A new way of solving the non-linear secular equation within the augmented plane wave (APW) method is presented. The traditional computation of the APW secular determinant is replaced by a scheme where we follow the eigenvalues of the secular matrix for a number of energies. These eigenvalues, referred to as the meta-eigenvalues of the original APW problem, provide more information about the system for each energy examined than just their product, i.e. the value of the secular determinant. The new scheme for finding the APW eigenvalues can be made considerably more efficient than the traditional one, and is useful in calculations where a linearized approximation to the APW secular matrix is insufficient, such as for optical calculations extending over a wide energy range or for energy dependent effective potentials.