Evaluate √30 to four decimal places by using Newton-Raphson method, (upto fourth iterations) with all steps
Answers
Given : √30 , Newton-Raphson method
To find : Evaluate √30 to four decimal places upto fourth iterations
Solution:
x² = 30
f(x) = x² - 30
f'(x) = 2x
x₀ = 5 or 6
xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ).
x₁ = 5 - ( 5² - 30) /(2 * 5) or 6 - ( 6² - 30) /(2 * 6) = 5.5
x₁ = 5 + 1/2
x₁ = 5.5
we take x₀ = 5 or 6 x₁ is same 5.5
x₂ = 5.5 - ( 5.5² - 30) /(2 * 5.5)
x₂ = 5.5 - 0.0227
x₂ = 5.4773
x₃ = 5.4773 - ( 5.4773² - 30) /(2 * 5.4773)
x₃ = 5.4773 - 0.00007
x₃ = 5.47723
x₄= 5.47723 - ( 5.47723² - 30) /(2 * 5.47723)
x₄ = 5.47723 - 0.000006
x₄ = 5.477224
√30 = 5.477224
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