Block Krylov subspace methods for shifted systems with different right-hand sides

Many Krylov subspace methods for shifted linear systems take advantage of the invariance of the Krylov subspace under a shift of the matrix. However, exploiting this fact introduces restrictions; e.g., initial residuals must be collinear and this collinearity must be maintained at restart. Thus we cannot simultaneously solve (in general) shifted systems with unrelated right-hand sides using this strategy, and all shifted residuals cannot be simultaneously minimized over a Krylov subspace such that collinearity is maintained. We present two methods which circumvent this problem. Block Krylov subspaces are shift invariant just as their single-vector counterparts. Thus by collecting all initial residuals into one block vector, we can generate a block Krylov subspace. Due to shift invariance, we can define block FOM- and GMRES-type projection methods to simultaneously solve all shifted systems. These are not block versions of the shifted FOM method of Simoncini [BIT '03] or the shifted GMRES method of Frommer and Gl\assner [SISC '98]. These methods are compatible with unrelated right-hand sides, and residual collinearity is no longer a requirement at restart. Furthermore, we realize the benefits of block sparse matrix operations which arise in the context of high-performance computing applications. In this paper, we show that the block Krylov subspace built from an appropriate block starting vector is compatible with solving individual shifted systems and use this to derive our block FOM and GMRES methods for shifted systems. Numerical experiments demonstrate the effectiveness of the methods.