bisektiomenetelmä

The bisection method is a root-finding algorithm used in numerical analysis to find a root of a continuous function. It relies on the intermediate value theorem, which states that if a continuous function has values of opposite sign at two points, then it must have at least one root between those two points.

The method begins by selecting an interval [a, b] such that f(a) and f(b) have opposite signs. The midpoint of the interval, c = (a + b) / 2, is then calculated. The value of the function at the midpoint, f(c), is evaluated. If f(c) is zero, then c is the root. If f(c) has the same sign as f(a), then the root must lie in the interval [c, b]. If f(c) has the same sign as f(b), then the root must lie in the interval [a, c]. This process is repeated, halving the interval in each step, until the interval becomes sufficiently small or a desired level of accuracy is reached.

The bisection method is guaranteed to converge to a root if an initial interval containing a root is provided. However, its convergence rate is relatively slow compared to other root-finding methods like Newton's method. It is considered a robust and reliable method due to its guaranteed convergence, making it suitable for situations where other methods might fail or diverge. The simplicity of the algorithm also makes it easy to implement.