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# 03 open methods

## by Mohammad Tawfik, University Professor at Cairo University on Nov 08, 2010

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## 03 open methodsPresentation Transcript

• Roots of Nonlinear Equations Open Methods
• 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
• Fixed Point Iterations
• Fixed Point Iterations
• Solve
• Rearrange terms:
• OR
• In some cases you do not get a solution!
• Example
• Example Which has the solutions -1 & 2 To get a fixed-point form, we may use:
• First trial!
• No matter how close your initial guess is, the solution diverges !
• Second trial
• The solution converges in this case!!
• Condition of Convergence
• For the fixed point iteration to ensure convergence of solution from point x k we should ensure that
• Fixed Point Algorithm
• Rearrange f(x) to get f(x)=x-g(x)
• If | g’(x 0 )|>=1, goto step 2
• Evaluate x k+1 =g(x k )
• If ( x k+1 -x k )/x k+1 <  s ; end
• Let x k =x k+1 ; goto step 4
• Newton-Raphson Method
• Newton’s Method: Line Equation The slope of the line is given by:
• Newton’s Method: Line equation Newton-Raphson Iterative method
• Newton’s Method: Taylor’s Series Newton-Raphson Iterative method
• Example
• Newton-Raphson Algorithm
• From f(x) get f’(x)
• Evaluate x k+1 =x k -f(x k )/f’(x k )
• If ( x k+1 -x k )/x k+1 <  s ; end
• Let x k =x k+1 ; goto step 4
• Convergence condition!
• Try to derive a convergence conditions similar to that of the fixed point iteration!
• Secant Method
• Secant Method The line equation is given by:
• Secant Method
• Secant Algorithm
• Select x 1 and x 2
• Evaluate f(x 1 ) and f(x 2 )
• Evaluate x k+1
• If ( x k+1 -x k )/x k+1 <  s ; end
• Let x k =x k+1 ; goto step 3
• Why Secant Method?
• The most important advantage over Newton-Raphson method is that you do not need to evaluate the derivative !
• Comparing with False-Position
• Actually, false position ensures convergence, while secant method does not !!!
• 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
• Homework #2
• Chapter 6, p 157, numbers: 6.1,6.2,6.3
• Homework due next week