Which of the following statements regarding the Newton-Raphson iterative method $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ for finding real roots of $f(x) = 0$ are **CORRECT**?

(A) The method exhibits quadratic convergence (order of convergence $p = 2$) near a simple real root where $f'(r) \neq 0$

(B) If the root $r$ has multiplicity $m > 1$ ($f'(r) = 0$), the convergence rate of the standard Newton-Raphson method degenerates to linear ($p = 1$)

(C) The method fails completely if the derivative $f'(x_n) = 0$ at any iteration point

(D) The Newton-Raphson method is guaranteed to converge to the desired root for any arbitrary starting initial guess $x_0 \in \mathbb{R}$

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles 1. **Order of Convergence:** For a simple root ($f'(r) \neq 0$), expanding $f(x_n)$ via Taylor series gives error relation $\epsilon_{n+1} \approx \frac{f''(r)}{2 f'(r)} \epsilon_n^2$, confirming quadratic convergence ($p = 2$). 2. **Multiple Roots:** For a root of multiplicity $m > 1$, $f'(r) = 0$. The error equation becomes $\epsilon_{n+1} \approx \left(1 - \frac{1}{m}\right) \epsilon_n$, which is only linear convergence ($p = 1$). 3. **Division by Zero:** If $f'(x_n) = 0$, the tangent line is horizontal and never intersects the $x$-axis, causing division by zero and breakdown. 4. **Global Convergence:** Newton-Raphson has only *local* convergence. For a poor starting guess $x_0$, it may diverge, cycle indefinitely, or converge to an unintended root. ### 2. Statement Evaluation - **Statement A:** **CORRECT.** Quadratic convergence for simple roots. - **Statement B:** **CORRECT.** Degenerates to linear convergence for multiple roots. - **Statement C:** **CORRECT.** Undefined division when $f'(x_n) = 0$. - **Statement D:** **INCORRECT.** Newton-Raphson is not globally convergent. ### 3. Exam Trap & Standard Reference - **Trap Alert:** Distinguish local vs global convergence; Newton-Raphson is highly sensitive to the initial guess $x_0$. - **Standard Reference:** Chapra, S.C. & Canale, R.P., *Numerical Methods for Engineers*, Chapter 6.

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