  Many data-analysis problems involve large dense matrices that describe the
covariance of stationary noise processes; the computational cost of inverting
these matrices, or equivalently of solving linear systems that contain them, is
often a practical limit for the analysis. We describe two general, practical,
and accurate methods to approximate stationary covariance matrices as low-rank
matrix products featuring carefully chosen spectral components. These methods
can be used to greatly accelerate data-analysis methods in many contexts, such
as the Bayesian and generalized-least-squares analysis of pulsar-timing
residuals.
