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Bisection Method

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Last updated date: 24th Feb 2024
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An Introduction to Bisection Method


In Mathematics, the bisection method is used to find the root of a polynomial function. For further processing, it bisects the interval and then selects a sub-interval in which the root must lie and the solution is iteratively reached by narrowing down the values after guessing, which encloses the actual solution. In this method, the interval distance between the initial values is treated as a line segment. It then successively divides the interval in half and replaces one endpoint with the midpoint so that the root is bracketed. This method is based on The Intermediate Value Theorem and is very simple robust and easy to implement. There are various names given to this method such as “ the interval halving method”, “the binary search method”, “the dichotomy method”, and “Bolzano’s Method”.


Finding Root by Bisection Method

As stated above, the Bisection method program is a root-finding method that we often come across while dealing with numerical analysis. It is also known as the Bisection Method Numerical Analysis. It is based on Bolzano's theorem for continuous functions. So let us understand what Bolzano’s theorem says.


Theorem (Bolzano): If on an interval a,b and f(a)·f(b) < 0, a function f(x) is found to be continuous, then there exists a value c such that c ∈ (a, b) or which f(c) = 0. The graph given below shows a continuous function.


The bisection method problems can be solved by using the bisection method formula to find the value c of the function f(x) that crosses the x-axis. In this case, the value c is an approximate value of the root of the function f(x). In this bisection method program, the value of the tolerance we set for the algorithm determines the value of c where it gets to the real root. One such bisection method is explained below.


Bisection Method Procedure

To solve bisection method problems, given below is the step-by-step explanation of the working of the bisection method algorithm for a given function f(x):


Step 1:

Choose two values, a and b such that f(a) > 0 and f(b) < 0 .


Step 2:

Calculate a midpoint c as the arithmetic mean between a and b such that c = (a + b) / 2. This is called interval halving.


Step 3:

Evaluate the function f for the value of c.


Step 4:

The root of the function is found only if the value of f(c) = 0.


Step 5:

If (c) ≠ 0, then we need to check the sign:


  1. we replace a with c if f(c) has the same sign as f(a) and we keep the same value for b

  2. we replace b with c if f(c) has the same sign as f(b), and we keep the same value for a


To get the right value with the new value of a or b, we go back to step 2 And recalculate c. 


Advantages of Bisection Method

  • Guaranteed convergence. The bracketing approach is known as the bisection method, and it is always convergent.

  • Errors can be managed. Increasing the number of iterations in the bisection method always results in a more accurate root.

  • Doesn't demand complicated calculations. There are no complicated calculations required when using the bisection method. To use the bisection method, we only need to take the average of two values.

  • Error bound is guaranteed. There is a guaranteed error bound in this technique, and it reduces with each repetition. Each cycle reduces the error bound by 12 per cent.

  • The bisection method is simple and straightforward to programme on a computer.

  • In the case of several roots, the bisection procedure is quick.


Disadvantages of Bisection Method

  • Although the Bisection method's convergence is guaranteed, it is often slow.

  • Choosing a guess that is close to the root may necessitate numerous iterations to converge.

  • Some equations' roots cannot be found. Because there are no bracketing values, like f(x) = x².

  • Its rate of convergence is linear.

  • It is incapable of determining complex roots.

  • If the guess interval contains discontinuities, it cannot be used.

  • It cannot be applied over an interval where the function returns values of the same sign.


Bisection Method Problems

The best way of understanding how the algorithm works are by looking at a bisection method example and solving it by using the bisection method formula.


Example 1:  Find the root of f(x) = 10 − x².


Solution:

The calculation of the value is described below in the table:


i

a

b

c

f(a)

f(b)

f(c)

0

−2

5

1.5

6

−15

7.75

1

1.5

5

3.25

7.75

−15

−0.5625

2

1.5

3.25

2.375

7.75

−0.5625

4.359375

3

2.375

3.25

2.8125

4.359375

−0.5625

2.0898438

4

2.8125

3.25

3.03125

2.0898438

−0.5625

0.8115234

5

3.03125

3.25

3.140625

0.8115234

−0.5625

0.1364746

6

3.140625

3.25

3.1953125

0.1364746

−0.5625

−0.210022

7

3.140625

3.1953125

3.1679688

0.1364746

−0.210022

−0.036026

8

3.140625

3.1679688

3.1542969

0.1364746

−0.036026

0.0504112

9

3.1542969

3.1679688

3.1611328

0.0504112

−0.036026

0.0072393


At initialization (i = 0), we choose a = −2 and b = 5. After evaluation of the function in both points, we find that f(a) is positive while f(b) is negative. This means that between these points, the plot of the function will cross the x-axis at a particular point, which is the root we need to find.


The value of f(c) after 9 iterations is less than our defined tolerance (0.0072393 < 0.01). This means that the value that approximates best the root of the function f is the last value of c = 3.1611328.


By solving our quadratic equations in a classic way, we can check our result:


10 − x2 =0


x² = 10


x = √10


x = ±3.16227766


The article discusses all the important points related to the bisection method such as its advantages, disadvantages and solved problems etc. Step by step bisection method procedure is also given in the article.


FAQs on Bisection Method

1. When is it asserted that a function is continuous?

A function is said to be continuous when small changes in the argument bring about small changes in the result too. For example, small changes in x will give small changes in f(x) too. If the change in x is in small steps, then the change in f(x) will also be in small steps and not big “jumps”. This shows that the argument and result are directly proportional to each other, such that if one increases then the other increases too. This makes a function continuous.

2. What does f(a)·f(b) < 0 means?

The statement f(a)·f(b) < 0 means that f(a)·f(b) is less than zero such that f(a) and f(b) both have different signs where one of them is above the x-axis and the other one is below the x-axis. To explain this, we can take an example of a function that crosses the x-axis. Suppose if we plot a function f(x) at some point, it will cross the x-axis such that the value of x is the root of the equation f(x) = 0 for which the plot is crossing the x-axis.

3. What is the condition when you can not use the bisection method?

When the root is a double root, bisection fails because the function keeps the same sign except for hitting zero at one point. In other words, at each step, f(a) and f(b) have the same sign.


Then, at each stage, it's unclear which half of the interval to take. A method for determining the lowest or maximum is preferable in this circumstance. But, of course, this isn't always evident when the problem is first presented.

4. What are the disadvantages of the bisection method?

Below is the disadvantages of the bisection method:

  • Although the bisection method's convergence is guaranteed, it is often slow.

  • Choosing a guess that is close to the root may necessitate numerous iterations to converge.

  • Some equations have no root that is found. Because there are no bracketing values, e.g. f(x) = x².

  • Its rate of convergence is linear.

  • It is incapable of determining complex roots.

  • If the guess interval contains discontinuities, it cannot be used.

  • It can't be used throughout a range of values where the function returns the same sign.

5. Where could be the possible applications of the bisection method?

The bisection method divides the interval in which the root of the problem is located. The intermediate theorem for continuous functions is the foundation of this principle. 


It has several applications too as listed below:

  • The bisection method can be used to detect short segments in video content for a digital video library.
  • The bisection method is used to determine the appropriate population size.
  • In a molecular system, the bisection method is used to locate and compute periodic orbits.