Question

. Establish an iteration

formula to find the

reciprocal of a positive number N by

N-R method? ​

Answer 1

Given : reciprocal of a positive number N

Newton raphson Method

To Find : Establish an iteration

Solution:

x = 1/N

=> 1/x  = N      

f(x)  = 1/x  -  N

f'(x)  = -1/x²  

xₙ₊₁  = xₙ ​− f(xₙ​) / f'(xₙ​)​

x₁  =  x₀  - f(x₀)/f'(x₀)

=> x₁  =  x₀  - (1/x₀  -  N)/(-1/x₀²)

=> x₁  =  x₀ +  x₀²(1/x₀  -  N)

=> x₁  =  x₀ +  x₀   - x₀² N

=> x₁  =  2x₀   - x₀² N

x₂ =  x₁  - f(x₁)/f'(x₁)

=> x₂ =  x₁  -  (1/x₁  -  N)/(-1/x₁²)

=> x₂ =  x₁  + x₁  -  x₁²N

=> x₂ =  2x₁ -  x₁²N

xₙ₊₁ = 2xₙ  - xₙ²N

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