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).
Example:
Input: A function of x, for example x3 - x2 + 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) = a1x3 + a2x2 + ….. + e where aa1, a2, … are constants and x is a variable.
Transcendental function are non algebraic functions, for example f(x) = sin(x)*x – 3 or f(x) = ex + x2 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 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 greater than and equal to a value ? (A very small value).
Below is implementation of above steps.
C++
// 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 + 2double func(double x){ return x*x*x - x*x + 2;}// Prints root of func(x) with error of EPSILONvoid 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 functionint main(){ // Initial values assumed double a =-200, b = 300; bisection(a, b); return 0;} |
Java
// Java program for implementation of Bisection Method // for solving equationsclass GFG{ static final float EPSILON = (float)0.01; // An example function whose solution is determined using // Bisection Method. The function is x^3 - x^2 + 2 static double func(double x) { return x*x*x - x*x + 2; } // Prints root of func(x) with error of EPSILON static void bisection(double a, double b) { if (func(a) * func(b) >= 0) { System.out.println("You have not assumed" + " right a and b"); 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; } //prints value of c upto 4 decimal places System.out.printf("The value of root is : %.4f" ,c); } // Driver program to test above function public static void main(String[] args) { // Initial values assumed double a =-200, b = 300; bisection(a, b); } // This code is contributed by Nirmal Patel} |
Python3
# Python program for implementation# of Bisection Method for# solving equations # An example function whose# solution is determined using# Bisection Method.# The function is x^3 - x^2 + 2def func(x): return x*x*x - x*x + 2 # Prints root of func(x)# with error of EPSILONdef bisection(a,b): if (func(a) * func(b) >= 0): print("You have not assumed right a and b\n") return c = a while ((b-a) >= 0.01): # 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 if (func(c)*func(a) < 0): b = c else: a = c print("The value of root is : ","%.4f"%c) # Driver code# Initial values assumeda =-200b = 300bisection(a, b)# This code is contributed# by Anant Agarwal. |
C#
// C# program for implementation // of Bisection Method for // solving equationsusing System;class GFG{static float EPSILON = (float)0.01;// An example function whose // solution is determined using// Bisection Method. The function// is x^3 - x^2 + 2static double func(double x){ return x * x * x - x * x + 2;}// Prints root of func(x) // with error of EPSILONstatic void bisection(double a, double b){ if (func(a) * func(b) >= 0) { Console.WriteLine("You have not assumed" + " right a and b"); 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; } // prints value of c // upto 4 decimal places Console.WriteLine("The value of " + "root is : "+ c);}// Driver Codestatic public void Main (){ // Initial values assumed double a = -200, b = 300; bisection(a, b);}}// This code is contributed by ajit |
PHP
<?php// PHP program for implementation // of Bisection Method for solving// equations$EPSILON = 0.01;// An example function whose // solution is determined // using Bisection Method. // The function is x^3 - x^2 + 2function func($x){ return $x * $x * $x - $x * $x + 2;}// Prints root of func(x)// with error of EPSILONfunction bisection($a, $b){ global $EPSILON; if (func($a) * func($b) >= 0) { echo "You have not assumed " . "right a and b","\n"; return; } $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; } echo "The value of root is : " , $c;}// Driver Code// Initial values assumed$a =-200;$b = 300;bisection($a, $b);// This code is contributed by ajit?> |
Javascript
<script>// JavaScript program for implementation // of Bisection Method for // solving equationslet EPSILON = 0.01; // An example function whose solution is determined using // Bisection Method. The function is x^3 - x^2 + 2 function func(x) { return x*x*x - x*x + 2; } // Prints root of func(x) with error of EPSILON function bisection(a, b) { if (func(a) * func(b) >= 0) { document.write("You have not assumed" + " right a and b"); return; } let 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; } //prints value of c upto 4 decimal places document.write("The value of " + "root is : "+ c.toFixed(4)); }// Driver program // Initial values assumed let a =-200, b = 300; bisection(a, b); // This code is contributed by susmitakundugoaldanga.</script> |
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
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