Given a self-adjoint operator (A) on a Hilbert space, suppose that one wishes to compute the spectrum of (A) numerically. In practice, these problems often arise in such a way that the matrix of (A) relative to a natural basis is "sparse." For example, discretized second-order differential operators can be represented by doubly infinite tridiagonal matrices. In these cases it is easy and natural to compute the eigenvalues of large (n \times n) submatrices of the infinite operator matrix, and to hope that if (n) is large enough then the resulting distribution of eigenvalues will give a good approximation to the spectrum of (A). Numerical analysts call this the Galerkin method. While this hope is often realized in practice it often fails as well, and it can fail in spectacular ways. The sequence of eigenvalue distributions may not converge as (n \rightarrow \infty), or they may converge to something that has little to do with the original operator (A). At another level, even the meaning of "convergence" has not been made precise in general. In this paper we determine the proper general setting in which one can expect convergence, and we describe the asymptotic behavior of the (n \times n) eigenvalue distrubutions in all but the most pathological cases. Under appropriate hypotheses we establish a precise limit theorem which shows how the spectrum of (A) is recovered from the sequence of eigenvalues of the (n \times n) compressions. In broader terms, our results have led us to the conclusion that numerical problems involving infinite dimensional operators require a reformulation in terms of (C^{})-algebras. Indeed, it is only when the single operator (A) is viewed as an element of an appropriate (C^{})-algebra (\mathscr{A}) that one can see the precise nature of the limit of the (n \times n) eigenvalue distributions; the limit is associated with a tracial state on (\mathscr{A}). Normally, (\mathscr{A}) is highly noncommutative, and in our main applications it is a simple (C^{*})-algebra having a unique tracial state. We obtain precise asymptotic results for operators which represent discretized Hamiltonians of one-dimensional quantum systems with arbitrary continuous potentials. C 1994 Academic Press, Inc.