The density matrix renormalization group (DMRG) method introduced by White for the study of strongly interacting electron systems is reviewed; the method is variational and considers a system of localized electrons as the union of two adjacent fragments A,B. A density matrix rho is introduced, whose eigenvectors corresponding to the largest eigenvalues are the most significant, the most probable states of A in the presence of B; these states are retained, while states corresponding to small eigenvalues of rho are neglected. It is conjectured that the decreasing behavior of the eigenvalues is Gaussian. The DMRG method is tested on the Pariser-Parr-Pople Hamiltonian of a cyclic polyene (CH), up to N = 34. A Hilbert space of dimension 5. X 10(18) is explored. The ground state energy is 10(-3) eV within the full CI value in the case N = 18. The DMRG method compares favorably also with coupled cluster approximations. The unrestricted Hartree-Fock solution (which presents spin density waves) is briefly reviewed, and a comparison is made with the DMRG energy values. Finally, the spin-spin and density-density correlation functions are computed; the results suggest that the antiferromagnetic order of the exact solution does not extend up to large distances but exists locally. No charge density waves are present.