mws_gen_nle_ppt_newton(1)

mws_gen_nle_ppt_newton(1) - Newton-Raphson Method Major All...

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Newton-Raphson Method Major: All Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates 1/10/2010 1 http://numericalmethods.eng.usf.edu
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Newton-Raphson Method http://numericalmethods.eng.usf.edu
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Newton-Raphson Method ) (x f ) f(x - = x x i i i i + 1 f(x) f(x i ) f(x i-1 ) x i+2 x i+1 x i X θ ( ) [ ] i i x f x , Figure 1 Geometrical illustration of the Newton-Raphson method. http://numericalmethods.eng.usf.edu 3
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Derivation f(x) f(x i ) x i+1 x i X B C A α ) ( ) ( 1 i i i i x f x f x x = + 1 ) ( ) ( ' + = i i i i x x x f x f AC AB = ) α tan( Figure 2 Derivation of the Newton-Raphson method. 4 http://numericalmethods.eng.usf.edu
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Algorithm for Newton- Raphson Method 5 http://numericalmethods.eng.usf.edu
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Step 1 ) ( x f Evaluate symbolically. http://numericalmethods.eng.usf.edu 6
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Step 2 ( ) ( ) i i i i x f x f - = x x + 1 Use an initial guess of the root, , to estimate the new value of the root, , as i x 1 + i x http://numericalmethods.eng.usf.edu 7
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Step 3 0 10 1 1 x - x x = i i i a × + + Find the absolute relative approximate error as a http://numericalmethods.eng.usf.edu 8
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Step 4 Compare the absolute relative approximate error with the pre-specified relative error tolerance . Also, check if the number of iterations has exceeded the maximum number of iterations allowed. If so, one needs to terminate the algorithm and notify the user. s Is ? Yes No Go to Step 2 using new estimate of the root. Stop the algorithm s a >∈ http://numericalmethods.eng.usf.edu 9
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Example 1 You are working for ‘DOWN THE TOILET COMPANY’ that makes floats for ABC commodes. The floating ball has a specific gravity of 0.6 and has a radius of 5.5 cm. You are asked to find the depth to which the ball is submerged when floating in water. Figure 3 Floating ball problem. http://numericalmethods.eng.usf.edu 10
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Example 1 Cont. The equation that gives the depth x in meters to which the ball is submerged under water is given by ( ) 4 2 3 10 993 3 165 0 - . + x . - x x f × = Use the Newton’s method of finding roots of equations to find a) the depth ‘x’ to which the ball is submerged under water. Conduct three
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida.

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mws_gen_nle_ppt_newton(1) - Newton-Raphson Method Major All...

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