How to detect a cycle in newton raphson method in C - c

Question

Failure analysis of newton raphson method says that "For some functions, some starting points may enter an infinite cycle, preventing convergence". I want to keep a check within the program whether it is entering into an infinite cycle or not using assert statement. If it enters then the program will terminate saying convergence is not possible using this initial guess. How can I detect this cycle within the program?
Code:
int user_power, i=0, cnt=0, flag=0;
int coef[10]={0};
float x1=0, x2=0, t=0;
float fx1=0, fdx1=0;
void main()
{
printf("\n\n\t\t\t PROGRAM FOR NEWTON RAPHSON GENERAL");
printf("\n\n\n\tENTER THE MAXIMUM POWER:");
scanf("%d",&user_power);
for(i=0;i<=user_power;i++)
{
printf("\n\t x^%d:",i);
scanf("%d",&coef[i]);
}
printf("\n");
printf("\n\tINTIAL X1---->");
scanf("%f",&x1);
printf("\n ******************************************************");
printf("\n ITERATION X1 FX1 F'X1 ");
printf("\n **********************************************************");
do
{
cnt++;
fx1=fdx1=0;
for(i=user_power;i>=1;i--)
{
fx1+=coef[i] * (pow(x1,i)) ; //calculating f(x1)
}
fx1+=coef[0];
for(i=user_power;i>=0;i--)
{
fdx1+=coef[i]* (i*pow(x1,(i-1))); //calculating f'(x1)
}
t=x2;
assert(fdx1!=0);
x2=(x1-(fx1/fdx1));
x1=x2;
printf("\n %d %.3f %.3f %.3f ",cnt,x2,fx1,fdx1);
} while((fabs(t - x1))>=0.0001);
printf("\n\t THE ROOT OF EQUATION IS %f",x2);
printf("\n");
}

Answer 1

You don't check for periodic orbits. Determining the period and then convergence to the orbit is too expensive.
What you can do is to check after 5 Newton iterations if the condition for quadratic convergence is approximately satisfied. For that you check over 3 steps if the decrease of the function value is more than by a divisor of 4 (or in the step length of the Newton step, it should be more than a divisor of 2).
Failing that, restart with some (random) modification of the initial point.
For most problems, the application of Newton's method in double precision has 2-3 steps of global orientation and then 4-6 steps of quadratic convergence before the precision of the double floating point format is exceeded. Thus, one can be even bolder, if after 10 steps the iteration does not converge, the initial point was bad, be it that it leads to a periodic orbit or divergence to infinity. Most likely, close-by initial points will behave similarly, so make a non-trivial change in the initial point for the start of the next run of the iteration.
Additional remarks:
Investigate the Horner scheme and the double Horner scheme for the evaluation of polynomials (and their derivatives). The use of the power function is ill-advised.
Spend some thought on the situation that there might be no real roots of the polynomial.
For a general idea on finding all roots of a polynomial via Newtons method, see J. H. Hubbard, D. Schleicher, S. Sutherland: How to Find All Roots of Complex Polynomials by Newton's Method, Inventiones Mathematicae vol. 146 (2001) - with a discussion of the global structure of Newton fractals