Bisection Method

Definition

The bisection method is a numerical technique used to approximate the roots of a continuous function. It involves repeatedly dividing an interval in half and selecting the subinterval that contains the root, thereby narrowing down the search for the root.

5 Must Know Facts For Your Next Test

1. The bisection method is an iterative process that relies on the Intermediate Value Theorem, which states that if a continuous function changes sign over an interval, it must have a root within that interval.
2. The method starts with an initial interval that is known to contain a root, and then repeatedly halves the interval, keeping the subinterval that contains the root.
3. The accuracy of the bisection method is determined by the number of iterations performed, with each iteration halving the interval and improving the approximation of the root.
4. The bisection method is guaranteed to converge to a root, provided that the initial interval contains a single root and the function is continuous within that interval.
5. The bisection method is a simple and robust numerical method, but it may converge slowly compared to other root-finding algorithms, especially for functions with rapidly changing slopes near the root.

Review Questions

• Explain how the bisection method is used to approximate the roots of a polynomial function.
  • To use the bisection method to approximate the roots of a polynomial function, you first need to identify an interval that is known to contain a single root. This can be done by analyzing the sign changes of the function over the interval. Once the initial interval is established, the method repeatedly divides the interval in half and selects the subinterval that contains the root, based on the function's sign at the midpoint. This process continues until the desired level of accuracy is achieved, at which point the midpoint of the final subinterval is the approximation of the root.
• Describe the relationship between the bisection method and the Intermediate Value Theorem.
  • The bisection method relies on the Intermediate Value Theorem, which states that if a continuous function changes sign over an interval, it must have a root within that interval. By repeatedly bisecting the interval and keeping the subinterval that contains the root, the bisection method is able to converge to the root. The Intermediate Value Theorem ensures that the method will always find a root, provided that the initial interval contains a single root and the function is continuous within that interval.
• Analyze the advantages and limitations of the bisection method compared to other root-finding algorithms, such as the Newton-Raphson method.
  • The main advantage of the bisection method is its simplicity and robustness. It is guaranteed to converge to a root, provided that the initial interval contains a single root and the function is continuous. Additionally, the bisection method does not require the computation of the function's derivative, which can be advantageous for functions where the derivative is difficult to obtain. However, the bisection method may converge slowly compared to other root-finding algorithms, especially for functions with rapidly changing slopes near the root. In contrast, the Newton-Raphson method can converge more quickly, but it requires the computation of the function's derivative and may not converge if the initial guess is not sufficiently close to the root.

Related terms

Root of a Function: The value of the independent variable for which the function's output is zero, indicating where the graph of the function crosses the x-axis.

Interval Bisection: The process of dividing an interval into two equal parts, creating two smaller intervals.

Convergence:

The process of approaching a specific value or set of values as the number of iterations increases.