03 open methods
TRANSCRIPT
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Roots of Nonlinear Equations
Open Methods
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Objectives
• Be able to use the fixed point method to find a root of an equation
• Be able to use the Newton Raphson method to find a root of an equations
• Be able to use the Secant method to find a root of an equations
• Write down an algorithm to outline the method being used
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Fixed Point Iterations
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
kk xgx 1
Fixed Point Iterations
• Solve 0xf
0 xgxxf
• Rearrange terms:
• OR
xgx
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
In some cases you do not get a solution!
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example
22 xxxf Which has the solutions -1 & 2
To get a fixed-point form, we may use:
22 xxg
xxg 21 2 xxg
12
22
x
xxg
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
First trial!
• No matter how close your initial guess is, the solution diverges!
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Second trial
• The solution converges in this case!!
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Condition of Convergence
• For the fixed point iteration to ensure convergence of solution from point xk we should ensure that
1' kxg
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Fixed Point Algorithm
1. Rearrange f(x) to get f(x)=x-g(x)
2. Start with a reasonable initial guess x0
3. If |g’(x0)|>=1, goto step 2
4. Evaluate xk+1=g(xk)
5. If (xk+1-xk)/xk+1< s; end
6. Let xk=xk+1; goto step 4
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Newton-Raphson Method
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Newton’s Method: Line Equation
121
21 ' xfxx
yym
The slope of the line is given by:
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Newton’s Method: Line equation
121
1 ' xfxx
xf
11
12 ' xf
xfxx
kk
kk xf
xfxx
'1
Newton-RaphsonIterative method
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Newton’s Method: Taylor’s Series
1121 ' xfxxxf 11
12 ' xf
xfxx
kk
kk xf
xfxx
'1
Newton-RaphsonIterative method
11212 ' xfxxxfxf
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Newton-Raphson Algorithm
1. From f(x) get f’(x)
2. Start with a reasonable initial guess x0
3. Evaluate xk+1=xk-f(xk)/f’(xk)
4. If (xk+1-xk)/xk+1< s; end
5. Let xk=xk+1; goto step 4
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Convergence condition!
• Try to derive a convergence conditions similar to that of the fixed point iteration!
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Secant Method
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Secant Method
21
21
2
2
xx
yy
xx
yy
The line equation is given by:
2
21
221 0xx
yy
yxx
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Secant Method
2
21
221 0xx
yy
yxx
21
2122 yy
xxyxx
kk
kkkkk xfxf
xxxfxx
1
11
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Secant Algorithm
1. Select x1 and x2
2. Evaluate f(x1) and f(x2)
3. Evaluate xk+1
4. If (xk+1-xk)/xk+1< s; end
5. Let xk=xk+1; goto step 3
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Why Secant Method?
• The most important advantage over Newton-Raphson method is that you do not need to evaluate the derivative!
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Comparing with False-Position
• Actually, false position ensures convergence, while secant method does not!!!
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Conclusion
• The fixed point iteration, Newton-Raphson method, and the secant method in general converge faster than bisection and false position methods
• On the other hand, these methods do not ensure convergence!
• The secant method, in many cases, becomes more practical than Newton-Raphson as derivatives do not need to be evaluated
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Homework #2
• Chapter 6, p 157, numbers:6.1,6.2,6.3
• Homework due next week