Lecture+16--Fixed+point+iteration+&+Newton

Lecture+16--Fixed+point+iteration+&+Newton

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Fixed-Point Iteration Idea: original equation : f(x) = 0 rewrite x = g(x) Use iteration x i+1 = g ( x i ) to find a value that reaches convergence Example: 2 ( ) 2 3 0 f x x x = - + = ( ) sin 0 f x x = = sin ( ) x x g x x + = = 2 3 ( ) 2 x g x x + ⇒ = =
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Fixed-Point Iteration Example: f ( x ) = x 3- x -1 = 0 Option i : Rearrange x = x 3 - 1 g ( x ) g ( x ) = x 3 - 1 Initial guess: x 0 = 0 x 1 = 03 - 1 = -1 , x 2 = ( -1 )3 - 1 = -2 , x 3 = ( -2 )3 - 1 = -9 . ..  diverges!
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Fixed-Point Iteration Example: f ( x ) = x 3- x -1 = 0 Option ii : Rearrange x ( x 2 -1) = 1 x =1/( x 2 -1); g ( x )  1/( x 2 -1) Initial guess 1: x 0 = 0 x 1 = 1/( 0 2 -1)= -1 x 2 = 1/(( -1 )2 -1)  ∞ How about initial guess 2: x 0 = 0.5?
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Fixed-Point Iteration Example: f ( x ) = x 3- x -1 = 0 Option ii: x =1/( x 2 -1); Initial guess 2: x 0 = 0.5? n x n n x n 1 -1.333333 2 1.285714 3 1.531252 4 0.743643 5 -2.237161 6 0.249695 7 -1.066493 8 7.277602 9 0.019244 10 -1 .000371 11 1349.3016 12 0.000001 13 -1.00000 14 1/0  ∞
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Fixed-Point Iteration Example: f ( x ) = x 3- x -1 = 0 Option iii: Rearrange x = ( x +1)1/3 ; g ( x ) = ( x +1)1/3 Initial guess: x 0 = 0 n xn n xn 1 1.0000000 2 1.2599211 3 1.3122939 4 1.3223538 5 1.3242688 6 1.3246326 7 1.3247018 8 1.3247149 9 1.3247174 10 1.3247179 11 1.3247180 12 1.3247180
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Fixed-Point Iteration Example: f ( x ) = x 3- x -1 = 0 WHY three different kinds of behavior?
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Lecture+16--Fixed+point+iteration+&+Newton

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