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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.
See Figure 6.1
Figure 6.1
Simple Fixedpoint Iteration
•Bracketing methods are “convergent”.
•Fixedpoint 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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Figure 6.3
Fig 6.4
Conclusion
• Fixedpoint 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.”
NewtonRaphson Method
• Most widely used method.
• Based on Taylor series expansion:
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 Spring '07
 seminario
 Linear Equations, Derivative, Taylor series expansion, Rootfinding algorithm

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