Abstract. The problem of finding a rank-revealing QR (RRQR) factorisation of a matrix A consists of permuting the columns of A such that the resulting QR factorisation contains an upper triangular matrix whose linearly dependent columns are separated from the linearly independent ones. In this paper a systematic treatment of algorithms for determining RRQR factorisations is presented.In particular, the authors start by presenting precise mathematical formulations for the problem of determining a RRQR factorisation, all of them optimisation problems. Then a hierarchy of "greedy" algorithms is derived to solve these optimisation problems, and it is shown that the existing RRQR algorithms correspond to particular greedy algorithms in this hierarchy. Matrices on which the greedy algorithms, and therefore the existing RRQR algorithms, can fail arbitrarily badly are presented.Finally, motivated by an insight from the behaviour of the greedy algorithms, the authors present "hybrid" algorithms that solve the optimisation problems almost exactly (up to a factor proportional to the size of the matrix). Applying the hybrid algorithms as a follow-up to the conventional greedy algorithms may prove to be useful in practice.

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We present randomized algorithms for estimating the trace and determinant of Hermitian positive semi-definite matrices. The algorithms are based on subspace iteration, and access the matrix only through matrix vector products. We analyse the error due to randomization, for starting guesses whose elements are Gaussian or Rademacher random variables. The analysis is cleanly separated into a structural (deterministic) part followed by a probabilistic part. Our absolute bounds for the expectation and concentration of the estimators are non-asymptotic and informative even for matrices of low dimension. For the trace estimators, we also present asymptotic bounds on the number of samples (columns of the starting guess) required to achieve a user-specified relative error. Numerical experiments illustrate the performance of the estimators and the tightness of the bounds on low-dimensional matrices; and on a challenging application in uncertainty quantification arising from Bayesian optimal experimental design.

We present a qualitative model for the convergence behaviour of the Generalised Minimal Residual (GMRES) method for solving nonsingular systems of linear equations Ax = b in nite and in nite dimensional spaces. One application of our methods is the solution of discretised in nite dimensional problems, such as integral equations, where the constants in the asymptotic bounds are independent of the mesh size. Our model provides simple, general bounds that explain the convergence of GMRES as follows: If the eigenvalues of A consist of a single cluster plus outliers then the convergence factor is bounded by the cluster radius, while the asymptotic error constant re ects the non-normality of A and the distance of the outliers from the cluster. If the eigenvalues of A consist of several close clusters, then GMRES treats the clusters as a single big cluster, and the convergence factor is the radius of this big cluster. We exhibit matrices for which these bounds are tight. Our bounds also lead to a simpler proof of existing r-superlinear convergence results in Hilbert space.

A common challenge faced in quantum physics is finding the extremal eigenvalues and eigenvectors of a Hamiltonian matrix in a vector space so large that linear algebra operations on general vectors are not possible. There are numerous efficient methods developed for this task, but they generally fail when some control parameter in the Hamiltonian matrix exceeds some threshold value. In this Letter we present a new technique called eigenvector continuation that can extend the reach of these methods. The key insight is that while an eigenvector resides in a linear space with enormous dimensions, the eigenvector trajectory generated by smooth changes of the Hamiltonian matrix is well approximated by a very low-dimensional manifold. We prove this statement using analytic function theory and propose an algorithm to solve for the extremal eigenvectors. We benchmark the method using several examples from quantum many-body theory.