Newton s Method for One - Dimensional Optimization - Theory
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1 Numercal Methods Newton s Method for One - Dmensonal Optmzaton - Theory
2 For more detals on ths topc Go to Clck on Keyword Clck on Newton s Method for One- Dmensonal Optmzaton
3 You are free to Share to copy, dstrbute, dsplay and perform the work to Remx to make dervatve works
4 Under the followng condtons Attrbuton You must attrbute the work n the manner specfed by the author or lcensor (but not n any way that suggests that they endorse you or your use of the work. Noncommercal You may not use ths work for commercal purposes. Share Alke If you alter, transform, or buld upon ths work, you may dstrbute the resultng work only under the same or smlar lcense to ths one.
5 Newton s Method-Overvew Open search method A good ntal estmate of the soluton s requred The objectve functon must be twce dfferentable Unlke Golden Secton Search method Lower and upper search boundares are not requred (open vs. bracketng May not converge to the optmal soluton 5
6 Newton s Method-How t works The dervatve of the functon f Opt. x,nonlnear ' root fndng equaton f ( x = 0 = F( x at the functon s maxmum and mnmum. The mnma and the maxma can be found by applyng the Newton-Raphson method to the dervatve, essentally obtanng x + 1 = x f f Next slde wll explan how to get/derve the above formula ' '' ( x ( x ( 6
7 F(x F(x F(x +1 Newton s Method-To fnd root of a Slope = F ( E A B x +1 x Hence: nonlnear equaton C F x x +1 D F(x = x pt. C We wsh that n the next teraton x +1 wll be the root, or. Thus: pt. C = Or F( F ( I F( 1 = 0 F ( + = F( + 1 F( F( + 1 N-R Equaton F(
8 Newton s Method-To fnd root of a nonlneat equaton f ( f ( If F ( x f ( x,then + 1 =. For Mult-varable case,then N-R method becomes = [ f ( ] f (
9 Newton s Method-Algorthm Intalzaton: Determne a reasonably good estmate for the maxma or the mnma of the functon f ( x. Step 1. Determne f ' ( x and f '' ( x. Step 2. Substtute x (ntal estmate x0 for the frst teraton and f ' ( x nto f '' ( x ' f ( x x+ 1 = x '' f ( x to determne x +1 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 2 wth the new value of x 9
10 THE END
11 Acknowledgement Ths nstructonal power pont brought to you by Numercal Methods for STEM undergraduate Commtted to brngng numercal methods to the undergraduate
12 For nstructonal vdeos on other topcs, go to Ths materal s based upon work supported by the Natonal Scence Foundaton under Grant # Any opnons, fndngs, and conclusons or recommendatons expressed n ths materal are those of the author(s and do not necessarly reflect the vews of the Natonal Scence Foundaton.
13 The End - Really
14 Numercal Methods Newton s Method for One - Dmensonal Optmzaton - Example
15 For more detals on ths topc Go to Clck on Keyword Clck on Newton s Method for One- Dmensonal Optmzaton
16 You are free to Share to copy, dstrbute, dsplay and perform the work to Remx to make dervatve works
17 Under the followng condtons Attrbuton You must attrbute the work n the manner specfed by the author or lcensor (but not n any way that suggests that they endorse you or your use of the work. Noncommercal You may not use ths work for commercal purposes. Share Alke If you alter, transform, or buld upon ths work, you may dstrbute the resultng work only under the same or smlar lcense to ths one.
18 Example. 2 2 θ 2 θ The cross-sectonal area A of a gutter wth equal base and edge length of 2 s gven by A = 4snθ (1 + cosθ Fnd the angle θ whch maxmzes the cross-sectonal area of the gutter. 18
19 Soluton The functon to be maxmzed s f ( θ = 4snθ (1 + cosθ f ( θ = 4(cosθ + cos 2 θ sn 2 θ f ( θ = 4snθ (1 + 4 cosθ Iteraton 1: Use the soluton θ = π 4 0 = rad π 2 π 2 π 4(cos + cos sn π θ = π π 4sn (1 + 4cos = f ( = as the ntal estmate of
20 Soluton Cont. Iteraton 2: 2 2 4(cos cos sn θ2 = = sn1.0466(1 + 4 cos Summary of teratons Iteraton θ f '( θ f ''( θ θ estmate f (θ E E E Remember that the actual soluton to the problem s at 60 degrees or radans. 20
21 THE END
22 Acknowledgement Ths nstructonal power pont brought to you by Numercal Methods for STEM undergraduate Commtted to brngng numercal methods to the undergraduate
23 For nstructonal vdeos on other topcs, go to Ths materal s based upon work supported by the Natonal Scence Foundaton under Grant # Any opnons, fndngs, and conclusons or recommendatons expressed n ths materal are those of the author(s and do not necessarly reflect the vews of the Natonal Scence Foundaton.
24 The End - Really
Chapter Newton s Method
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