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This paper develops a class of algorithms for general linear systems and eigenvalue problems. These algorithms apply fast randomized dimension reduction ("sketching") to accelerate standard subspace projection methods, such as GMRES and Rayleigh-Ritz. This modification makes it possible to incorporate nontraditional bases for the approximation subspace that are easier to construct. When the basis is numerically full rank, the new algorithms have accuracy similar to classic methods but run faster and may use less storage. For model problems, numerical experiments show large advantages over the optimized MATLAB routines, including a 70x speedup over gmres and a 10x speedup over eigs. [ABSTRACT FROM AUTHOR]

The implicit trace estimation problem asks for an approximation of the trace of a square matrix, accessed via matrix-vector products (matvecs). This paper designs new randomized algorithms, XTrace and XNysTrace, for the trace estimation problem by exploiting both variance reduction and the exchangeability principle. For a fixed budget of matvecs, numerical experiments show that the new methods can achieve errors that are orders of magnitude smaller than existing algorithms, such as the Girard--Hutchinson estimator or the Hutch++ estimator. A theoretical analysis confirms the benefits by offering a precise description of the performance of these algorithms as a function of the spectrum of the input matrix. The paper also develops an exchangeable estimator, XDiag, for approximating the diagonal of a square matrix using matvecs. [ABSTRACT FROM AUTHOR]

This paper introduces the Nyström preconditioned conjugate gradient (PCG) algorithm for solving a symmetric positive-definite linear system. The algorithm applies the randomized Nyström method to form a low-rank approximation of the matrix, which leads to an efficient preconditioner that can be deployed with the conjugate gradient algorithm. Theoretical analysis shows that the preconditioned system has constant condition number as soon as the rank of the approximation is comparable with the number of effective degrees of freedom in the matrix. The paper also develops adaptive methods that provably achieve similar performance without knowledge of the effective dimension. Numerical tests show that Nyström PCG can rapidly solve large linear systems that arise in data analysis problems, and it surpasses several competing methods from the literature. [ABSTRACT FROM AUTHOR]

This paper discusses a new class of matrix nearness problems that measure approximation error using a directed distance measure called a Bregman divergence. Bregman divergences offer an important generalization of the squared Frobenius norm and relative entropy, and they all share fundamental geometric properties. In addition, these divergences are intimately connected with exponential families of probability distributions. Therefore, it is natural to study matrix approximation problems with respect to Bregman divergences. This article proposes a framework for studying these problems, discusses some specific matrix nearness problems, and provides algorithms for solving them numerically. These algorithms apply to many classical and novel problems, and they admit a striking geometric interpretation. [ABSTRACT FROM AUTHOR]

In this paper, we present new algorithms that can replace the diagonal entries of a Hermitian matrix by any set of diagonal entries that majorize the original set without altering the eigenvalues of the matrix. They perform this feat by applying a sequence of (N-1) or fewer plane rotations, where N is the dimension of the matrix. Both the Bendel--Mickey and the Chan--Li algorithms are special cases of the proposed procedures. Using the fact that a positive semidefinite matrix can always be factored as $\mtx{X^\adj X}$, we also provide more efficient versions of the algorithms that can directly construct factors with specified singular values and column norms. We conclude with some open problems related to the construction of Hermitian matrices with joint diagonal and spectral properties. [ABSTRACT FROM AUTHOR]