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03 open methods
This document outlines numerical methods for finding roots of nonlinear equations presented by Dr. Eng. Mohammad Tawfik. It introduces the fixed point, Newton-Raphson, and secant methods. The fixed point method rearranges the equation to an iterative form where the next estimate is a function of the previous. Newton-Raphson linearizes the function to get faster convergence. The secant method does not require derivatives by using the slope between previous points. Convergence conditions and algorithms are provided for each method. Students are assigned homework problems from the textbook.
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03 open methods
- 1. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik Roots of Nonlinear Equations Open Methods
- 2. 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
- 3. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik Fixed Point Iterations
- 4. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik ( )kk xgx =+1 Fixed Point Iterations • Solve ( ) 0=xf ( ) ( ) 0=−= xgxxf • Rearrange terms: • OR ( )xgx =
- 5. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik In some cases you do not get a solution!
- 6. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik Example
- 7. 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 ( ) x xg 21+= ( ) 2+= xxg ( ) 12 22 − + = x x xg
- 8. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik First trial! • No matter how close your initial guess is, the solution diverges!
- 9. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik Second trial • The solution converges in this case!!
- 10. 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
- 11. 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< εs; end 6. Let xk=xk+1; goto step 4
- 12. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik Newton-Raphson Method
- 13. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik Newton’s Method: Line Equation ( )1 21 21 ' xf xx yy m = − − = The slope of the line is given by:
- 14. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik 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
- 15. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik 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 −+≈
- 16. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik Example
- 17. 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< εs; end 5. Let xk=xk+1; goto step 4
- 18. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik Convergence condition! • Try to derive a convergence conditions similar to that of the fixed point iteration!
- 19. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik Secant Method
- 20. 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 0 xx yy yxx −= − −−
- 21. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik Secant Method ( )( ) 2 21 221 0 xx yy yxx −= − −− ( ) 21 212 2 yy xxy xx − − −= ( )( ) ( ) ( )kk kkk kk xfxf xxxf xx − − −= − − + 1 1 1
- 22. 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< εs; end 5. Let xk=xk+1; goto step 3
- 23. 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!
- 24. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik Comparing with False-Position • Actually, false position ensures convergence, while secant method does not!!!
- 25. 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
- 26. ENEM602 Spring 2007 Dr.Eng. Mohammad Tawfik Homework #2 • Chapter 6, p 157, numbers: 6.1,6.2,6.3 • Homework due next week