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Chapter-06 - Open Methods Chapter 6 Open methods are based...

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Open Methods Chapter 6 Open methods are based on formulas that require only a single starting value of x or two starting values that do not necessarily bracket the root. Figure 6.1
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Fig 6.1
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Simple Fixed-point Iteration ... 2, 1, k , given ) ( ) ( 0 ) ( 1 = = = = - o k k x x g x x x g x f Bracketing methods are “convergent”. Fixed-point methods may sometime “diverge”, depending on the stating point (initial guess) and how the function behaves. Rearrange the function so that x is on the left side of the equation:
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x x g or x x g or x x g x x x x f 2 1 ) ( 2 ) ( 2 ) ( 0 2 ) ( 2 2 + = + = - = - - = Example:
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Convergence x=g(x) can be expressed as a pair of equations: y 1 =x y 2 =g(x) (component equations) Plot them separately. Figure 6.2 ; 6.3 and 6.4
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Fig 6.2
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Fig 6.3
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Fig 6.4
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Conclusion Fixed-point iteration converges if x) f(x) line the of (slope 1 ) ( = x g When the method converges, the error is roughly proportional to or less than the error of the previous step, therefore it is called “linearly convergent.”
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Newton-Raphson Method Most widely used method.
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