and therefore it is a purely numerical RG approach. Although this needs not to be a disadvantage we like to have an analytical formulation of the DMRG method. In such a reformulation the numerical DMRG scheme will occur as one possible realization of a more general description. We therefore expect a deeper insight to successful working RG approaches.

3.  DMRG is restricted to zero temperature and is usually applied for calculating ground state properties like the ground state magnetization or even the ground state itself. Finite temperature results were obtained only in the low lying spectrum but with very limited accuracy. In comparison to other real-space RG methods DMRG is different because it is designed to calculate ground state quantities. Recently, based on the idea of Xiang et al, a thermodynamic method was applied successfully, which combines White’s DMRG idea with the [*transfer-matrix*]{} technique[@krs99] and which is now called TMRG. Although TMRG is also purely numerical since it shares the basic idea with DMRG it is an even more complicated algorithm [@krs99]. Due to the close relationship to DMRG, the aim of TMRG is to give numerical accurate results for physical quantities and does not predict a RG flow-behaviour. In 