C Program for Birge-Vieta Method

Basic Theory:

The Birge-Vieta method investigates the real root of a polynomial, say x_r, by assuming that (x-x_r) is a factor of that polynomial. Starting with an initial value or approximation x₀ to x_r, we use some iterative method to improve the value of the approximation such that R(x_r)®0. It makes the use of the Newton-Raphson method for iteration in order to improve the value of x_r.

x_n+1=x_n – [f(x_n)/f ’(x_n)]

We deploy the Synthetic Division method in order to find out R(x_r) in this method. The remainder of first division gives the functional value while the remainder second division gives the value of derivative. We continue the iteration until the result of desired precision is obtained.

Example:  Solve the equation x³+x²-3x-3 using Birge-Vieta Method where x₀ = 2.

Using the synthetic division,

2|1         1              -3            -3

  |           2              6              6

  |1         3              3              3¬f(x₀)

  |           2              10

  |1         5              13¬f ’(x₀)

Now, x₁ = 2 – 3/13 = 1.7692

1.7692|1              1              -3            -3

           |                 1.7692   4.8992   3.3602

           |1              2.7692   1.8992   0.3602¬f(x₁)

           |                 1.7692   8.0292

           |1              4.5384   9.9285¬f ’(x₁)

Now, x₂ = 1.7692 – 0.3602/9.9285 = 1.7329

1.7329|1              1              -3            -3

           |                 1.7329   4.7358   3.0079

           |1              2.7329   1.7358   0.0079¬f(x₂)

           |                 1.7329   7.7387

           |1              4.4658   9.4745¬f ’(x₂)

Now, x₃ = 1.7329 – 0.0079/9.4745 = 1.7320

Therefore, Approximate Root = 1.732 correct to 3 decimal places.

Program:

#include<stdio.h>

#include<conio.h>

#include<math.h>

float p[6], ply[6],q[6];

float synth(int m, float r){

                int i;

                q[0] = p[0];

                for(i=1;i<=m;i++){

                                q[i] = (q[i-1]*r)+p[i];

                }

                printf("\n");

                for(i=0;i<m;i++){

                                printf("\t%f",q[i]);

                }

                printf("\t%f",q[m]);

                return(q[m]);

}

void main(){

                clrscr();

                int m,i,flag=0;

                float r, x,x1, fx, fdx;

                printf("\t\tBIRGE-VIETA METHOD");

                printf("\n Enter the highest degree of the equation (max 5): ");

                scanf("%d",&m);

                                for(i=0;i<=m;i++){

                                printf("\n Coefficient x[%d] = ",m-i);

                                scanf("%f",&p[i]);

                                ply[i] = p[i];

                                }

                printf("\n Enter the initial value x0 : ");

                scanf("%f",&r);

                x = r;

                do{

                   printf("\n%f\n",x);

                   fx = synth(m,x);

                                for(i=0;i<=m;i++){

                                                p[i]=q[i];

                                }

                   fdx = synth(m-1,x);

                   x1 = x - (fx/fdx);

                                if(fabs(x1-x) <= 0.0009){

                                                flag = 1;

                                 }

                   x = x1;

                                for(i=0;i<=5;i++){

                                                p[i]=ply[i];

                                }

                }while(flag!=1);

                printf("\nApproximate root = %f", x1);

getch();

}

Sample Output:

                        BIRGE-VIETA METHOD
Enter the highest degree of the equation (max 5): 3
Coefficient x[3] = 1
Coefficient x[2] = 1
Coefficient x[1] = -3
Coefficient x[0] = -3
Enter the initial value x0 : 2
2.000000
              1.000000   3.000000   3.000000   3.000000
              1.000000   5.000000   13.000000
1.769231
              1.000000   2.769231   1.899408   0.360492
              1.000000   4.538462   9.928994
1.732924
              1.000000   2.732924   1.735948   0.008266
              1.000000   4.465847   9.474921
Approximate root = 1.732051

©Dixit Bhatta 2013