Bisección

Bisección is a numerical method used for finding the roots of a function. It consists of repeatedly dividing an interval where the function changes sign into two subintervals and choosing the one that contains the root. The process continues until the desired level of precision is achieved.

The bisección algorithm begins with an initial interval [a, b], where a < b and f(a) and f(b) have opposite signs. The midpoint of this interval, c, is calculated. If f(c) is within a predetermined tolerance of zero, then c is the approximated root. If not, then c is used to create a new interval [a, c] or [c, b] depending on the sign of f(c). This process is repeated until the interval [a, b] contains the desired precision.

One of the advantages of bisección is that it is simple to implement. However, it can be slow for finding roots of complex functions or functions with multiple roots in the same interval. Additionally, it is not guaranteed to converge to a root if there are multiple roots within the initial interval.

Bisección is an iterative method that can be used with many types of functions. It is often used in conjunction with other numerical methods to improve the accuracy and efficiency of root finding. The method is named after the process of repeatedly dividing the interval by two, similar to a binary search.

The bisección method has various applications in fields such as physics, engineering, and economics, where finding the roots of equations is crucial for solving problems. Its iterative nature makes it a useful tool for approximating roots in situations where a direct solution is not possible.