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In this text I present a couple of new principles and thereon based iterativemethods for numerical solution of sequences of systems of linear equations withfixed system matrix and changing right-hand-sides. The use of the new methodsis to recycle all subspace information that is obtained anyway in the solutionprocess of a former system, to solve subsequent systems.  All these principles and methods are based on short recurrences and smallstorage requirements. The principles are based on the IDR-theorem and theHorner scheme for polynomials.

Short-Recurrence and -Storage Recycling of large Krylov-Subspaces for Sequences of Linear Systems with changing Right-Hand-Sides