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