Secant Method

Newton Raphson is good general purpose root finding method, but sometimes if function is very complicated then computing derivates will take much computational time, so to overcome this issue, in secant method we approximate the first order derivative term f’(r). Algorithm is more or less similar to secant method

Given an equation f(x) = 0
Let the initial guesses be x0 and x1
   Do
      xi+1= xi -       ( f(xi) * (xi - xi-1) ) / ( f(xi) - f(xi-1) )
       while (none of the convergence criterion C1 or C2 is met)

Convergence criterias are:-

• C1. Fixing apriori the total number of iterations N.
• C2. By testing the condition |xi+1−xi| (where i is the iteration number) less than some tolerance limit, say epsilon, fixed threshold.

C Implementation

#include< stdio.h>
#include< math.h>
#define f(x) (pow(x,3)-18)
int main(){
float x0,x1,x2,error;
int i=0;
printf("Input Two Approximations: ");
scanf("%f %f",&x0,&x1);
printf("Ite\tX0\t\tX1\t\tf(X0)\t\tf(X1)\t\tError\n");
do{
        x2=((x0*f(x1))-((x1)*f(x0)))/((f(x1)-f(x0)));
        printf("%2d\t%4.6f\t%4.6f\t%4.6f\t%4.6f\t%4.6f\n",i++,x0,x1,f(x0),f(x1),error);
        error=fabs((x2)-(x1));
        x0=x1;
        x1=x2;
}while(error>0.00005);
return 0;
}

Output

Input Two Approximations: 1 2
Ite    X0              X1                    f(X0)             f(X1)           Error
0      1.000000        2.000000        -17.000000      -10.000000      0.000000
1      2.000000        3.428571        -10.000000      22.303208       1.428571
2      3.428571        2.442238        22.303208       -3.433201       0.986333
3      2.442238        2.573814        -3.433201       -0.949728       0.131575
4      2.573814        2.624131        -0.949728       0.069927        0.050317
5      2.624131        2.620680        0.069927        -0.001263       0.003451
6      2.620680        2.620741        -0.001263       -0.000001       0.000061

With this discussion, this series on solution of non linear equations with Iterative Methods concludes.