Direct Iteration Method

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Mechanical Engineering
ME 303
Serhiy Yarusevych

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 g(x) on dg the right hand side satisfiedx <1 near the root. dg (i) <1 dx y y=x (slope=1) y=x y=g(x) y y=g(x) length=x2 root root P g(x ) 1 0 x1 x2 x x1 Calculations: 1) guess x1 2) compute g(x 1 3) x2= g(x1), i.e., the length corresponding 2o x equals to the length given b1 g(x ), as shown on the sketch above. Thus, the point P(x=x new y=g(xold) falls on the line y=x. In this case, repeating the process moves P along y=x toward
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