Lecture-06 - 2/10/10 Open Methods Chapter 6 Open methods...

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2/10/10 1 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. See Figure 6.1 Figure 6.1 Simple Fixed-point Iteration •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: Example: 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 and 6.3 Figure 6.2
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2/10/10 2 Figure 6.3 Fig 6.4 Conclusion • Fixed-point iteration converges if •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.” Newton-Raphson Method • Most widely used method. • Based on Taylor series expansion:
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Lecture-06 - 2/10/10 Open Methods Chapter 6 Open methods...

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