Predicting the electronic structure of complex systems is an outstanding problem in materials science. If the electronic structure of a given material is known, then many physical and chemical properties can be accurately determined without resorting to experiment. At present, the optical and structural properties of simple solid phases can be predicted solely from a knowledge of the atomic species present. Of course, while it is intriguing to speculate about designing complex materials on the computer, many technical and fundamental questions remain to be addressed. The chief obstacle faced by computational materials and computer scientists in computing the electronic structure of matter is the solution of a complex eigenvalue problem. Often, these eigenvalue problems are large and sparse, and thousands of eigenvalues are required. Their solution depends increasingly on complex data structures that reduce memory and time requirements, and on parallel computing. We present a ..