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.

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Karen Luckhurst

Created: 2014-06-27 11:03:11, Last updated: 2020-11-06 11:08:20

This is a calculator that finds a function root using the bisection method, or interval halving method. A brief method description can be found below the calculator.

PLANETCALC, Bisection method

Bisection method

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Bisection method

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] that satisfies f (a) * f (b) < 0 must have a zero in the interval [a,b].
Methods that uses this theorem are called dichotomy methods, because they divide the interval into two parts (which are not necessarily equal).

We have alreadyy explored False position method and Secant method, now it is time for the simplest method – bisection, also know as interval halving. As you can guess from its name, this method uses division of an 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 — the difference between the two subsequent хk is less than ε. Note that since the interval is halved on each step, you can instead 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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PLANETCALC, Bisection method