An iterative root-finding numerical algorithm produces successive error approximations $\epsilon_k = |x_k - r|$ satisfying $\lim_{k \to \infty} \frac{\epsilon_{k+1}}{\epsilon_k^p} = C$ for asymptotic error constant $C > 0$. Match the numerical methods with their exact order of convergence $p$: 1. Bisection Method 2. Secant Method 3. Standard Newton-Raphson (simple root) 4. Halley's Method

(A) 1: $p = 1.0$, 2: $p \approx 1.618$, 3: $p = 2.0$, 4: $p = 3.0$

(B) 1: $p = 2.0$, 2: $p = 1.0$, 3: $p = 2.0$, 4: $p = 4.0$

(C) 1: $p = 1.0$, 2: $p = 2.0$, 3: $p = 1.618$, 4: $p = 3.0$

(D) 1: $p = 0.5$, 2: $p = 1.0$, 3: $p = 2.0$, 4: $p = 3.0$

Answer Key:

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### 1. Strategy & Theoretical Principles Orders of convergence $p$ of standard numerical root-finding methods: - **Bisection Method:** Halves the interval width at each step ($\epsilon_{k+1} = 0.5 \epsilon_k$). Hence, it has **linear convergence** ($p = 1.0$). - **Secant Method:** Super-linear convergence with order $p = \frac{1 + \sqrt{5}}{2} \approx 1.618$. - **Newton-Raphson Method:** For a simple root ($f'(r) \neq 0$), convergence is **quadratic** ($p = 2.0$). - **Halley's Method:** Uses second-derivative information ($f''$) to achieve **cubic convergence** ($p = 3.0$). ### 2. Analytical Verification - 1 $\to p = 1.0$ - 2 $\to p \approx 1.618$ - 3 $\to p = 2.0$ - 4 $\to p = 3.0$ ### 3. Distractor Forensics - **Option A:** Correct classical ranking. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Halley's method is 3rd order; Newton-Raphson is 2nd order; Secant is 1.618; Bisection is 1st order. - **Standard Reference:** Chapra, S.C. & Canale, R.P., *Numerical Methods for Engineers*, Chapter 6.

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