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Numerical Analysis
1
EE, NCKU
Tien-Hao Chang (Darby Chang)
In the previous slide Rootfinding
multiplicity
Bisection method
Intermediate Value Theorem
convergence measures
False position
yet another simple enclosure method
advantage and disadvantage in comparison with bisection method
2
In this slide Fixed point iteration scheme
what is a fixed point?
iteration function
convergence
Newtons method
tangent line approximation
convergence
Secant method
3
Rootfinding Simple enclosure
Intermediate Value Theorem
guarantee to converge
convergence rate is slow
bisection and false position
Fixed point iteration
Mean Value Theorem
rapid convergence
loss of guaranteed convergence
4
2.3
5
Fixed Point Iteration Schemes
6
7
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 theorem
proof the inequality
sin sin
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Fixed points Consider the function sin
thought of as moving the input value of
6
to the output value 1
2
the sine function maps 0 to 0
the sine function fixes the location of 0
= 0 is said to be a fixed point of the
function sin
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11
Number of fixed points
According to the previous figure, a
trivial question is
how many fixed points of a given
function?
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13
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< 1
Only sufficient conditions
Namely, not necessary conditions
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?
21
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)?
22
hint
Iteration function Algebraically transform to the form
=
= 3 + 2 3 3
= 3 + 2 2 3
=3+23
3
Every rootfinding problem can be transformed into any number of fixed point problems
(fortunately or unfortunately?)
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In action
http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg
26
Analysis #1 iteration function converges
but to a fixed point outside the interval 1,2
#2 fails to converge
despite attaining values quite close to #1
#3 and #5 converge rapidly
#3 add one correct decimal every iteration
#5 doubles correct decimals every iteration
#4 converges, but very slow
27
Convergence This analysis suggests a trivial question
the fixed point of is justified in our previous
theorem
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31
11 0 demonstrates
the importance of the parameter
when 0, rapid
when 1, dramatically slow
when 1
2, 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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36
37
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Stopping condition
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Two steps
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The first step
lim
+1
=
lim
+1
= lim
1
lim
= 0
lim
+1
= 0 when > 1
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The second step
+1
=
+1+
+1
+1
+ +1
lim
+1
= 0
1 0 lim
+1
1 + 0
lim
+1
= 1 when > 1
43
Any Questions?
44
2.3 Fixed Point Iteration Schemes
2.4
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Newtons Method
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Newtons Method
Definition
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In action
http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg
50
In the previous example
Newtons method used 8 function
evaluations
Bisection method requires 36
evaluations starting from (1,2)
False position requires 31
evaluations starting from (1,2)
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Any Questions?
53
Initial guess
Are these comparisons fair?
= tan 6
0 = 0.48, converges to 0.4510472613
after 5 iterations
0 = 0.4, fails to converges after 5000
iterations
0 = 0, converges to 697.4995475 after 42
iterations
54
example
answer
Initial guess
Are these comparisons fair?
= tan 6
0 = 0.48, converges to 0.4510472613
after 5 iterations
0 = 0.4, fails to converges after 5000
iterations
0 = 0, converges to 697.4995475 after 42
iterations
55
answer
Initial guess
Are these comparisons fair?
= tan 6
0 = 0.48, converges to 0.4510472613
after 5 iterations
0 = 0.4, fails to converges after 5000
iterations
0 = 0, converges to 697.4995475 after 42
iterations
56
0 in Newtons method Not guaranteed to converge
0 = 0.4, fails to converge
May converge to a value very far
from 0
0 = 0, converges to 697.4995475
Heavily dependent on the choice of
0
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Convergence analysis for Newtons 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?
65
Newtons Method
Guaranteed to Converge?
Why sometimes Newtons method
does not converge?
This theorem guarantees that
exists
But it may be very small
66
hint
answer
Newtons Method
Guaranteed to Converge?
Why sometimes Newtons method
does not converge?
This theorem guarantees that
exists
But it may be very small
67
answer
Newtons Method
Guaranteed to Converge?
Why sometimes Newtons method
does not converge?
This theorem guarantees that
exists
But it may be very small
68
69
Oh no! After these annoying analyses, the Newtons method is
still not guaranteed to converge!?
http://img2.timeinc.net/people/i/2007/startracks/071008/brad_pitt300.jpg
Dont worry Actually, there is an intuitive method
Combine Newtons method and
bisection method
Newtons method first
if an approximation falls outside current
interval, then apply bisection method to
obtain a better guess
(Can you write an algorithm for this
method?)
70
Newtons Method
Convergence analysis
At least quadratic
=
2
= 0, since = 0
Stopping condition
1 <
71
72 http://www.dianadepasquale.com/ThinkingMonkey.jpg
Recall that
73
Is Newtons method always faster?
74
75
In action
http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg
76
Any Questions?
77
2.4 Newtons Method
2.5
78
Secant Method
Secant method Because that Newtons method
2 function evaluations per iteration
requires the derivative
Secant method is a variation on either false position or Newtons method
1 additional function evaluation per iteration
does not require the derivative
Lets see the figure first
79
answer
80
Secant method Secant method is a variation on
either false position or Newtons
method
1 additional function evaluation per
iteration
does not require the derivative
does not maintain an interval
+1 is calculated with and 1
81
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Any Questions?
86
2.5 Secant Method