ON THE ORDER AND RATE OF CONVERGENCE FOR PSEUDO-SECANT-NEWTON’S METHOD LOCATING A

By combining the classical Newton’s method with the pseudo-secant method, pseudo-secant-Newton’s method is constructed and its order and rate of convergence are investigated. Given a function f : R → R that has a simple real zero α and is sufficiently smooth in a small neighborhood of α, the convergence behavior is analyzed near α for pseudo-secant-Newton’s method. The order of convergence is shown to be cubic and the rate of convergence is proven to be . Numerical experiments show the validity of the theory presented here and are confirmed via high-precision programming in Mathematica.