8. Convergence_of_the_direct_iteration_meth

8. Convergence_of_the_direct_iteration_meth - as shown on...

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Convergence of the direct iteration method The ) ( x g x = rearrangement that we choose can have a strong effect on convergence of the iteration. We can show graphically that the iteration will converge only if the function ) ( x g on the right hand side satisfies 1 < dx dg near the root. (i) 1 < dx dg Calculations: 1) guess x 1 2) compute g(x 1 ) 3) x 2 = g(x 1 ), i.e., the length corresponding to x 2 equals to the length given by g(x 1 ), as shown on the sketch above. Thus, the point P(x=x new , y=g(x old ) ) falls on the line y=x. In this case, repeating the process moves P along y=x towards the root and the iterations eventually converge,
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Unformatted text preview: as shown on the sketch. (ii) 1 dx dg Calculations here are the same as above. In this case, however, repeating the process moves P along y=x away from the root and the iterations do not converge on the root, as shown in the sketch. The Rule: always choose an ) ( x g x = form corresponding to 1 &lt; dx dg near the root. x 2 length=x 2 x y y=g(x) P root g(x 1 ) y=x (slope=1) x 1 x y root x 1 y=g(x) y=x length=x 2 x y x 1 y=g(x) x 2 P root g(x 1 ) y=x (slope=1) y y=x root x 1 y=g(x)...
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