We present a new fast algorithm to compute the real stability radius with respect to the open left half plane which is an important problem in many engineering applications. The method is based on a well-known formula for the real stability radius and the correspondence of singular values of a transfer function to pure imaginary eigenvalues of a three-parameter Hamiltonian matrix eigenvalue problem. We then apply the implicit determinant method, used previously by the authors to compute the complex stability radius, to find the critical point corresponding to the desired singular value. This corresponds to a two-dimensional Jordan block for a pure imaginary eigenvalue in the parameter dependent Hamiltonian matrix. Numerical results showing quadratic convergence of the algorithm are given.