Rectified approximations for the solution of nonlinear equations (Q1063387)

The authors introduce a general method for solving nonlinear equations \(f(x)=0\), with x a real variable, and call it ''rectification method''. In fact, this is not an algorithm but a method for the construction of algorithms of pre-designed order. The idea is to construct an associated function, g(x), which at the point of a simple root \(x=\rho\) of f(x) also has a simple root and, in addition, has a contact of order r-1 with the tangent line of f(x) at \(x=\rho\), i.e. \(g(\rho)=f(\rho)=0,\quad g'(\rho)=f'(\rho),\quad g^{(k)}(\rho)=0\) for \(k=2,3,...,(r-1);\) hence, the term ''rectified function''. Starting from the Padé approximation to f(x) the authors arrive at a general formula of the form \(g=[(C/f)^{(r- 2)}]^{-1/(r-1)}\), where C is an arbitrary constant. The root-finding is then performed by applying the Newton-Raphson method for the function g(x) instead of f(x). As shown, the convergence of this iteration is of order r. Explicit forms of the function g have been given for the values \(r=3,4,5\), while \(g(x)=f(x)\) for \(r=2\). The individual iterative formulas obtained in terms of the rectified function g(x) are not novel. In fact, for \(r\geq 3\) they are the known formulas of Halley, Kiss, etc. The main contribution of the rectification method is that it links higher order, \(r\geq 3\), root-finding algorithms to the rectified function g(x), i.e. to a common theory, and, thus, produces insight into numerical root approximation. In addition, when f(x) is replaced by g(x), the routine sufficient condition for the convergence of Newton-Raphson method can be replaced by a less severe one in many cases. There is given a numerical example to illustrate this phenomenon. Finally, the authors state that the case of multiple roots of f(x) can be treated replacing f by f/f'. The authors promise that the extension of the rectification method to systems of nonlinear equations will be published in a future paper.