Given a function f(x) on floating number x and two numbers ‘a’ and ‘b’ such that f(a)*f(b) < 0 and f(x) is continuous in [a, b]. Here f(x) represents algebraic or transcendental equation. Find root of function in interval [a, b] (Or find a value of x such that f(x) is 0).
Input: A function of x, for example x^{3} - x^{2} + 2. And two values: a = -200 and b = 300 such that f(a)*f(b) < 0, i.e., f(a) and f(b) have opposite signs. Output: The value of root is : -1.0025 OR any other value with allowed deviation from root.
What are Algebraic and Transcendental functions?
Algebraic function are the one which can be represented in the form of polynomials like f(x) = a_{1}x^{3} + a_{2}x^{2} + ..... + e where aa_{1}, a_{2}, ... are constants and x is a variable.
Transcendental function are non algebraic functions, for example f(x) = sin(x)*x - 3 or f(x) = e^{x} + x^{2} or f(x) = ln(x) + x ....
What is Bisection Method?
The method is also called the interval halving method, the binary search method or the dichotomy method. This method is used to find root of an equation in a given interval that is value of 'x' for which f(x) = 0 .
The method is based on The Intermediate Value Theorem which states that if f(x) is a continuous function and there are two real numbers a and b such that f(a)*f(b) < 0. Then f(x) has at least one zero between a and b. If for a function (f(a) < 0 and f(b) > 0) or (f(a) > 0 and f(b) < 0), then it is guaranteed that it has at least one root between them.
Assumptions:
- f(x) is a continuous function in interval [a, b]
- f(a) * f(b) < 0
Steps:
- Find middle point c= (a + b)/2 .
- If f(c) == 0, then c is the root of the solution.
- Else f(c) != 0
- If value f(a)*f(c) < 0 then root lies between a and c. So we recur for a and c
- Else If f(b)*f(c) < 0 then root lies between b and c. So we recur b and c.
- Else given function doesn't follow one of assumptions.
Since root may be a floating point number, we repeat above steps while difference between a and b is less than a value ε (A very small value).
Below image is taken from wiki page.
Below is C++ implementation of above steps.
// C++ program for implementation of Bisection Method for // solving equations #include<bits/stdc++.h> using namespace std; #define EPSILON 0.01 // An example function whose solution is determined using // Bisection Method. The function is x^3 - x^2 + 2 double func(double x) { return x*x*x - x*x + 2; } // Prints root of func(x) with error of EPSILON void bisection(double a, double b) { if (func(a) * func(b) >= 0) { cout << "You have not assumed right a and b\n"; return; } double c = a; while ((b-a) >= EPSILON) { // Find middle point c = (a+b)/2; // Check if middle point is root if (func(c) == 0.0) break; // Decide the side to repeat the steps else if (func(c)*func(a) < 0) b = c; else a = c; } cout << "The value of root is : " << c; } // Driver program to test above function int main() { // Initial values assumed double a =-200, b = 300; bisection(a, b); return 0; }
Output:
The value of root is : -1.0025
Time complexity :- Time complexity of this method depends on the assumed values and the function.
What are pros and cons?
Advantage of the bisection method is that it is guaranteed to be converged. Disadvantage of bisection method is that it cannot detect multiple roots.
In general, Bisection method is used to get an initial rough approximation of solution. Then faster converging methods are used to find the solution.
We will soon be discussing other methods to solve algebraic and transcendental equations
References:
Introductory Methods of Numerical Analysis by S.S. Sastry
https://en.wikipedia.org/wiki/Bisection_method
This article is contributed by Abhiraj Smit. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above