Numerical Analysis
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EE, NCKUTien-Hao Chang (Darby Chang)
In the previous slide Rootfinding
â multiplicity
Bisection methodâ Intermediate Value Theorem
â convergence measures
False positionâ yet another simple enclosure method
â advantage and disadvantage in comparison with bisection method
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In this slide Fixed point iteration scheme
â what is a fixed point?
â iteration function
â convergence
Newtonâs methodâ tangent line approximation
â convergence
Secant method3
Rootfinding Simple enclosure
â Intermediate Value Theorem
â guarantee to convergeâą convergence rate is slow
â bisection and false position
Fixed point iterationâ Mean Value Theorem
â rapid convergenceâą loss of guaranteed convergence
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2.3
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Fixed Point Iteration Schemes
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There is at least one point on the graph at which the tangent lines is parallel to the secant line
Mean Value Theorem
We use a slightly different formulation
An example of using this theoremâ proof the inequality
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Fixed points Consider the function
â thought of as moving the input value of to the output value
â the sine function maps to âą the sine function fixes the location of
â is said to be a fixed point of the function
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Number of fixed points According to the previous figure, a
trivial question isâ how many fixed points of a given
function?
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đ âČ (đ„ )â€đ<1
Only sufficient conditions
Namely, not necessary conditionsâ it is possible for a function to violate one or more of the
hypotheses, yet still have a (possibly unique) fixed point
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Fixed point iteration
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Fixed point iteration If it is known that a function has a
fixed point, one way to approximate the value of that fixed point is fixed point iteration scheme
These can be defined as follows:
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In action
http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg
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Any Questions?
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About fixed point iteration
Relation to rootfinding Now we know what fixed point
iteration is, but how to apply it on rootfinding?
More precisely, given a rootfinding equation, f(x)=x3+x2-3x-3=0, what is its iteration function g(x)?
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hint
Iteration function Algebraically transform to the form
â âŠ
Every rootfinding problem can be transformed into any number of fixed point problemsâ (fortunately or unfortunately?)
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In action
http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg
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Analysis #1 iteration function converges
â but to a fixed point outside the interval
#2 fails to convergeâ despite attaining values quite close to #1
#3 and #5 converge rapidlyâ #3 add one correct decimal every iteration
â #5 doubles correct decimals every iteration
#4 converges, but very slow
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Convergence This analysis suggests a trivial question
â the fixed point of is justified in our previous theorem
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đ demonstrates the importance of the
parameter â when , rapid
â when , dramatically slow
â when , roughly the same as the bisection method
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Order of convergence of fixed point iteration schemes
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All about the derivatives,
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Stopping condition
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Two steps
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The first step
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The second step
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Any Questions?
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2.3 Fixed Point Iteration Schemes
2.4
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Newtonâs Method
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Newtonâs Method
Definition
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In action
http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg
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In the previous example Newtonâs method used 8 function
evaluations Bisection method requires 36
evaluations starting from False position requires 31 evaluations
starting from
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Any Questions?
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Initial guess Are these comparisons fair?
â , converges to after 5 iterations
â , fails to converges after 5000 iterations
â , converges to after 42 iterations
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example
answer
Initial guess Are these comparisons fair?
â , converges to after 5 iterations
â , fails to converges after 5000 iterations
â , converges to after 42 iterations
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answer
Initial guess Are these comparisons fair?
â , converges to after 5 iterations
â , fails to converges after 5000 iterations
â , converges to after 42 iterations
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in Newtonâs method Not guaranteed to converge
â , fails to converge
May converge to a value very far from â , converges to
Heavily dependent on the choice of
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Convergence analysis for Newtonâs method
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The simplest plan is to apply the general fixed point iteration convergence theorem
Analysis strategy To do this, it is must be shown that
there exists such an interval, , which contains the root , for which
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Any Questions?
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Newtonâs Method
Guaranteed to Converge?
Why sometimes Newtonâs method does not converge?
This theorem guarantees that exists But it may be very small
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hint
answer
Newtonâs Method
Guaranteed to Converge?
Why sometimes Newtonâs method does not converge?
This theorem guarantees that exists But it may be very small
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answer
Newtonâs Method
Guaranteed to Converge?
Why sometimes Newtonâs method does not converge?
This theorem guarantees that exists But it may be very small
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Oh no! After these annoying analyses, the Newtonâs method is still not guaranteed to converge!?
http://img2.timeinc.net/people/i/2007/startracks/071008/brad_pitt300.jpg
Donât worry Actually, there is an intuitive method Combine Newtonâs method and bisection
methodâ Newtonâs method first
â if an approximation falls outside current interval, then apply bisection method to obtain a better guess
(Can you write an algorithm for this method?)
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Newtonâs Method
Convergence analysis At least quadratic
â
â , since
Stopping conditionâ
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72http://www.dianadepasquale.com/ThinkingMonkey.jpg
Recall that
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Is Newtonâs method always faster?
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In action
http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg
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Any Questions?
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2.4 Newtonâs Method
2.5
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Secant Method
Secant method Because that Newtonâs method
â 2 function evaluations per iteration
â requires the derivative
Secant method is a variation on either false position or Newtonâs methodâ 1 additional function evaluation per iteration
â does not require the derivative
Letâs see the figure first
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answer
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Secant method Secant method is a variation on
either false position or Newtonâs methodâ 1 additional function evaluation per
iteration
â does not require the derivative
â does not maintain an interval
â is calculated with and
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Any Questions?
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2.5 Secant Method