A Novel Algorithm for Calculation of the Extreme Eigenvalues of Large Hermitian Matrices

A new fast algorithm for calculating a few maximum (or minimum) eigenvalues-&nd the corresponding eigenvectors of large N x N Hermitian m&rices is presented. The method is based on a molecular dynamics algorithm for N coupled harmonic-oscillators. The time step for iteration is chosen so that only the normal mode with _ _. _ the maximum eigenvalue grows exponentially. Other eigenvalues and eigenvectors are obtained one by one from the largest eigenvalue by repea.ting the process in subspaces orthogonal to the previous modes. The characteristics of the algorithm lie in the simplicity, speed (CPU time 0: N2), and memory efficiency (O(N) besides the matrix). The effectiveness of the algorithm is illustrated by calculation of the groundstate and first-excited state energies of the Heisenberg model for an antiferromagnetic chain with N up to 16384.