Numercal Methods Newton s Method for One - Dmensonal Optmzaton - Theory
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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
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
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 - 0 + 1 = x Slope @ pt. C We wsh that n the next teraton x +1 wll be the root, or. Thus: Slope @ pt. C = Or F( F ( I F( 1 = 0 F ( + = F( + 1 F( F( + 1 N-R Equaton F( 0 + 1 + 1 7
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 ( 1 + 1 8
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
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For nstructonal vdeos on other topcs, go to Ths materal s based upon work supported by the Natonal Scence Foundaton under Grant # 0717624. 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.
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Numercal Methods Newton s Method for One - Dmensonal Optmzaton - Example
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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
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 = 0.7854rad π 2 π 2 π 4(cos + cos sn π θ = 4 4 4 4 π π 4sn (1 + 4cos 4 4 1 = f ( 1.0466 = as the ntal estmate of 5.196151 1.0466 19
Soluton Cont. Iteraton 2: 2 2 4(cos1.0466 + cos 1.0466 sn 1.0466 θ2 = 1.0466 = 1.0472 4sn1.0466(1 + 4 cos1.0466 Summary of teratons Iteraton θ f '( θ f ''( θ θ estmate f (θ 1 0.7854 2.8284-10.8284 1.0466 5.1962 2 1.0466 0.0062-10.3959 1.0472 5.1962 3 1.0472 1.06E-06-10.3923 1.0472 5.1962 4 1.0472 3.06E-14-10.3923 1.0472 5.1962 5 1.0472 1.3322E-15-10.3923 1.0472 5.1962 Remember that the actual soluton to the problem s at 60 degrees or 1.0472 radans. 20
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Acknowledgement Ths nstructonal power pont brought to you by Numercal Methods for STEM undergraduate Commtted to brngng numercal methods to the undergraduate
For nstructonal vdeos on other topcs, go to Ths materal s based upon work supported by the Natonal Scence Foundaton under Grant # 0717624. 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.
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