An exponential regula falsi method

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An exponential regula falsi method
By Ben Yu
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From an easy Start
S is the solution of
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Define a new function
Let U(s) be a sufficiently small neighborhood of
s and s ? 0, in U(s). We can
define a function in U(s)
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Do you remember
With and ,
do you remember something?
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The regula falsi method

• In regula falsi method, we have f(a)f(b) lt 0
• f(x) continuous on a,b
• Get
• If f(c) lt TOL, then stop
• If f(a)f(c) lt 0, then b c
• If f(b)f(c) lt 0, then a c
• Actually, c is the estimate of the root s after
  each iteration

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Combine together
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Algorithm

• Still need to have f(a)f(b) lt 0 and f(x)
  continuous on a,b
• If c ? a, b, then xn1 c and jump to the last
  step
• If c?a, b, then xn1 c
• If f(a)f(c) lt 0, then b c else a c
• If f(xn1) lt TOL, then stop

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Example

• f(c) ln(x), a, b 0.5, 5 and start with x
  1.5

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Reference

• Jinhai Chen Weiguo Li An exponential regula
  falsi method for solving nonlinear
  equations.(2006). 41, 327-338.