-step approah, we see to evaluate the verse of the B matr. B B B B The soluto s otaed y arryg out the matr multplato B. a. Elmato You may have see ths hgh shool algera. For revty s sae, let s let =. I essee, we wsh to elmate uows from the equatos y a sequee of algera steps. Normalzato Reduto ) multply eq. y ) multply eq y ad add to eq. ; replae eq.. ad add to eq. ; replae eq.. 7

8 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa ) multply eq. y ad add to eq. ; replae eq.. We have elmated ad from eq. ad from eq.. a susttuto v) solve eq. for, susttute eq. &. solve eq. for, susttute eq.. solve eq. for.. Pvotg Due to the fte umer of dgts arred alog y the mahe, we have to worry aout the relatve magtudes of the matr elemets, espeally the dagoal elemets. I other words, the verse matr, may e effetvely sgular eve f ot atually so. To mmze ths posslty, we ommoly rearrage the set of equatos to plae the largest oeffets o the dagoal, to the etet possle. Ths proess s alled pvotg. e.g. Rearrage or B 7 = = = = 99 7 = = -9 8

9 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa = -9 7 = = 99. Matr Operatos I preparato for wrtg a omputer program, we ll ast the elmato ad a susttuto the form of matr multplatos. a. Augmeted matr B : A. Elemetary matres Eah sgle step s represeted y a sgle matr multplato. The elmato steps: A The frst a susttuto step: Q

10 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa A Q Ths ompletes oe yle. Net we elmate oe uow from the seod row usg A Q Q A Q Q Ths ompletes the seod yle. The fal yle s 5 6 Q 5 6 Q Q Q

11 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa We detfy the verse matr B Q 65QQ. Note that the order of the matr multplatos s sgfat. Naturally, we wat to automate ths proess, ad geeralze to equatos.. Gauss-Jorda Elmato a. Iverse matr We mght multply all the elemetary matres together efore multplyg y the augmeted matr. That s, arry out the evaluato of B, the perform B A. % rpt to mplemet Gauss-Jorda Elmato. Algorthm =[ ; ; 5]; =[6;;]; a=[ ]; a j a a j j aj a a a j,, j, p=sze(); =p(); for =: for m=+:+ = umer of equatos = de of the step or yle aj = elemets of the orgal augmeted matr, A. a(,m)=a(,m)/a(,); ed a(,)=; for l=: f l~= For eah value of, do the = le frst. Eample = ad + = 6 for m=+:+ a(l,m)=a(l,m)-a(l,)*a(,m); ed a(l,)=; ed ed 5

12 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa 6 = A 5 e.g., for =, =, j = & j = a a 6 a a a a a a aa = A = A = A

13 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa Iteratve Methods For > aout, the oe-step methods tae too log ad aumulate too muh roud-off error.. Jao Method a. Reurso formula Eah equato s solved for oe of the uows. I short j j j j, =,,,...,. Of ourse, we aot have = for ay. o efore startg the teratve program, we may have to reorder the equatos. Further, t a e show that f for eah, the the method wll overge, though t may e slowly. Here s a outle of the showg. The frst terato s: A V After several teratos, A V A A A A A A AV A A V We wat lm A j, whh wll happe f.. Algorthm We eed four arrays:,, B, ad. j

14 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa Frstly, selet a tal guess ( = ) eodly, ompute a ew ( + = ).. j j j j Thrdly, test for overgee.. Note that all the must pass the test. If all the do ot pass the test, the repeat utl they do.. Gauss-edel Method The Gauss-edel Method hopes to speed up the overgee y usg ewly omputed values of at oe, as soo as eah s avalale. Thus, omputg ew(), for stae, the values of ew(), ew(),..., ew() are used o the rght had sde of the formula. We stll eed to eep separate sets of ew ad old order to perform the overgee tests. Applatos A ouple of ases egeerg that gve rse to smultaeous lear equatos.. Eletral Crut (7++6) 6 = - + (+5++) = -6 + (+9+6) 9 = (9++) =

15 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa A ; oluto: Truss ystem 5

16 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa A 5 ; oluto: % rpt to mplemet Gauss-edel =[ ; ; 5]; =[6;;]; old=[;;]; ew=old; p=sze(); =p(); flag=; whle flag > for =: sum=; for l=: f ~=l sum=sum+(,l)*ew(l); ed ed ew()=(()-sum)/(,); ed for =: f as((ew()-old())/old()) >.5 old=ew;

17 Fall emester- 6 Numeral Methods Mehaal Egeerg - ME Dr. aeed J. Almalow, smalow@taahu.edu.sa rea else flag=; ed ed ed ew 7