Bisection

Bisectors

Mittelsenkr-ab-e

Mittelsenkr-ab-konstr-e

Mittelloteb-ab-3d-e

Bisection construction

Bisection is a method of mathematical optimization, root-finding, and numerical analysis that divides an interval into two halves and selects the subinterval in which a root exists. This process is repeated on the chosen subinterval until a satisfactory approximation to the root is found or the interval is narrowed down to a predefined precision. The bisection method is one of the simplest and most reliable numerical methods for solving equations of the form \(f(x) = 0\), where \(f\) is a continuous function.

Overview

The bisection method is based on the Intermediate Value Theorem, which states that if \(f\) is a continuous function on the interval \([a, b]\) and \(N\) is a number between \(f(a)\) and \(f(b)\), then there exists at least one \(c\) in \([a, b]\) such that \(f(c) = N\). In the context of root-finding, \(N\) is typically set to zero.

To apply the bisection method, one starts with two points, \(a\) and \(b\), such that \(f(a)\) and \(f(b)\) have opposite signs, indicating that a root lies between \(a\) and \(b\). The midpoint \(c = \frac{a + b}{2}\) is then calculated, and the value of \(f(c)\) is evaluated. If \(f(c)\) is close enough to zero for the purposes of the problem, \(c\) is accepted as the root. Otherwise, if \(f(c)\) has the same sign as \(f(a)\), the interval \([c, b]\) is chosen for the next iteration. If \(f(c)\) has the same sign as \(f(b)\), the interval \([a, c]\) is chosen instead. This process is repeated until the root is found to the desired accuracy.

Algorithm

The algorithm for the bisection method can be summarized as follows: 1. Choose initial endpoints \(a\) and \(b\) such that \(f(a) \cdot f(b) < 0\). 2. Compute the midpoint \(c = \frac{a + b}{2}\). 3. If \(f(c) = 0\) or the size of the interval \([a, b]\) is below a predetermined tolerance, stop and return \(c\) as the root. 4. If \(f(c) \cdot f(a) < 0\), set \(b = c\); otherwise, set \(a = c\). 5. Go back to step 2.

Advantages and Disadvantages

The bisection method has several advantages, including simplicity, reliability, and guaranteed convergence for continuous functions over a closed interval where the function changes sign. However, it also has disadvantages, such as potentially slow convergence compared to other methods like Newton's method or the Secant method, and the requirement that the initial interval must be chosen such that the function changes sign over it.

Applications

The bisection method is widely used in various fields of science and engineering for solving equations where no analytical solution is available. It is particularly useful in situations where the equation to be solved is complex or when an approximate solution is sufficient.