The Newton-Raphson method is used to find the root of the equation x² - 2 = 0.…
1997
The Newton-Raphson method is used to find the root of the equation x² - 2 = 0. If the iterations are started from x₀ = -1, the iterations will
- A.
converge to -1
- B.
converge to √2
- C.
converge to -√2
- D.
not converge
Show answer & explanation
Correct answer: C
For f(x) = x² - 2, we have f′(x) = 2x.
Newton-Raphson iteration is
xₙ₊₁ = xₙ - f(xₙ)/f′(xₙ)
= xₙ - (xₙ² - 2)/(2xₙ)
= (xₙ + 2/xₙ)/2.
Starting with x₀ = -1:
x₁ = (-1 + 2/(-1))/2 = -3/2.
The iterates remain negative and approach the negative root of x² - 2 = 0. Therefore, the sequence converges to -√2.