Eiciecy o a ethod Lecture Solutio o Noliear Equatios - III The eiciecy ide o a iterative ethod is deied by / E r r: rate o covergece o the ethod : total uber o uctios ad derivative evaluatios at each step o iteratio Obviously, i the value o the ide is larger, the the ethod is ore eiciet Eaple: Method r E Secat 6 6 Newto 4 Methods or ultiple roots Deiitio:Iwecawrite()as ( ) ( *) g( ), where g() is bouded ad g(*), the * is called a ultiple root o ultiplicity I * is a ultiple root o ultiplicity o equatio (), the we have ro deiitio o ultiple root: ( *) ( ) ( *) LL ( *) ad ( *) Note: It ca be veriied that all iterative ethods discussed have oly liear rate o covergece whe > For eaple, or Newto-Raphso ethod, we get the error equatio as + ( *) 3 + + ( ) + O ( + ) ( *) i, + + O( ), Which shows the liear covergece Whe the ultiplicity o the root is ow i advace we ca odiy the ethods by itroducig paraeters depedet o the ultiplicity o the root to icrease their order o covergece For eaple Newto-Raphso ethod
+ α a: arbitrary paraeter to be deteried + α α ( *) 3 Error equatio or this ethod: + ( ) + O ( + ) ( *) + I the ethod has quadratic rate o covergece, the coeiciet o gives α α ust vaish, which Thus the ethod: + I the ultiplicity is ot ow i advace, the use the ollowig procedure: It is ow that () has a root * o ultiplicity, the ( ) has the sae root ( ) * ad its o ultiplicity (-) Hece g( ) has a siple root *, we ca ow ( ) g use Newto-Raphso ethod + to id approiate value o the ultiple g root * Sipliicatio gives + Veriy this ethod has secod order covergece Zeros o Polyoials Till ow we have discussed ethods or idig sigle zero o a arbitrary uctio i oe diesio For a special case o a polyoial p() o degree, oe ote ay eed to id all o its zeros, which ay be cople eve i the coeiciets o the polyoial are real There are several approaches available: Use oe o the ethods such as Newto to id a sigle root, the cosider a delated polyoial p( ) o degree oe less Repeat util all zeros have ( ) bee oud
Its is a good idea to go bac ad reie each root usig origial polyoial p() to avoid cotaiatio due to roudig error i orig the delated polyoial For copaio atri o polyoial ad copute the eige values This is used by MATLA 3 Use ethod desiged speciically or idig all roots o polyoial Syste o No-liear Equatios Syste o equatios ted to be ore diicult to solve tha sigle oliear equatios or a uber o reasos: A uch wider rage o behavior is possible So that theoretical aalysis o the eistece ad uber o solutios is uch ore cople No sigle way, i geeral, to guaratee covergece to desire solutio ot to bracet solutio to produce absolutely sae ethod 3 Coputatioal overhead icreases rapidly with diesio o the proble Fied Poit Iteratio or Syste o equatios Fied poit proble or g : R R is to id a vector such that g() correspodig ied poit iteratio is siply + g( ) give soe startig poit Newto s Method May ethods or solvig oliear equatios i oe-diesio do ot geeralized directly to -diesios The ost popular ethod that does geeralized is Newto s ethod For a dieretiable uctio : R R Trucated Taylor series: ( + s) ( ) + J ( ) s,where J () is the Jacobia atri i ( ) { J ( ) } ij j I s satisies the liear syste J () s-(),the +s is tae as a approiate zero o
I this sese Newto s ethod replaces a syste o oliear equatios with a syste o liear equatios, but sice the solutio o the two systes are ot idetical i geeral, the process ust be repeated util the approiate solutio is reached as accurate as desired Algorith Iitial guess or,,, 3, Solve J ( ) s ( ) or s (Copute Newto Step) + + s (Update solutio) ed Cost o Newto s Method Cost per iteratio o Newto s ethod or dese proble i diesios is substatial Coputig Jacobia atri costs scalar uctio evaluatios Solvig liear syste costs O( 3 ) operatios Secat Updatig Method The partial derivatives that ae up the Jacobia atri could be replaced by iite dierece approiatio alog each coordiate directio, but this would etail additioal uctio evaluatio purely or the purpose o obtaiig derivative ioratio Istead we tae our ow cue ro the secat ethod or oliear equatios i oe diesio which avoids eplicitly coputig derivatives by approiatig the derivatives based o the chage i uctio values betwee successive iterates Secat updatig ethods reduce cost o the Newto s ethod by Usig uctio values at successive iterates to build approiate Jacobia ad avoidig eplicit evaluatio o derivatives Updatig actorizatio o approiate Jacobia rather tha reactorig it at each iteratio Note: Most secat updatig ethods have superliear but ot quadratic covergece rate; ote cost less overall tha Newto s ethod Oe o the Siples ad ost eective secat updatig ethod or solvig oliear systes is royde s Methods
royde s Methods This ethod begis with a approiate Jacobia atri ad updates it (or a actorizatio o it) at each iteratio, Algorith Iitial guess Iitial Jacobia approiatio (Ca be true Jacobia or Fiite dierece approiatio or to avoid derivatives we siply start with I) or,,, 3, Solve s ( ) or s (Copute Newto lie Step) + + s (Update solutio) y ) ( ) ed ( + + T ( y s ) s ) + T (Update approiate Jacobia) s s The otivatio or the orula or the update Jacobia approiatio + is that it gives theleastchageto subject to satisyig the secat equatio + ( + ) ( + ) ( ) I this way the sequece o atrices gais ad aitais ioratio about the behavior o the uctio alog the various directors geerated by the algorith, without the eed or the uctio to be sapled purely or the purpose o obtaiig derivative ioratio Updatig as just idicated would still leave o eedig to solve a liear syste at each iteratio at a cost o O( 3 ) arithetic Thereore, i practice a actorizatio o is updated istead updatig directly, so that total cost per iteratio is oly O ( 3 )