Theory of Krylov Subspace Methods

This chapter discusses the theoretical foundations of the most important available methods known for solving large and spars linear systems. These methods are based on orthogonal or oblique projection processes. The chapter focuses solely on the mathematical foundations, theory, and derivation of Krylov subspace methods. It describes the general principles of the projection method as the cornerstone of the Krylov subspace methods and defines the Krylov subspace. The chapter studies the principles, derivation approaches and convergence theory of the conjugate gradient method. It also describes the minimal residual method, generalized minimal residual method, conjugate residual method, and bi-conjugate residual method. The chapter also explains the principles of extracting conjugate gradient squared and bi-conjugate gradient stabilized methods, as a subset of the transpose-free methods.