Today we continue our discussion on matrix methods of optimization. We mentioned linear and second order methods. Newton's method is a good example of the latter. It finds a solution to an equation in one unknown. This method is an iterative method that seeks a critical point of the cost function f ( the gradient to f tends to zero ) by selecting an update vector at xk along which the directional derivative of grad f is equal to -grad f(xk). The second order information on the cost function is incorporated through the directional derivative of the gradient.
The iteration of this method now follows:
From an initial point x0 for the function F in domain R superscript N, it constructs a sequence of iterates according to xk+1 = xk - F(xk) / F'(xK+1), where F' denotes the derivative of F.
xk+1 corresponds to the intersection of the tangent to the graph of F at xk with the horizontal axis.
xk+1 is the solution to first order Taylor expansion of F around xk.  which is written in a linear form as F(xk)  + F'(xk) [xk+1  -xk] = 0
Thus this method is a linear approximation to a non-linear equation and has a convergence near the roots of the equations.
This method is then applied to finding the critical point of a cost function f on R n simply by taking F(x) as grad f which is the transpose of the partial derivatives of the function f from 1 to n. On a plane this is the Euclidean gradient of f. Instead of finding the difference the xk+1 - xk, we find xk+1 from the tangent vector in the tangent space xk The step control is either line search or trust region
Newton's method is used as the default method in computing FindRoot.

#coding exercise
Write an RBAC open source program
along the lines of  Permission.Demand()
and declarative [PermissionAttribute(SecurityAction.Demand Role="User")]

function ( role, permissions)
{
   switch (role)
   {
          case user:
                  if (has_permission(permission)) // lookup table
                     return Operation.Allowed;
                  return Operation.Disallowed;
        case Admin:
                if (for_trustedbase(permission))
                    return Operation.Disallowed;
                return Operation.Allowed;
       default:
                 throw;
   }
}