Before revision: We present a Conjugate Gradient (CG) implementation of the probabilistic numerical solver BayesCG , whose error estimates are a fully integrated design feature, easy to compute, and competitive with the best existing estimators. More specifically, we extend BayesCG to singular prior covariances, derive recursions for the posterior covariances, express the posteriors as projections, and establish that BayesCG retains the minimization properties over Krylov spaces regardless of the singular priors. We introduce a possibly singular Krylov prior covariance, under which the BayesCG posterior means coincide with the CG iterates and the posteriors can be computed efficiently.
After revision: We present a Conjugate Gradient (CG) for solving linear systems with real symmetric positive definite coefficient matrices. We present a CG-based implementation of BayesCG with a structure-exploiting prior distribution. The BayesCG output consists of CG iterates and the posteriors can be computed efficiently.