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# 08 - Lecture 8 Todays class The method of false position...

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Lecture 8 Today’s class: The method of false position for finding zeros Fixed-point iteration The Newton-Raphson method The secant method Determining convergence of an iterative method

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Method of False Position Input : f , x 0 , x 0 u with x 0 < x 0 u and f ( x 0 ) f ( x 0 u ) < 0 n 0 while not converged x n r x n u - f ( x n u ) ( x n - x n u ) f ( x n ) - f ( x n u ) x n +1 , x n +1 u ( [ x n , x n r ] , if f ( x n ) f ( x n r ) 0 [ x n r , x n u ] , otherwise n ( n + 1) Test for convergence end while Output : Final bracket [ x n , x n u ]
Example Apply the method of false position (using a calculator) to compute an estimated solution x 3 r of the cubic equation x 3 + 4 x 2 = 10 together with a bracket x 3 , x 3 u . Start from the initial bracket x 0 , x 0 u = [1 , 2]. -1.602274384 1.26315789474 1.33882783883 1.00000000000 2.00000000000 1.26315789474 2.00000000000 0 1 2 -0.430364748 2.00000000000 1.33882783883 1.35854634182 -0.110008788 3 1.35854634182 2.00000000000

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Fixed-point iteration Rearrange nonlinear equation into form x = g ( x ) for some function g , and define iteration of the form x n +1 = g ( x n ) ( n = 0 , 1 , 2 , . . . ) n x n 0 9.000000000000000 1 8.5 55555555555555 2 8.5440 11544011543 3 8.5440037453 21090 4 8.544003745317532 5 8.544003745317532 e.g., solve x 2 = 73 using the fixed-point iteration x n +1 = 1 2 x n + 73 x n
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08 - Lecture 8 Todays class The method of false position...

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