Extensions to generalized quasilinearization versus Newton's method for convex-concave functions

Abstract

In this paper we use the Method of Generalized Quasilinearization to obtain Newton-like comparative schemes to solve the equation $f(x)=0$, which has an isolated zero, $x=r$ in $[a_0,b_0]\subset \Omega$, where $f(x) \in C[\Omega,\mathbb{R}]$. Two sets of results are presented. In the first cases $f(x)$ is neither concave nor convex, but by the addition of the convex function $\phi(x)$, convexity properties are then used on $F(x)=f(x)+\phi(x)=0$ to show that an iterative scheme based on Generalized Quasilinearization generates two monotone sequences $\{a_n\}$ and $\{b_n\}$ that converge quadratically to $r$, the isolated zero of $f(x)=0$. The first set of results are then extended to the case where $f(x)$ admits the decomposition $f(x)=F(x)+G(x)$, where $F(x)$ and $G(x)$ are not naturally convex and concave, but are forced by adding the functions $\Phi(x)$ and $\Psi(x)$ with $\Phi_{xx}(x)>0$ and $\Psi_{xx}(x)\leq 0$ in $\Omega$. The existence of monotone sequences that converge quadratically to the isolated root of $f(x)=0$ in $[a_0,b_0]\subset\Omega$ is shown via iterative schemes relevant to Generalized Quasilinearization.