  We present a numerical method for computing the solution of a partial
differential equation (PDE) for modeling acoustic pressure, known as an
extra-wide angle parabolic equation, that features the square root of a
differential operator. The differential operator is the negative of an
indefinite Schr\"{o}dinger operator with a piecewise constant potential. This
work primarily deals with the 3-piece case; however, a generalization is made
the case of an arbitrary number of pieces. Through restriction to a judiciously
chosen lower-dimensional subspace, approximate eigenfunctions are used to
obtain estimates for the eigenvalues of the operator. Then, the estimated
eigenvalues are used as initial guesses for the Secant Method to find the exact
eigenvalues, up to roundoff error. An eigenfunction expansion of the solution
is then constructed. The computational expense of obtaining each eigenpair is
independent of the grid size. The accuracy, efficiency, and scalability of this
method is shown through numerical experiments and comparisons with other
methods.
