Lecture15 - Fixed-Point Iteration f(x) = 0 => x = g(x)...

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y x y=g(x) y=x x 0 g(x ) 0 1 x g(x ) 1 2 x |g'(x)| > 1 y x y=g(x) y=x x 0 g(x ) 0 1 x g(x ) 1 2 x |g'(x)| < 1 g(x ) 2 3 x r r Diverges Converges Fixed-Point Iteration x 1 = f(x) = 0 => x = g(x) x 2 = x 1 = x 2 =
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Steps of Fixed-Point Iteration Step 1: Guess x 0 and calculate y 0 = g(x 0 ). Step 2: Let x 1 = g(x 0 ) Step 3:Examine if x 1 satisfies f(x 1 ) = 0. Step 4. If not, repeat the iteration, x 0 = x 1
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Another Example Example: Find the root for x = ln(x)+2 using fixed point iteration method Soln. * x 0 = 1 n x n x n+1 Error e n Error e n = | x n x_exact |; E xact = 3.146193220620580… 0 1 2 1.1461932 1 2 2.693147181 0.453046 2 2.693147181 2.990710465 0.1554828 3 2.990710465 3.095510973 0.0506822 4 3.095510973 3.129952989 0.0162402 5 3.129952989 3.141017985 0.0051752 6 3.141017985 3.144546946 0.0016463 7 3.144546946 3.145669825 0.0005234 8 3.145669825 3.146026848 0.0001664 9 3.146026848 3.146140339 5.29E-05
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1.E-16 1.E-14 1.E-12 1.E-10 1.E-08 1.E-06 1.E-04
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Lecture15 - Fixed-Point Iteration f(x) = 0 =&gt; x = g(x)...

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