fixed_point - Numerical Methods Finding Roots Fixed Point...

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Numerical Methods Finding Roots
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Fixed Point Iteration ° Rewrite f(x) = 0 to x = g(x) ° To solve x = g(x) , we iteratively calculate ± x i+1 = g(x i ) ° The problem is how to choose g(x) so as to ensure convergence
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y = f(x) 4 6 8 10 12 f ( x ) y = g(x) and y = x 1 2 3 4 5 -6 -4 -2 0 2 -3 -2 -1 0 1 2 3 x f -4 -3 -2 -1 0 -3 -2 -1 0 1 2 3 x y
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Fixed Point Iteration The The equation f( x ) = 0, where f( x ) = x 3 - 7 x + 3, may be re- arranged to give x = ( x 3 + 3)/7. 3 4 y = ( x 3 + 3)/7 Intersection of the graphs of y = x and y = ( x 3 + 3)/7 represent roots of the original equation x 3 - 7 x + 3 = 0. -4 -3 -2 -1 0 1 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 x y y = x
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The rearrangement x = ( x 3 + 3)/7 leads to the iteration To find the middle root α , let initial approximation x 0 = 2. Fixed Point Iteration ... , 3 , 2 , 1 , 0 , 7 3 3 1 = + = + n x x n n 57143 . 1 7 3 2 7 3 3 3 0 1 = + = + = x x 98292 . 0 3 57143 . 1 3 3 3 1 = + = + = x x The iteration slowly converges to give α = 0.441 (to 3 s.f.) etc. 7 7 2 56423 . 0 7 3 98292 . 0 7 3 3 3 2 3 = + = + = x x 45423 . 0 7 3 56423 . 0 7 3 3 3 3 4 = + = + = x x
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Fixed-point Iteration Example(1)
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Fixed-point Iteration Example(2)
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Convergence of Fixed Point Iteration ° |g’(x)| < 1 : (a), (b) ° |g’(x)| > 1
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