Roots of Nonlinear Equations - Open Methods

Roots of Nonlinear Equations - Open Methods

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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Roots of Nonlinear Equations Open Methods
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • 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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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Fixed Point Iterations
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com  kk xgx 1 Fixed Point Iterations • Solve   0xf     0 xgxxf • Rearrange terms: • OR  xgx 
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com In some cases you do not get a solution!
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example   22  xxxf Which has the solutions -1 & 2 To get a fixed-point form, we may use:   22  xxg   x xg 21   2 xxg   12 22    x x xg
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com First trial! • No matter how close your initial guess is, the solution diverges!
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Second trial • The solution converges in this case!!
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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< es; end 6. Let xk=xk+1; goto step 4
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Newton-Raphson Method
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Newton’s Method: Line Equation  1 21 21 ' xf xx yy m     The slope of the line is given by:
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Newton’s Method: Line equation    1 21 1 ' xf xx xf      1 1 12 ' xf xf xx     k k kk xf xf xx ' 1  Newton-Raphson Iterative method
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Newton’s Method: Taylor’s Series      1121 ' xfxxxf     1 1 12 ' xf xf xx     k k kk xf xf xx ' 1  Newton-Raphson Iterative method        11212 ' xfxxxfxf 
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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< es; end 5. Let xk=xk+1; goto step 4
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Secant Method
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Secant Method 21 21 2 2 xx yy xx yy      The line equation is given by:    2 21 221 0 xx yy yxx   
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Secant Method    2 21 221 0 xx yy yxx      21 212 2 yy xxy xx          kk kkk kk xfxf xxxf xx       1 1 1
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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< es; end 5. Let xk=xk+1; goto step 3
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Comparing with False-Position • Actually, false position ensures convergence, while secant method does not!!!
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Roots of Nonlinear System of Equations
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Be able to use the fixed point method to find a root of a set of equations • Be able to use the Newton Raphson method to find a root of a set equations
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Fixed Point Iterations
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com  kk xgx 1 Fixed Point Iterations • Solve   0xf     0 xgxxf • Rearrange terms: • OR  xgx 
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com    kkk kkk yxgy yxgx , , 21 11     Fixed Point Iterations (cont’d) • Solve     0, 0, 2 1   yxf yxf         0,, 0,, 22 11   yxgyyxf yxgxyxf• Rearrange terms: • OR    yxgy yxgx , , 2 1  
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example     0573 010 2 2 2 1   xyyxf xyxxf Which has a solution x=2 & y=3 To get a fixed-point form, we may use: With initial values: x=1.5 and y=3.5 kkk k k k yxy y x x 2 1 2 1 357 10      38.245.3*5.1*357 214.2 5.3 5.110 2 1 2 1     y x 7.42938.24*214.2*357 209.0 38.24 214.210 2 2 2 2      y x
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Diverging !
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Another trial k k k kkk x y y yxx 3 57 10 1 1      861.2 5.1*3 5.357 179.25.3*5.110 1 1     y x 05.3 179.2*3 861.257 941.1861.2*179.210 2 2     y x
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Condition of Convergence • For the fixed point iteration to ensure convergence of solution from point xk and yk we should ensure that 1 1 21 21             y g y g and x g x g
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Newton-Raphson Method
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Newton’s Method: Taylor’s Series    1 21 1 ' xf xx xf      1 1 12 ' xf xf xx     k k kk xf xf xx ' 1  Newton-Raphson Iterative method      112 ' xxfxfxf 
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Taylor’s series in multiple variables         y f y x f xyxfyxf y f y x f xyxfyxf             22 112222 11 111221 ,, ,,    112 22 111 11 , , yxf y f y x f x yxf y f y x f x              
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Manipulating the equations                                         112 111 22 11 , , yxf yxf y x y f x f y f x f Solve for x and y then evaluate:                       y x y x y x 1 1 2 2 Repeat until convergence
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example     0573 010 2 2 2 1   xyyxf xyxxf Which has a solution x=2 & y=3 With initial values: x=1.5 and y=3.5 Get the derivatives xy y f y x f x y f yx x f 613 2 222 11            
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example Using initial values: x=1.5 and y=3.5 Get the derivatives 5.3275.36 5.15.6 22 11             y f x f y f x f                       625.1 5.2 5.3275.36 5.15.6 y x                 656.0 536.0 y x              844.2 036.2 2 2 y x
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Conclusion • The fixed point iteration and Newton- Raphson methods were used to find a solution for a system of nonlinear equations in a manner similar to that used in single-variable problems
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  Roots of Non-LinearEquations Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Homework #2 • Chapter 6, p 171, numbers: 6.1,6.2,6.3,6.16,6.17 • Homework due next week