QMRCGstab Algorithm for Families of Shifted Linear Systems

This study is mainly focused on iterative solutions to shifted linear systems arising from a Quantum Chromodynamics (QCD) problem. For solving such systems efficiently, we explore a new shifted QMRCGstab (SQMRCGstab) method, which is derived by extending the quasi-minimum residual to the shifted BiCGstab. The shifted QMRCGstab method takes advantage of the shifted invariant property, so that it could handle multiple shifts simultaneously using only as many matrix-vector multiplications as the solution of a single system required. Moreover, the SQMRCGstab achieves a smoothing of the residual compared to the shifted BiCGstab, and the SQMRCGstab is more competitive than the MS-QMRIDR(s) and the shifted BiCGstab on the QCD problem. Numerical examples show the efficiency of the method when one applies it to the real problems.