Diagonalization of large matrices: a new parallel algorithm

On the basis of a dressed matrices formalism, a new algorithm has been devised for obtaining the lowest eigenvalue and the corresponding eigenvector of large real symmetric matrices. Given an N × N matrix, the proposed algorithm consists in the diagonalization of (N - 1)2 × 2 dressed matrices. Both sequential and parallel versions of the proposed algorithm have been implemented. Tests have been performed on a Hilbert matrix, and the results show that this algorithm is up 340 times faster than the corresponding LAPACK routine for N = 10(4) and about 10% faster than the Davidson method. The parallel MPI version has been tested using up to 512 nodes. The speed-up for a N = 10(6) matrix is fairly lineal until 64 cores. The time necessary to obtain the lowest eigenvalue and eigenvector is nearly 5.5 min with 512 cores. For an N = 10(7) matrix, the speed-up is nearly linear to 256 cores and the calculation time is 5.2 h with 512 nodes. Finally, in order to test the new algorithm on MRCI matrices, we have calculated the ground state and the π → π* excited state of the butadiene molecule, starting from both SCF and CASSCF wave functions. In all the cases considered, correlation energies and wave functions are the same as obtained with the Davidson algorithm.