On an iterative method for solving systems of linear equations

In this work, we aim to solve a system of linear equations, $Ax=b$, with a semi-iterative technique. Using a control parameter $\alpha$ the original system is rewritten in an equivalent form $x=Gx+c$, where $G$ is a diagonally dominant matrix. For solving this new system we use a Taylor series based method which uses the Taylor expantion of $x$. In this manner $G$ plays the role of an iterative matrix. We discuss the properties of G and present a sufficient condition for the general method to be convergent. Also a specific class of matrices is introduced for which this method converges