Secant Method, is a Numerical Technique to find the root of an algebraic or transcendental equation.

The root is approximated by drawing secant lines repeatedly.

A secant line is a line joining two points on a function. Secant method requires two initial guesses(x0 and x1), to draw the first secant line. The root of this line(x2), that is, where this line touches the x-axis, becomes the new point, and now a secant line is drawn between the new point(x2) and one of the last points(x1).

This process is repeated until a root is found upto a certain tolerance.

The method is similar to Bisection Method, in that it requires two initial guesses, but still a lot different, as the guesses don’t require to bracket(enclose) the root. Moreover, unlike Bisection Method, Secant Method may not always converge, so it might be a good idea to have a limit for the maximum iterations to be performed.

So, the program would ask the user to enter two initial guesses:x1 and x2.

Then, it will calculate the new point(x3) using the formula:

### PROGRAM (Simple Version):

/*********************************** *********SECANT METHOD************** ***********************************/ #include<stdio.h> #include<math.h> /*Function whose root is to be determined*/ double f(double x){ return x*x-4; } main(){ int iter=1,maxSteps; double x1,x2,x3,eps; printf("Enter the accuracy desired: \n"); scanf("%lf",&eps); printf("Enter the intial guesses: \nx1 = "); scanf("%lf",&x1); printf("x2 = "); scanf("%lf",&x2); printf("Enter the max number of iterations to be performed: "); scanf("%d",&maxSteps); printf("iter\tx1\t\tx2\t\tx3\t\tf(x3)\n"); printf("___________________________________________________________________\n"); do{ x3=(x1*f(x2)-x2*f(x1))/(f(x2)-f(x1)); printf("%d\t%lf\t%lf\t%lf\t%lf\n",iter,x1,x2,x3,f(x3)); x1=x2; x2=x3; iter++; }while(fabs(f(x3))>eps&&iter<=maxSteps); printf("\nOne of the roots is: %lf",x3); }

### OUTPUT:

### PROGRAM (USING FUNCTIONS)

/*********************************** *********SECANT METHOD************** ***********************************/ #include<stdio.h> #include<math.h> /*Function whose root is to be determined*/ double f(double x){ return x*x-4; } /*Function that returns the root from Secant Method*/ double secant(double f(double x), double x1, double x2, double eps, int maxSteps){ int iter=1; double x3; do{ x3=(x1*f(x2)-x2*f(x1))/(f(x2)-f(x1)); x1=x2; x2=x3; iter++; }while(fabs(f(x3))>eps&&iter<=maxSteps); return x3; } /*Secant Method Function that tabulates the values at each iteration*/ double secantPrint(double f(double x), double x1, double x2, double eps, int maxSteps){ int iter=1; double x3; printf("___________________________________________________________________\n"); printf("iter\tx1\t\tx2\t\tx3\t\tf(x3)\n"); printf("___________________________________________________________________\n"); do{ x3=(x1*f(x2)-x2*f(x1))/(f(x2)-f(x1)); printf("%d\t%lf\t%lf\t%lf\t%lf\n",iter,x1,x2,x3,f(x3)); x1=x2; x2=x3; iter++; }while(fabs(f(x3))>eps&&iter<=maxSteps); printf("___________________________________________________________________\n"); return x3; } main(){ int maxSteps; double x1,x2,x3,eps; printf("Enter the accuracy desired: \n"); scanf("%lf",&eps); printf("Enter the intial guesses: \nx1 = "); scanf("%lf",&x1); printf("x2 = "); scanf("%lf",&x2); printf("Enter the max number of iterations to be performed: "); scanf("%d",&maxSteps); printf("\nOne of the roots is: %lf",secantPrint(f,x1,x2,eps,maxSteps)); }