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This article is about searching continuous function values. For searching a finite sorted array, see binary search algorithm . A few steps of the bisection method applied over the starting range [a 1 ;b 1 ]. The bigger red dot is the root of the function. The bisection method in mathematics is a root-finding method that repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods. [1] The method is also called the interval halving method, [2] the binary search method , [3] or the dichotomy method . [4] The method is applicable for numerically solving the equation f ( x ) = 0 for the real variable x , where f is a continuous function defined on an interval [ a , b ] and where f ( a ) and f ( b ) have opposite signs. In this case a and b are said to bracket a root since, by the intermediate value theorem , the continuous function f must have at least one root in the interval ( a , b ). At each step the method divides the interval in two by computing the midpoint c = ( a + b ) / 2 of the interval and the value of the function f ( c ) at that point. Unless c is itself a root (which is very unlikely, but possible) there are now only two possibilities: either f ( a ) and f ( c ) have opposite signs and bracket a root, or f ( c ) and f ( b ) have opposite signs and bracket a root. [5] The method selects the subinterval that is guaranteed to be a bracket as the new interval to The method
be used in the next step. In this way an interval that contains a zero of f is reduced in width by 50% at each step. The process is continued until the interval is sufficiently small.

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