  Quantum many-body scarred systems host nonthermal excited eigenstates
immersed in a sea of thermal ones. In cases where exact expressions for these
special eigenstates are not known, it is computationally demanding to
distinguish them from their exponentially many thermal neighbors. We propose a
matrix-product-state (MPS) algorithm, dubbed DMRG-S, to extract such states at
system sizes far beyond the scope of exact diagonalization. Using this
technique, we obtain scarred eigenstates in Rydberg-blockaded chains of up to
80 sites and perform a finite-size scaling study to address the lingering
question of the stability for the N\'eel state revivals in the thermodynamic
limit. Our method also provides a systematic way to obtain exact MPS
representations for scarred eigenstates near the target energy without a priori
knowledge. In particular, we find several new scarred eigenstates with exact
MPS representations in kinetically constrained spin and clock models. The
combination of numerical and analytical investigations in our work provides a
new methodology for future studies of quantum many-body scars.
