The bisection method is based on the following result from calculus:
The Intermediate Value Theorem: Assume f : IR→IR 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.
In other words, if a continuous function has different signs at two points, it has to go through zero somewhere in between!
The bisection method consists of finding two such numbers a and b, then halving the interval [a,b] and keeping the half on which f (x) changes sign and repeating the procedure until this interval shrinks to give the required accuracy for the root.
An algorithm could be defined as follows. Suppose we need a root for f (x) = 0 and we have an error tolerance of ε (the absolute error in calculating the root must be less that ε).
Bisection Algorithm:
Find two numbers a and b at which f has different signs.
Define c = a+b
If b−c ≤ε then accept c as the root and stop.
If f (a)f (c)≤0 then set c as the new b. Otherwise, set c as the new a. Return to step 1.
Error bounds
Let α be the value of the root, a≤α≤b. Let an, bn and cn be the values of a, b and c on the nth iteration of the algorithm.
Then the error bound for cn is given by
|α−cn| ≤ (1/2n) (b−a)
This inequality can give us the number of iterations needed for a required accuracy ε
n ≥ ((log(b−a)/ε)) / log2
Advantages and disadvantages of the bisection method
The method is guaranteed to converge.
The error bound decreases by half with each iteration.
The bisection method converges very slowly.
The bisection method cannot detect multiple roots
Example Code:
Bisection Method x2-5
root between 2 and 3
File name : bisection.c
#include<stdio.h>
#include<conio.h>
#include<stdlib.h>
#include<math.h>
float f(float x)
{
return (x*x)-5;
}
main()
{
float x0,x1,x2,E;
int count=1;
charch;
clrscr();
xyz:
printf("\nEnter value of x0");
scanf("%f",&x0);
printf("\nEnter value of x1");
scanf("%f",&x1);
printf("\nEnter value of error interation:");
scanf("%f",&E);
clrscr();
printf("\n NO\tx0\tx1\tx2\tf(x0)\tf(x1)\tf(x2)");
if(f(x0)*f(x1)<0)
{
x2=(x0+x1)/2;
while(fabs(x1-x0)>E)
{
printf("\n %d\t%.3f\t%.3f\t%.3f\t%.3f\t%.3f\t%.3f",count,x0,x1,x2,f(x0),f(x1 ),f(x2));
if(f(x0)*f(x2)<0)
{
x1=x2;
}
else
{
x0=x2;
}
count++;
x2=(x0+x1)/2;
}
printf("\nRoot is %f",x2);
}
else
{
clrscr();
printf("\nYour value is incorrect...\nPlease enter a new value then Press 'Y' key otherwise any key press to exit!....");
fflush(stdin);
scanf("%c",&ch);
if(ch=='y'||ch=='Y')
{
goto xyz;
}
else
{
exit(0);
}
}
getch();
}
A quadratic equation is a second-order polynomial equation in a single variable x with ax2+bx+c=0, a!=0.
The Quadratic Formula uses the "a", "b", and "c" from "ax2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients" of the quadratic equation they've given you to solve.
The Quadratic Formula is derived from the process of completing the square, and is formally stated as:
For ax2 + bx + c = 0, the value of x is given by:
x = [ -b ± sqrt(b2 - 4ac) ] / 2a
Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has two solutions. These solutions may be both real, or both complex.
For Example Code:
File name : qurdation.c
#include<stdio.h>
#include<conio.h>
#include<math.h>
void main()
{
float a,b,c,d,x1=0,x2=0;
clrscr();
printf("Enter the value of A:=");
scanf("%f",&a);
printf("Enter the value of B:=");
scanf("%f",&b);
printf("Enter the value of C:=");
scanf("%f",&c);
d=((b*b)-(4*a*c));
clrscr();
printf("\n| A | %.2f ",a);
printf("\n| B | %.2f ",b);
printf("\n| C | %.2f ",c);
printf("\n-----------------");
printf("\n| D | %.2f",d);
if(d==0)
{
printf("\n\n| Roots are real and equal");
x1=((-b+sqrt(d))/(2*a));
x2=((-b-sqrt(d))/(2*a));
}
else if(d>0)
{
printf("\n\n| Roots are real and distict");
x1=((-b+sqrt(d))/(2*a));
x2=((-b-sqrt(d))/(2*a));
}
else if(d<0)
{
printf("\n\n| Roots are imegnand ");
}
printf("\n\n| Value of X1 | %.2f",x1);
printf("\n| Value of X2 | %.2f",x2);
getch();
}
