The *bisection method* is one of the simplest and most reliable of iterative methods for the solution of nonlinear equations. This method, also known as *binary chopping* or *half-interval* method, relies on the fact that if *f(x)* is real and continuous in the interval *a < x < b*, and *f(a)* and *f(b)* are of opposite signs, that is,

*f(a)*f(b) < 0*

then there is at least one real root in the interval between a and b.

**Bisection method** is a closed bracket method and requires two initial guesses. It is the simplest method with slow but steady rate of convergence.

**Features of Bisection Method:**

- Type – closed bracket
- No. of initial guesses – 2
- Convergence – linear
- Rate of convergence – slow but steady
- Accuracy – good
- Programming effort – easy
- Approach – middle point

### Bisection Method Algorithm and Flowchart:

**Algorithm:**

Start

- Decide initial values for x1 and x2 and stopping criterion, E.
- Compute f1 = f(x1) and f2 = f(x2).
- If f1 * f2>0, x1 and x2 do not bracket any root and go to step 7;

Otherwise continue. - Compute x0 = (x1+x2)/2 and compute f0 = f(x0)
- If f1*f0 < 0 then

set x2 = x0

else

set x1 = x0

set f1 = f0 - If absolute value of (x2 – x1)/x2 is less than error E, then

root = (x1 + x2)/2

write the value of root

go to step 7

else

go to step 4 - Stop.

**Flow Chart**

Above given Algorithm and Flowchart of Bisection Methods Root computation is a simple and easier way of understanding how the bracketing system works, algorithm and flowchart may not follow same procedure, yet they give the same outputs.

### Tabular Example of Bisection Method Numerical Computation

Consider **f(x) = x ^{3} + 3x – 5**, where

**[x1 = 1, x2 = 2]**and E

**= 0.001.**

i | x1 | x0 | x2 | f(x1) | f(x0) | f(x2) |
---|---|---|---|---|---|---|

1 | 1 | 1.5 | 2 | –1 | 2.875 | 9 |

2 | 1 | 1.25 | 1.5 | –1 | 0.703125 | 2.875 |

3 | 1 | 1.125 | 1.25 | –1 | –0.201171875 | 0.703125 |

4 | 1.125 | 1.1875 | 1.25 | –0.201171875 | 0.237060546875 | 0.703125 |

5 | 1.125 | 1.15625 | 1.1875 | –0.201171875 | 0.014556884765625 | 0.237060546875 |

6 | 1.125 | 1.140625 | 1.15625 | –0.201171875 | –0.0941429138183594 | 0.014556884765625 |

7 | 1.140625 | 1.1484375 | 1.15625 | –0.0941429138183594 | –0.0400032997131348 | 0.014556884765625 |

8 | 1.1484375 | 1.15234375 | 1.15625 | –0.0400032997131348 | –0.0127759575843811 | 0.014556884765625 |

9 | 1.15234375 | 1.154296875 | 1.15625 | –0.0127759575843811 | 0.000877253711223602 | 0.014556884765625 |

**Note**: Bisection method guarantees the convergence of a function f(x) if it is continuous on the interval [a,b] (denoted by x1 and x2 in the above algorithm. For this, f(a) and f(b) should be of opposite nature i.e. opposite signs. The slow convergence in bisection method is due to the fact that the absolute error is halved at each step. Due to this the method undergoes linear convergence, which is comparatively slower than the Newton-Raphson, secant or false-position method.

### Bisection Method – Code in C Programming

Method 1:

This program in C is used to demonstarte bisection method.

Bisection method is one of the many root finding methods.

In this method we are given a function f(x) and we approximate 2 roots a and b for the function such that f(a).f(b)<0.

Then we find another point

c=(a+b)/2

if f(c)==0

then root=c;

else

if f(a).f(c)<0

b=c;

if f(b).f(c)<0

a=c;

and we repeat these steps for the given number of iterations.

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#include"stdio.h" #include"math.h" double F(double x) { return(pow(x,3)+3*x-5);//This return the value of the function } int main() { printf("This program illustrates the bisection method in C\n"); printf("x^3 + 3*x - 5 = 0\n"); double x0,x1; printf("Enter the first approximation to the root\n"); scanf("%lf",&x0); printf("Enter the second approximation to the root\n"); scanf("%lf",&x1); int iter; printf("Enter the number of iterations you want to perform\n"); scanf("%d",&iter); int ctr=1; double l1=x0; double l2=x1; double r,f1,f2,f3; //We check if the initail approximations are the root or not if(F(l1)==0) r=l1; else if(F(l2)==0) r=l2; else { while(ctr <= iter) {//this is an implementation of the algorithm mentioned above f1=F(l1); r=(l1+l2)/2.0; f2=F(r); f3=F(l2); if(f2==0) { r=f2; break; } printf("The root after %d iteration is %lf\n",ctr,r); if(f1*f2<0) l2=r; else if(f2*f3<0) l1=r; ctr++; } } printf("The approximation to the root is %lf",r); getch(); } /*A sample run of the program was carried out and the results were found as:- This program illustrates the bisection method in C x^3 + 3*x - 5 = 0 Enter the first approximation to the root 1 Enter the second approximation to the root 2 Enter the number of iterations you want to perform 9 The root after 1 iteration is 1.500000 The root after 2 iteration is 1.250000 The root after 3 iteration is 1.125000 The root after 4 iteration is 1.187500 The root after 5 iteration is 1.156250 The root after 6 iteration is 1.146025 The root after 7 iteration is 1.148438 The root after 8 iteration is 1.152344 The root after 9 iteration is 1.154297 The root is 1.154297 */ |

**Method 2**: Source Code for Bisection Method

for equation* f(x) = x^3 – 4*x – 9*

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#include "math.h" #include "conio.h" float fun (float x) { return (x*x*x - 4*x - 9); } void bisection (float *x, float a, float b, int *itr) /* this function performs and prints the result of one iteration */ { *x=(a+b)/2; ++(*itr); printf("Iteration no. %3d X = %7.5f\n", *itr, *x); } void main () { int itr = 0, maxmitr; float x, a, b, allerr, x1; printf("\nEnter the values of a, b, allowed error and maximum iterations:\n"); scanf("%f %f %f %d", &a, &b, &allerr, &maxmitr); bisection (&x, a, b, &itr); do { if (fun(a)*fun(x) < 0) b=x; else a=x; bisection (&x1, a, b, &itr); if (fabs(x1-x)<allerr) { printf("After %d iterations, root = %6.4f\n", itr, x1); return 0; } x=x1; } while (itr<maxmitr); printf("The solution does not converge or iterations are not sufficient"); return 1; } |

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