Which of the following statements regarding the Secant method for finding real roots of non-linear equations $f(x) = 0$ are **CORRECT**?

(A) The order of convergence of the Secant method is approximately equal to the golden ratio $\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$

(B) Unlike the Newton-Raphson method, the Secant method does not require analytical evaluation of the derivative $f'(x)$

(C) The Secant method requires two initial starting approximations to initiate iteration

(D) The Secant method is a bracketing method that is strictly guaranteed to converge to a bracketed root

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles 1. **Formula:** $x_{n+1} = x_n - f(x_n) \frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})}$. 2. **Convergence Order:** The error satisfies $\epsilon_{n+1} \propto \epsilon_n \epsilon_{n-1}$. The characteristic equation for convergence order $p$ is $p^2 - p - 1 = 0$, giving the golden ratio $p = \frac{1 + \sqrt{5}}{2} \approx 1.618$ (super-linear convergence). 3. **Derivative Evaluation:** Approximates the tangent line by a secant line between the two most recent points, completely eliminating the need to evaluate $f'(x)$. 4. **Starting Values:** Requires two points $x_0, x_1$ to draw the initial secant line. 5. **Bracketing Nature:** The Secant method is an **open method**, not a bracketing method. The root does not need to remain bracketed between $x_n$ and $x_{n-1}$, and the method may diverge for poorly chosen initial points (unlike the Bisection or Regula Falsi methods). ### 2. Statement Evaluation - **Statements A, B, C:** **CORRECT.** - **Statement D:** **INCORRECT.** Secant method is an open method, not a guaranteed bracketing method. ### 3. Exam Trap & Standard Reference - **Trap Alert:** Do not confuse the Secant method (open, order 1.618) with the Regula Falsi / False Position method (bracketing, order 1.0). - **Standard Reference:** Chapra, S.C. & Canale, R.P., *Numerical Methods for Engineers*, Chapter 6.

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