Bisection Method

Introduction

The bisection method is a root-finding method that applies to any continuous function for which one knows two values with opposite signs . There are different ways to find the roots of different equations(like simple, quadratic) and functions. Bisection method is the way to find the roots of a function which is polynomial in nature. This method is quite interesting.

We first select an interval or range where the roots may lie, then we bisect and go on dividing the interval into a number of sub intervals until we finally reach the root. The main idea of this method is to find the mid points to find the root. Bisection method is known worldwide in a various of names like the Dichotomy method, Bolzano’s Method etc.

Bisection Method

In this method, the possible range of values or the entire distance between the initial point and the root or the actual point is treated as a line segment. This segment is bisected and re-bisected repeatedly. In this way, the range of values within which the root lie decreases gradually. Let us consider an interval x, y and $\mathrm{f(x)\times\:f(y)<0}$.

In this interval, f(x) is continuous and there is another point v which satisfies

$$\mathrm{v\:\varepsilon\:(x,y),f(v)\:=\:0}$$

Process of solving roots of polynomials by bisection method

The different steps of finding roots by bisection method are

• Take two values x, y such that $\mathrm{f(x)>0\:\&\:f(y)<0}$

• Interval halving is that process by which we choose another point v such that it is the mid point in between the interval x,y. So $\mathrm{v\:=\:\frac{x\:+\:2}{2}}$

• The value of v would help us to solve the value of the function f(x).

• The root of this function is zero. So, the equation $\mathrm{f(x)\:=\:0}$ would be useful in solving the function.

• In case, if we come across a situation where $\mathrm{f(v)\:\neq\:0}$, the sign needs to be checked and put carefully.

• If f(v) has the same sign as f(x) we replace x with v and we keep the same value for y

• If f(v) has the same sign as f(y), we replace y with v and we keep the same value for x

Solved Examples

1) What will be the roots of the polynomial function $z\mathrm{f(x)\:=\:10\:-\:x^{2}}$?

Ans.

The given function is polynomial is nature. So, bisection method must be applied in this case. The chart given below shows the different values of this function.

Hence, we can say that $\mathrm{f(x)\:=\:10\:-\:x^{2}\:=\:0;\:x\:=\:3.16227766.}$

So, 𝑥 = 3.16227766 are the roots of the polynomial function $\mathrm{f(x)\:=\:10\:-\:x^{2}}$

2)What will be the roots of the polynomial function $\mathrm{f(x)\:=\:x^{2}\:-\:3\:=\:0\:,\:x\varepsilon\:[1,2]}$?

Ans.

The given function is polynomial in nature. So, the bisection method must be applied in this case.

If we put (1,2) as the values of x in the function, we get 1 − 3 = −2; 4 − 3 = 1.

The chart given below shows the different values of this function.

Hence, we can say that $\mathrm{f(x)\:=\:x^{2}\:-\:3\:=\:0;\:x\:=\:1.7344.}$

This is because, at the 7th I, we get [1.7266, 1.7344].

So, 𝑥 = 1.7344 are the roots of the polynomial function $\mathrm{f(x)\:=\:5\:-\:x^{2}}$

Conclusion

The bisection method is the method that helps us to solve the functions which are polynomial in nature. It enables us to find the roots of the polynomial functions. A line segment which is basically the distance between the initial and the final points is taken into consideration. It basically means that there are two points which form an interval and the functional value of these two points is continuous, graphically speaking, a simple straight line. Our job is to bisect this line repeatedly until we get the root/s. This can be done by finding the midpoint of the two points that we had considered earlier. This mid point would help us to solve the main function in cosideration which in turn gives us the root of this function.

FAQs

1. If we write $\mathrm{f(a)\:\times\:f(b)<0}$, then what would a person understand by this expression?

It simply means that the product of the two completely different functions f(a) and the function f(b) is less than zero. But it is the external meaning. If we dig deeper, we will see that another meaning of this expression is that the two functions must be of opposite signs, which is to say, they will be on the opposite side of the same axis. To explain this, we know that if we multiply ‘+’ with ‘+’, we will get another positive number. Similarly, multiplying ‘-’ and ‘-’ gives a negative result. Only when the functions are of the opposite signs, that we will get a negative product which is to say a product which is less than zero.

2. We have all used the term continuous function. But what does this expression actually mean?

In a nutshell, a continuous function is that function whose graph is always a straight line and never a graph like parabola, hyperbola, circle etc. So, a continuous function must always satisfy the equation of a straight line which is 𝑦 = 𝑚𝑥 + 𝑐. If there is any point which satisfies the function but not the equation of the straight line, then the function will be considered a discontinuous one. It will no longer be a continuous function in such a case.

3. Can we always use the bisection method? What are its constraints?

No, we can’t always use the bisection method. It is only applicable for polynomial functions and also the functions which are continuous.

4. What are the fields in which we can apply bisection method to get the best possible results?

We can use the bisection method to find the roots of polynomial functions.

5. What is the importance of the middle term in bisection method?

The middle term is probably the single most important thing in the bisection method. Without the middle term, the entire concept of bisection completely vanishes because bisection refers to that thing in which a line or an angle gets divided into ywo equal parts. So, bisection method is all about finding the middle term repeatedly until we finally find the roots of the function.