The Newton-Raphson method is to be used to find the root of the equation f(x)…
1999
The Newton-Raphson method is to be used to find the root of the equation f(x) = 0, where x_0 is the initial approximation and f' is the derivative of f. The method converges:
- A.
Always
- B.
Only if f is a polynomial
- C.
Only if f(x_0) < 0
- D.
None of the above
Show answer & explanation
Correct answer: D
The correct answer is: None of the above.
Newton-Raphson iteration is given by:
x_(k+1) = x_k - f(x_k) / f'(x_k)
The method may converge rapidly when the initial approximation is sufficiently close to a simple root and f'(x) is non-zero near that root. However, convergence is not guaranteed in general.
So:
It does not always converge.
It is not restricted only to polynomial functions.
The condition f(x_0) < 0 alone says nothing about convergence.
Therefore none of the first three statements is a correct general condition, so the answer is None of the above.