**Regula Falsi Method Numerical Problem using C Language**.

Regula Falsi Method in C |

## Overview:

## What is Regula Falsi Method?

In this method, we first sufficiently choose a small interval [a0,b0], such that f(a0)f(b0)<0, by tabulation or graphical method, and which contains only one root a' (say) of f(x)=0, i.e., f'(x) maintains same sign in [a0,b0]. This method is based on the assumption that the graph of y=f(x) in small interval [a0,b0] can be represented by the chord joining (a0,f(a0)) and (b0,f(b0)). Therefore, the point x=a1=a0+h0 at which chord meets the x-axis gives us an approximate value of the root a' of the equation f(x)=0.Thus, we obtain two intervals [a0,x1] and [x1,b0], one of which must contain the root a', depending upon the conditions f(a0)f(a')<0 or f(x1).f(b0)<0. If f(x1).f(b0)<0, then a' lies in the interval [x1,b0], which we rename as [a1,b1]. Again we consider that the graph of y=f(x) in [a1,b1] as the chord joining (a1,f(a1)) and (b1,f(b1)), thus, the point of intersection of the chord with the x-axis (say) x2=a1+h1 gives us an approximate value of the root a' and x2 is called second approximation of the root a'. Proceeding in this manner, we shall get a sequence {x1,x2,x3,...,xn}. Each member of which is the successive approximation of the exact root a' of the equation f(x)=0, geometrically.

## Computation Scheme

1. Find an interval [a0,b0], where f(a0).f(b0)<0 and f'(x) maintains the same sign.2. Write n (number of iterations), an, bn, f(an), f(bn), hn={|f(an)|(bn-an)}/{|f(an)|+|f(bn)|} , xn+1=an+hn and f(xn+1) in a row.

3. Insert, +ve or -ve sign with an, as an(+) or an(-) according as f(a0)>0 or f(a0)<0 and -ve or +ve sign with bn as bn(-) or bn(+) according as f(b0)<0 or f(b0)>0.

4. In the (r+1)th iteration, write xr+1, in the column of an(+) if f(xr+1)>0 and keep br fixed in the column bn(-). Otherwise, write xr+1 in the column of bn(-) if f(xr+1)<0 and keep ar fixed in the column of an(+).

## Algorithm

**STEP 1:**Start

**STEP 2:**Read values of x0, x1 and e

Here x0 and x1 are the two initial guesses e is the degree of accuracy or the absolute error i.e. the stopping criteria

**STEP 3:**Computer function values f(x0) and f(x1)

**STEP 4:**Check whether the product of f(x0) and f(x1) is negative or not. If it is positive take another initial guesses. If it is negative then goto

**STEP 5.**

**STEP 5:**Determine: x2 = [x0*f(x1) – x1*f(x0)] / (f(x1) – f(x0))

**STEP 6:**Check whether the product of f(x1) and f(x) is negative or not. If it is negative, then assign x0 = x2; If it is positive, assign x1 = x2;

**STEP 7:**Check whether the value of f(x) is greater than 0.00001 or not. If yes, goto

**STEP 5**. If no, goto

**STEP 8**.

Here the value 0.00001 is the desired degree of accuracy, and hence the stopping criteria.

**STEP 8:**Display the root as x.

**STEP 9:**Stop.

## Flowchart

Regular Falsi Flowchart |

## Sample Program

WAP in C to find the root of the equation 3x-cos(x)-1=0 using Method of Regular Falsi.#### Code:

```
#include<stdio.h>
#include<conio.h>
#include<math.h> /*included header files*/
#define esp 0.0001 /*defining error term*/
#define f(x) 3*x-1-cos(x) /*defining the function*/
void main()
{
float x0,x1,x2,f1,f2,f0;
int count=1;
clrscr();
do
{
printf("\n Enter the value of x0: ");
scanf("%f",&x0); /*loop to get x0*/
}while(f(x0)>0);
do
{
printf("\n Enter the value of x1: ");
scanf("%f",&x1); /*loop to get x1*/
}while(f(x1)<0);
printf("\n ****************************************************** \n");
printf(" n x0\t x1\t x2\t f0\t f1\t f2");
printf("\n ****************************************************** \n"); /*formatting the output*/
do
{
f0=f(x0);
f1=f(x1);
x2=x0-((f0*(x1-x0))/(f1-f0)); /*working formula loop*/
f2=f(x2);
printf("\n %d %f %f %f %f %f %f",count,x0,x1,x2,f0,f1,f2);
if(f0*f2<0)
x1=x2;
else
x0=x2;
count++;
}while(fabs(f2)>esp);
printf("\n ****************************************************** \n");
printf("\n\n Approximate Root = %f",x2); /*final answer*/
getch();
}
```

#### Compiler Window:

Compiler Window |

#### Output Window:

Output Screen |

## Video Tutorial:

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