A unified approach to Krylov subspace methods for solving linear systems

In this paper, we present a comprehensive framework for studying Krylov subspace methods used to solve the linear system \(Ax=f\). These methods aim to achieve convergence within a specified number of iterations, denoted by m, given a particular initial estimate vector \(x_0\) and its corresponding residual \(r_0=f-Ax_0\). Our analysis focuses on the minimal polynomial \({\varPhi }_m\) of degree m of A for the vector \(r_0\). We establish that these methods encompass Petrov-Galerkin methods and minimal seminorm methods as special cases. Additionally, we demonstrate that minimal seminorm methods satisfy implicit Petrov-Galerkin conditions. We provide a general formulation for the iterates based on generalized inverses. The choice of a specific left inverse and the method of constructing the Krylov basis are crucial distinguishing factors among different Krylov subspace methods. We describe and analyze the mathematical properties of these methods, emphasizing their dependency on two matrices. Notably, we prove that CMRH and QMR, as specific instances, also satisfy implicit Petrov-Galerkin orthogonality conditions. Furthermore, we explore techniques to improve the convergence behavior of these methods by carefully selecting vectors in their implementations. Through our investigation, we aim to deepen the understanding of Krylov subspace methods, provide insights into their convergence properties, and identify potential enhancements. We also consider some Krylov subspace methods, which are of product-type methods. In this case, the kth residual \(r_k\) associated with the approximation \(x_k\) of the exact solution is given by \(r_k={\varPsi }_k(A){\varPhi }_k(A)r_0\), and \({\varPsi }_k\) is a polynomial of fixed or variable degree. We will examine particular choices of \({\varPsi }_k\) involving local convergence, smoothing, fixed memory, and cost for each iteration. We will also give an enhancement of some product-type methods such as CGS. To illustrate the performance of the derived algorithms, we provide some numerical examples.