Chapter 9. Newton s Method After readng ths chapter, you should be able to:. Understand how Newton s method s dfferent from the Golden Secton Search method. Understand how Newton s method works 3. Solve one-dmensonal optmzaton problems usng Newton s method How s the Newton s method dfferent from the Golden Secton Search method? The Golden Secton Search method requres explctly ndcatng lower and upper boundares for the search regon n whch the optmal soluton les. Such methods where the boundares need to be specfed are known as bracketng approaches n the sense that the optmal soluton s bracketed by these boundares. Newton s method s an open (nstead of bracketng approach, where the optmum of the one-dmensonal functon f ( x s found usng an ntal guess of the optmal value wthout the need for specfyng lower and upper boundary values for the search regon. Unlke the bracketng approaches, open approaches are not guaranteed to converge. However, f they do converge, they do so much faster than bracketed approaches. Therefore, open approaches are more useful f there s reasonable evdence that the ntal guess s close to the optmal value. Otherwse, f there s doubt about the qualty of the ntal guess, t s advsable to use bracketng approaches to brng the guess closer to the optmal value and then use an open approach beneftng from the advantages presented by both technques. What s the Newton s method and how does t work? Newton s method s an open approach to fnd the mnmum or the maxmum of a functon f ( x. It s very smlar to the Newton-Raphson method http://numercalmethods.eng.usf.edu/topcs/newton_raphson.html to fnd the roots of a functon such that f ( x =. Snce the dervatve of the functon f ( x, f ( x = at the functons maxmum and mnmum, the mnma and the maxma can be found by applyng the Newton-Raphson method to the dervatve, essentally obtanng 9..
9.. Chapter 9. f ( x x+ = x ( f ( x We cauton that before usng Newton s method to determne the mnmum or the maxmum of a functon, one should have a reasonably good estmate of the soluton to ensure convergence, and that the functon should be easly twce dfferentable. Dervaton of the Newton-Raphson Equaton Slope at pont + C + We wsh that n the next teraton + wll be the root, or F ( + =. Thus: Slope at pont C = + or
Newton s Method 9..3 Hence : F ( + = = + F ( Remarks:. If F ( f (,then + = f ( f (. For Mult-varable case, then NR method becomes = [ f ( ] f ( + Step by step use of Newton s method The followng algorthm mplements Newton s method to determne the maxmum or f x. mnmum of a functon ( Intalzaton Determne a reasonably good estmate x for the maxma or the mnma of the functon f ( x. Step Determne f ( x and f ( x. Step Substtute x +, the ntal estmate x for the frst teraton, f ( x and f ( x nto Eqn. to determne x and the functon value n teraton. Step 3 If the value of the frst dervatve of the functon s zero, then you have reached the optmum (maxma or mnma, otherwse repeat Step wth the new value of x untl the absolute relatve approxmate error s less than the pre-specfed tolerance. Example Consder Fgure below. The cross-sectonal area A of a gutter wth equal base and edge length of s gven by A = 4sn ( + cos Fnd the angle whch maxmzes the cross-sectonal area of the gutter.
9..4 Chapter 9. Fgure : Cross secton of the gutter Soluton The functon to be maxmzed s f ( = 4sn ( + cos. The frst and second dervatve of the functon s shown below. f ( = 4(cos + cos sn f ( = 4sn ( + 4cos Let us use = π / 4 as the ntal estmate of. Usng Eqn. (, we can calculate the frst teraton follows: = f ( = f ( π f π 4 = 4 π f 4 π π π 4(cos + cos sn π = 4 4 4 4 π π 4sn ( + 4cos 4 4 =.466 The functon s evaluated at the frst estmate as f (.466 = 5. 96. The next teraton uses =.466 as the best estmate of. Usng Eqn( agan, the second teraton s calculated as follows: =
Newton s Method 9..5 f = f =.466 ( ( f (.466 f (.466 4(cos.466 + cos.466 sn.466 =.466 4sn.466( + 4 cos.466 =.47 The teratons wll contnue untl the soluton converges to a sngle optmal soluton. Summary results of all the teratons are shown n Table. Several mportant observatons regardng the 5th teraton can be made. At each teraton, the magntude of the frst dervatve gets smaller and approaches zero. A value of zero of the frst dervatve tells us that we have reached the optmal and we can stop. Also note that the sgn of the second dervatve s negatve whch tells us that we are at a maxmum. Ths value would have been postve f we had reached a mnmum. The soluton tells us that the optmal angle s.47. Remember that the actual soluton to the problem s at 6 degrees or.47 radans. See Example n Golden Search Method http://numercalmethods.eng.usf.edu/topcs/opt_goldensearch.html. Table. Summary of teratons for Example Iteraton f ( f ( + f ( +.7854.884 -.88.466 5.96.466.6898 -.396.47 5.96 3.47.63E-6 -.39.47 5.96 4.47 3.64E-4 -.39.47 5.96 5.47.333E-5 -.39.47 5.96 OPTIMIZATION Topc Newton s Method Summary Textbook notes for the Newton s method Major All engneerng majors Authors Al Yalcn Date August 7, Web Ste http://numercalmethods.eng.usf.edu