Krylov Methods Toolbox for Solving Linear Systems

This toolbox incorporates various Krylov subspace algorithms for solving linear systems, including classical methods like Conjugate Gradient (CG) and Generalized Minimal Residual (GMRES). These iterative methods are built upon Krylov subspaces and play a crucial role in solving large-scale linear systems. The CG method is particularly efficient for symmetric positive-definite matrices, using orthogonal direction vectors to minimize the quadratic form. GMRES handles non-symmetric systems by constructing an orthogonal basis through Arnoldi iteration and solving a least squares problem at each iteration. Beyond these fundamental approaches, the toolbox includes enhanced algorithms like preconditioned CG (which accelerates convergence through matrix transformations) and BiCGStab (a stabilized version of biconjugate gradient suitable for non-symmetric matrices). Advanced implementations also feature Lanczos algorithm based on Arnoldi iteration for Hermitian matrices, and minimal residual methods that optimize solution accuracy. These algorithms can be flexibly selected based on matrix properties and problem requirements, with implementations typically involving key functions for matrix-vector products, preconditioning setup, and orthogonalization procedures to achieve optimal solving performance.