Before revision:  With the aid of such high-order compact finite difference schemes, DAT can solve heterogeneous Helmholtz equations with arbitrarily large variable wave numbers accurately by solving small linear systems - 4 by 4 matrices in the extreme case - with tridiagonal coefficient matrices in a parallel fashion.
After revision:  With the aid of such high-order compact finite difference schemes, DAT can solve heterogeneous Helmholtz equations with arbitrarily large variable wave numbers accurately by solving small linear systems - 4 \times 4 matrices in the extreme case - with tridiagonal coefficient matrices in a parallel fashion.