Question

The order of the convergence of the bisection method is

• A. \( 4 \)
• B. \( 2 \)
• C. \( 1 \)
• D. \( 3 \)

Hint:

Bisection method converges slowly but reliably—it has **linear convergence**, hence order 1.

Answer 1

The Correct Option is C

The bisection method is a numerical technique used to find roots of a continuous function. It is a simple and robust method which iteratively reduces the interval where a root lies by dividing it in half and selecting the subinterval where the function changes sign.

The convergence order of an iterative method indicates how quickly the method approaches the solution as the iterations progress. For the bisection method, the convergence is linear. This means that the number of correct digits in the approximation grows linearly with the number of iterations.

Mathematically, the error in the bisection method is halved with each iteration. If \( I_n \) is the length of the interval after \( n \) iterations, then:

\( I_{n+1}=\frac{I_n}{2} \)

Thus, the convergence is first-order or linear, denoted by an order of convergence \( p=1 \).

Therefore, the correct answer indicating the order of convergence for the bisection method is: 1