A state-specific scheme for time-dependent density functional theory (SS-TDDFT) based on the Davidson algorithm is presented. SSTDDFT is a method devised for speeding up TDDFT calculations by screening transitions that contribute to a specific excitation. By applying this method to calculations of the low-lying excitation energies of test molecules (N-2, CO, H2CO, C2H4 and C6H6), water clusters and polyenes, we found that SS-TDDFT accurately reproduced the excitation energies of standard TDDFT while drastically reducing the rank of the TDDFT response matrix without loss of accuracy. We have thus formulated TDDFT that works more efficiently and economically for memory storage. (c) 2006 Elsevier B.V. All rights reserved.