/* An example of Newton's method       file: 4ex4.cpp
   FALL 1998
   ___________________________________
   Jacob Y. Kazakia   jyk0
   October 1, 1998
   Example 4 of week 4
   Recitation Instructor: J.Y.Kazakia
   Recitation Section  01
   ___________________________________

   Purpose: This program estimates the root of the function f(x) = x^2 - 3,
            near x = 2 [ which in fact is   + sqrt(3) ]using Newton's method
            of root finding.

   Algorithm:   The improvements to our initial guess x0  are obtained by
                repeated  use of the formula:

                    x_new = x_old - ( f(x_old) / f'(x_old) ).

                The initial guess ( x0 ), the maximum number of iterations(nmax),
                the root tolerance (xtol), the function value tolerance (ftol)
                are all inputed through the keyboard.

                The root tolerance is defined as follows:
                The iterations will stop if

                    | x_new - x_old | / | x_old | < xtol.


                The function value tolerance is defined as follows:
                The iterations will stop if

                    | f( x_new) | < ftol.

                For accuracy all variables will be declared as double.
                */

 #include <iostream.h>
 #include <iomanip.h>
 #include <math.h>

 void main()
 {
 double x_old = 0., x_new = 0 ;  //  estimates of the root
 double xtol = 0. , ftol = 0. ;  // tolerances
 double f = 0. , fprime = 0. ;   // values of the function and of its derivative
 int nmax = 0 ;                  // max number of iterations
 int n = 0 ;                     // running index

 cout <<" \n\n enter your initial guess of the root   ...  ";
 cin >> x_old;

 cout <<" \n\n enter the maximum number of iterations ...  ";
 cin >> nmax;

 cout <<" \n\n enter the root tolerance ...  ";
 cin >> xtol;

 cout <<" \n\n enter the function value tolerance ...  ";
 cin >> ftol;


 // Prepare the heading of the table

 cout <<"\n\n    x_old                 f(x_old)             f'(x_old)           x_new   \n\n";
 cout<< setiosflags(ios:: scientific);
 cout<< setprecision(10);

 //    START THE LOOP

 for( n = 1; n <= nmax ; n++ )
  {
   f = x_old * x_old - 3 ;
   fprime = 2 * x_old ;
   x_new = x_old - f / fprime ;
   cout<<setw(20)<< x_old << setw(20) << f << setw(20) << fprime;
   cout<<setw(20)<< x_new << endl;

   if( fabs((x_new - x_old ) / x_old ) < xtol )   break;

   if( fabs(f) < ftol )  break;

   x_old = x_new ;
   }
cout<<endl<<endl;
cout<<" enter e (exit) to terminate the program....";
char hold;
cin>>hold;
}