  Recently, we used the Sinc collocation method with the double exponential
transformation to compute eigenvalues for singular Sturm-Liouville problems. In
this work, we show that the computation complexity of the eigenvalues of such a
differential eigenvalue problem can be considerably reduced when its operator
commutes with the parity operator. In this case, the matrices resulting from
the Sinc collocation method are centrosymmetric. Utilizing well known
properties of centrosymmetric matrices, we transform the problem of solving one
large eigensystem into solving two smaller eigensystems. We show that only
1/(N+1) of all components need to be computed and stored in order to obtain all
eigenvalues, where (2N+1) corresponds to the dimension of the eigensystem. We
applied our result to the Schr\"odinger equation with the anharmonic potential
and the numerical results section clearly illustrates the substantial gain in
efficiency and accuracy when using the proposed algorithm.
