# NEWTON - NewtonRaphsonMethod NewtonRaphsonMethod f(x[x f x...

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Newton-Raphson Method

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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 ( 29 [ ] i i x f x , Figure 1  Geometrical illustration of the Newton-Raphson method. 2
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 = 29 α tan( Figure 2  Derivation of the Newton-Raphson method. 3

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Algorithm for Newton- Raphson Method 4
Step 1 ) ( x f Evaluate symbolically. 5

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Step 2 ( 29 ( 29 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 6
Step 3 0 10 1 1 x - x x = i i i a × + + Find the absolute relative approximate error        as a 7

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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 8
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.

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