AUGMENTED MGS-CGS BLOCK-ARNOLDI RECYCLING SOLVERS

The recycling of Krylov subspaces in iterative methods is a powerful strategy for reducing the computational burden of solving large-scale linear systems. However, traditional approaches often suffer from a loss of orthogonality among the basis vectors of the Krylov and recycled subspaces, leading to slower convergence rates. This paper introduces a novel hybrid Gauss-Seidel and Jacobi Gram-Schmidt algorithm that enhances both intra- and interblock orthogonalization, coupled with a block lower triangular correction matrix. Our approach ensures that Krylov vectors maintain orthogonality to working accuracy and demonstrates a significant reduction in the number of iterations required per linear system, outperforming established methods like HH-GMRES and the classical GCRO-DR. Additionally, we explore the utility of the recycle subspace eigenspectrum as an alternative to traditional preconditioners, with shifts that further decrease iteration counts. By employing the normwise relative backward error instead of the Arnoldi relative residual for convergence tests, our method avoids stagnation and enhances reliability. Applicable across a broad spectrum of recycling techniques, our algorithm offers substantial improvements in the efficiency and accuracy of solvers for large-scale linear systems.