  We report an attempt to calculate energy eigenvalues of large quantum systems
by the diagonalization of an effectively truncated Hamiltonian matrix. For this
purpose we employ a specific way to systematically make a set of orthogonal
states from a trial wavefunction and the Hamiltonian. In comparison with the
Lanczos method, which is quite powerful if the size of the system is within the
memory capacity of computers, our method requires much less memory resources at
the cost of the extreme accuracy.
  In this paper we demonstrate that our method works well in the systems of
one-dimensional frustrated spins up to 48 sites, of bosons on a chain up to 32
sites and of fermions on a ladder up to 28 sites. We will see this method
enables us to study eigenvalues of these quantum systems within reasonable
accuracy.
