Implementing Several Renowned Iterative Algorithms for Large-Scale Linear Systems of Equations

This article provides an in-depth exploration of several renowned iterative algorithms for solving large-scale linear systems of equations, enabling readers to thoroughly understand both the theoretical foundations and practical implementations. The covered algorithms include Conjugate Gradient (CG), Preconditioned Conjugate Gradient (PCG), Lanczos Algorithm, and Minimal Residual Method (MINRES). Each algorithm is presented with detailed mathematical formulations and code implementation insights, highlighting key computational steps such as Krylov subspace construction, orthogonalization processes, and convergence criteria. Through studying these algorithms, readers will gain deeper insights into fundamental concepts of linear algebra and computational mathematics, while learning how to apply these techniques to real-world problems. The discussion includes comparative analysis of each algorithm's advantages and limitations, along with guidance on selecting appropriate methods for different application scenarios. Implementation considerations cover aspects like preconditioner selection for PCG, handling indefinite matrices with MINRES, and efficient eigenvalue computation using Lanczos iteration. Finally, practical numerical examples demonstrate concrete applications and performance characteristics of these algorithms, including code snippets showing initialization parameters, iteration loops, and convergence monitoring techniques.