Subtopic: Solutions of linear and non-linear algebraic equations
Question Statement:
Which of the following statements is/are TRUE regarding iterative methods for solving non-linear algebraic equations?
Options:
(A) The bisection method is always guaranteed to converge if $f(x)$ is continuous and $f(a)f(b) < 0$
(B) The order of convergence of the secant method is approximately $1.618$
(C) The Newton–Raphson method fails if $f'(x_n) = 0$ at any iteration point
(D) The Regula-Falsi (false position) method retains bracketing at every iteration
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
1. The bisection method repeatedly halves a bracket containing an odd number of roots, guaranteeing linear convergence ($L = 0.5$) by the intermediate value theorem (A is true).
2. The secant method does not evaluate derivatives and has convergence order equal to the golden ratio $\frac{1+\sqrt{5}}{2} \approx 1.618$ (B is true).
3. Division by $f'(x_n) = 0$ produces an undefined step, meaning the tangent is parallel to the x-axis (C is true).
4. Regula-Falsi replaces one bracket boundary with the chord x-intercept such that $f(a)f(b) < 0$ is preserved (D is true).