The bisection method is a mathematical approach used to find out roots of values through recurring bisects. Bisects are repeated severally and then a subinterval in which the root should lie is chosen. It is a very robust and simple method s although it is relatively slow and tedious. As a result, bisection method is frequently used to achieve a rough approximate to a value that is then as a beginning point for other converging methods. This method is as well referred to as the interval halving method, the dichotomy approach or the binary search approach.
Numeric Applications of Bisection Method
Bisection method is numerically applicable in solving the equation f(x) = 0 for actual variable x, whereby f is the continuous function achieved through an interval (a, b) in which f(a) and f(b) possess negative signs, that negative and positive signs. The values at point (a, b) are said to brace a root because through the bisection method, the continuous function, f should at least have one of the roots of interval (a, b).
In every stage, bisection method divides the interval into equal halves which compute to c = ½(a+ b) of the interval and the value f(c) fiction at the same point. Unless c is a root by itself, which is not likely although possible, there are only two potentials; either f(c) and f(a) have bracket root and opposite signs or f(c) and f(b) have the bracket root and the opposite signs. The method choses a subinterval value which is guaranteed to be a bracket that is used as the new interval to be applied in the following step. B y so doing, an interval whereby the value for f is zero (0) is reduced by fifty percent in width in every step. The process is continued until when the interval becomes sufficiently small.
Whereby, f(b) and f(c) have the opposite value signs, bisection method uses the value of c instead in place of a. however, when f(c) = 0, the arithmetic solution is taken to be zero. In both instances, the new values for f(a) and f(b) have opposite signs. Nevertheless, this method is applicable only when dealing with small interval.
The bisection method uses a continuous system of input, function f, function values f(a) and f(b) and an interval (a, b). The values of the functions always have opposite signs. This means that, the intervals always have a zero value at certain point.
However, when used in computer arithmetic, there is always a challenge of finite precision and thus there is frequent more convergence test which limits the iterations to the numbers.