Zhi-an Shao, Wolfgang Porod, Craig S. Lent, and David J. Kirkner We present a numerical technique for open-boundary quantum transmission problems which yields, as the direct solutions of appropriate eigenvalue problems, the energies of (i) quasi-bound states and transmission poles, (ii) transmission ones, and (iii) transmission zeros. The eigenvalue problem results from reducing the inhomogeneous transmission problem to a homogenous problem, by forcing the in-coming source term to zero. Using the finite element method, this homogeneous problemt can be transformed to a standard linear eigenvalue problem. By treating either the transmission amplitude t(E) or the reflection amplitude r(E) as the known source term, this numerical method also can be used to calculate the positions of transmission zeros and transmission ones. We demonstrate the utility of this technique utilizing several example structures, such as single- and double-barrier resonant tunneling and quantum waveguide systems, including t-stubs and loops.