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StudyMath# Bisection method

##### 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. The method is also called the interval halving method.

This is calculator which finds function root using bisection method or interval halving method. Brief method description can be found below the calculator

This method is based on the intermediate value theorem for continuous functions, which says that any continuous function f (x) in the interval [a,b] which satisfies f (a) * f (b) < 0 must have a zero in the interval [a,b].

Methods which uses this theorem are called dichotomy methods, because they divide the interval into two parts (not necessary equal).

Here we already have False position method and Secant method, now it is time for simplest method - bisection or interval halving method. As you can guess from its name, this method uses division of interval into two equal parts.

That is, using the relation

x_{n+1} = \frac{x*n+x*{n-1}}{2}

the interval [x_{n-1},x*n] is replaced either with [x*{n-1},x*{n+1}] or with [x*{n+1},x*n] depending on the sign of f(x*{n-1}) * f (x_{n+1}). This process is continued until the zero is obtained. Since the zero is obtained numerically the value of c may not exactly match with all the decimal places of the analytical solution of f(x) = 0 in the given interval. Hence the following mechanisms can be used to stop the bisection iterations :

f(x_k)< \epsilon — function value is less than ε.

\left|x*k-x*{k-1}\right| < \epsilon — difference between two subsequent хk is less than ε. Note that since interval is halved on each step, instead of this you can compute the required number of iterations.

The absolute error is halved at each step so the method converges linearly, which is comparatively slow.

As can be seen from the recurrence relation, the false position method requires two initial values, x0 and x1, which should bracket the root.

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