function y = my_sqrt(x)
%Use Newton's method to compute the square root of x 
% -The essential point to understand is that Newton's Method can
% solve  for r, such that f(r)=0. 
% 
% -Hence, if r is the square root of x, then f(r) = r^2-x = 0,
% which gives us a function to solve for. 
%


% First, check that x>0
if(x<0)
  error('my_sqrt: the argument needs to be positive');
end

  

% Since we can't define a function in advance because we don't know
% x, we need to define it inside the function, this is called an
% 'inline function'. Here, we want 'x' to be just a parameter, and
% the function to be a function of the variable 'r'.
%
% We define it as follow: 
function_body = ['x.^2 - ' num2str(x)]; 
f = inline(function_body);

%For the precision of the result, we'll accept 'single-precision'
%results, i.e we want the relative error between  our solution
%and the real solution to be less than 2^(-32), i.e we should
%choose a tolerance smaller than   2^(-32) times x (not too small
%if not we won't be able to converge
tol = max(x * 2^(-40), 2^(-64)); 

%Maybe we could do better than that but an easy first guess for the
%square root of x is to use x itself.
y=simpleNewtonSolver(f,x,tol);