Question

Match List-I with List-II.

List-I (Method)	List-II (Rate of convergence)
A. Bisection method B. Secant method C. Newton-Raphson method D. Muller method	I. 2 II. 1 III. 1.84 IV. 1.618

• A. A-II, B-IV, C-I, D-III
• B. A-IV, B-II, C-I, D-III
• C. A-II, B-IV, C-III, D-I
• D. A-I, B-III, C-II, D-IV

Hint:

Bisection → slow (1), Newton → fastest (2), Secant ≈ 1.618, Muller ≈ 1.84.

Answer 1

The Correct Option is A

Concept: Rate of convergence determines how fast a numerical method approaches the exact solution.

Step 1: Bisection Method (A).

• Linear convergence
• Error reduces proportionally Thus: \[ \text{Order} = 1 \] Hence: \[ A \rightarrow II \]

Step 2: Secant Method (B).

• Superlinear convergence
• Order approximately equal to golden ratio \[ \approx 1.618 \] Thus: \[ B \rightarrow IV \]

Step 3: Newton-Raphson Method (C).

• Quadratic convergence
• Very fast near root \[ \text{Order} = 2 \] Thus: \[ C \rightarrow I \]

Step 4: Muller Method (D).

• Uses quadratic interpolation
• Faster than secant \[ \text{Order} \approx 1.84 \] Thus: \[ D \rightarrow III \]

Step 5: Final matching.
\[ A-II,\; B-IV,\; C-I,\; D-III \] Final Answer: \[ \boxed{A-II,\; B-IV,\; C-I,\; D-III} \]