function X = newtonraphson(f,df,R,h)
X = []; % variable to store solution
tol = 0.0001; % max error tolerance between two consecutive values in loop
for i=R(1):h:R(2)
% This loop iterates between the extremes of given range through
% prescribed increments-sub intervals. At a particular sub-interval, if
% it is found that function value at the ends of the sub-interval
% changes its sign, then a point inside that interval is chosen as the
% initial value for iteration. If values at both ends are of the same
% sign, then its derivative is checked. If the derivatives differ in
% sign, then midpoint of the interval is taken, and both the halfs of
% the sub-interval are verified the same way. The sub-interval is
% omitted if both the values of function as well as value of
% derivatives are the same in the sub-interval. Once the initial guess
% is obtained, Newton Raphson method is used to find the solution, and
% the program proceeds towards the next sub-interval
range = [i i+h];
flag = 0;
while flag == 0
   if ((f(range(1))<0)&&(f(range(2))>=0)) || ((f(range(1))>0)&&(f(range(2))<=0))
         x0 = mean(range); % function has different signs in the interval ends. Hence midpoint is selected
         flag = 1;
   else % funtion has same sign in the interval ends
         if ((df(range(1))<0)&&(df(range(2))>0)) || ((df(range(1))>0)&&(df(range(2))<0))
            % here, function derivative has different signs in the interval
            % ends

            if ((df(range(1))<0)&&(df(mean(range))>0)) || ((df(range(1))>0)&&(df(mean(range))<0))
               % here, function derivative has different signs in the first
               % end of interval and the middle of interval
               range(2) = mean(range);
            else
               range(1) = mean(range);
            end
            if abs(range(1)-range(2))<=tol
               flag = 2;
            else
               flag = 0;
            end

         else % here, derivatives are also of the same sign
            % the interval is skipped and another interval is selected
            flag = 2;
         end
   end

end

1.3.2. The theory of Taylor series tells us that for all real valu...