Efficient variants of the CMRH method for solving a sequence of   multi-shifted non-Hermitian linear systems simultaneously
  Multi-shifted linear systems with non-Hermitian coefficient matrices arise in numerical solutions of time-dependent partial/fractional differential equations (PDEs/FDEs), in control theory, PageRank problems, and other research fields. We derive efficient variants of the restarted Changing Minimal Residual method based on the cost-effective Hessenberg procedure (CMRH) for this problem class. Then, we introduce a flexible variant of the algorithm that allows to use variable preconditioning at each iteration to further accelerate the convergence of shifted CMRH. We analyse the performance of the new class of methods in the numerical solution of PDEs and FDEs, also against other multi-shifted Krylov subspace methods. 
Xian-Ming Gu, Ting-Zhu Huang, Bruno Carpentieri, Akira Imakura, Ke   Zhang, Lei Du
math.NA cs.NA