A fourth order method for finding a simple root of univariate function

In this paper, we describe an iterative method for approximating a simple zero $z$ of a real defined function. This method is a essentially based on the idea to extend Newton's method to be the inverse quadratic interpolation. We prove that for a sufficiently smooth function $f$ in a neighborhood of $z$ the order of the convergence is quartic. Using Mathematica with its high precision compatibility, we present some numerical examples to confirm the theoretical results and to compare our method with the others given in the literature

Pricing American bond options using a cubic spline collocation method

In this paper, American options on a discount bond are priced under the Cox-Ingrosll-Ross (CIR) model. The linear complementarity problem of the option value is solved numerically by a penalty method. The problem is transformed into a nonlinear partial differential equation (PDE) by adding a power penalty term. The solution of the penalized problem converges to the one of the original problem. To numerically solve this nonlinear PDE, we use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of trapezoidal method and a cubic spline collocation method, respectively. We show that this full discretization scheme is second order convergent, and hence the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed. Numerical results are presented and compared with other collocation methods given in the literature