Krylov space solvers for shifted linear systems
B. Jegerlehner
TL;DR
The paper addresses solving multiple shifted linear systems $(A+\sigma)x=b$ efficiently by reusing a single Krylov sequence, enabling simultaneous shifts with the cost of a single matrix-vector product. It introduces shifted polynomials $P_n^\sigma$ tied to the base Krylov polynomials $P_n$, enabling shifted solvers with short recurrences (CG, CR, BiCG, BiCGstab) and deriving parameter updates. The authors derive and implement shifted variants (CG-M, CR-M, BiCG-M, BiCG$\gamma_5$-M, BiCGstab-M) and analyze preconditioning, providing numerical tests in lattice QCD with Wilson and Clover fermions, plus an optimal shifted algorithm for staggered fermions. Results show practical speedups, especially when shifts (masses) are close, but roundoff and convergence checks remain important; the work offers a coherent framework for efficient shifted Krylov solvers in computational physics.
Abstract
We investigate the application of Krylov space methods to the solution of shifted linear systems of the form (A+σ) x - b = 0 for several values of σsimultaneously, using only as many matrix-vector operations as the solution of a single system requires. We find a suitable description of the problem, allowing us to understand known algorithms in a common framework and developing shifted methods basing on short recurrence methods, most notably the CG and the BiCGstab solvers. The convergence properties of these shifted solvers are well understood and the derivation of other shifted solvers is easily possible. The application of these methods to quark propagator calculations in quenched QCD using Wilson and Clover fermions is discussed and numerical examples in this framework are presented. With the shifted CG method an optimal algorithm for staggered fermions is available.