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The density-matrix renormalization group method (DMRG) has established itselfover the last decade as the leading method for the simulation of the staticsand dynamics of one-dimensional strongly correlated quantum lattice systems. Inthe further development of the method, the realization that DMRG operates on ahighly interesting class of quantum states, so-called matrix product states(MPS), has allowed a much deeper understanding of the inner structure of theDMRG method, its further potential and its limitations. In this paper, I wantto give a detailed exposition of current DMRG thinking in the MPS language inorder to make the advisable implementation of the family of DMRG algorithms inexclusively MPS terms transparent. I then move on to discuss some directions ofpotentially fruitful further algorithmic development: while DMRG is a verymature method by now, I still see potential for further improvements, asexemplified by a number of recently introduced algorithms.

The density-matrix renormalization group in the age of matrix product states