arXiv subjects
Publications and source records attributed to Michele Rinelli.
Periodic CMV matrices are unitary matrices that can be specified by $O(n)$ data. Their eigenvalues can be computed by standard methods, storing them as conventional matrices (using $O(n^{2})$ data) in $O(n^{3})$ time. Here a fast method that computes the eigenvalues in $O(n^{2})$ time (using $O(n)$ data) is presented.
We study the connection between block Krylov subspaces and matrix orthogonal functions. Under a no-deflation assumption, we show that polynomial block Krylov subspaces are isometrically isomorphic to spaces of matrix polynomials of bounded degree, providing a unified framework for the analysis and construction of orthonormal bases and recurrence relations. The same correspondence holds for rational block Krylov subspaces and matrix-valued rational functions, and in the extended Krylov setting this leads naturally to Laurent matrix polynomials. When the matrix $A$ is normal, we prove that the induced inner product admits a representation in terms of a discrete spectral matrix measure, extending a classical result for Hermitian matrices. In the unitary case, where the measure is supported on the unit circle, this connection allows us to transfer the Szegő recurrence for orthogonal matrix polynomials and the CMV framework for Laurent matrix polynomials to the block Krylov setting, yielding efficient procedures for the orthogonalization of polynomial and extended block Krylov subspaces.
In this paper, we develop algorithms for computing the recurrence coefficients corresponding to multiple orthogonal polynomials on the step-line. We reformulate the problem as an inverse eigenvalue problem, which can be solved using numerical linear algebra techniques. We consider two approaches: the first is based on the link with block Krylov subspaces and results in a biorthogonal Lanczos process with multiple starting vectors; the second consists of applying a sequence of Gaussian eliminations on a diagonal matrix to construct the banded Hessenberg matrix containing the recurrence coefficients. We analyze the accuracy and stability of the algorithms with numerical experiments on the ill-conditioned inverse eigenvalue problemshave related to Kravchuk and Hahn polynomials, as well as on other better conditioned examples.
In this paper we propose and analyze an algorithm for identifying spectral gaps of a real symmetric matrix $A$ by simultaneously approximating the traces of spectral projectors associated with multiple different spectral slices. Our method utilizes Hutchinson's stochastic trace estimator together with the Lanczos algorithm to approximate quadratic forms involving spectral projectors. Instead of focusing on determining the gap between two particular consecutive eigenvalues of $A$, we aim to find all gaps that are wider than a specified threshold. By examining the problem from this perspective, and thoroughly analyzing both the Hutchinson and the Lanczos components of the algorithm, we obtain error bounds that allow us to determine the numbers of Hutchinson's sample vectors and Lanczos iterations needed to ensure the detection of all gaps above the target width with high probability. In particular, we conclude that the most efficient strategy is to always use a single random sample vector for Hutchinson's estimator and concentrate all computational effort in the Lanczos algorithm. Our numerical experiments demonstrate the efficiency and reliability of this approach.
We consider the problem of estimating the trace of a matrix function $f(A)$. In certain situations, in particular if $f(A)$ cannot be well approximated by a low-rank matrix, combining probing methods based on graph colorings with stochastic trace estimation techniques can yield accurate approximations at moderate cost. So far, such methods have not been thoroughly analyzed, though, but were rather used as efficient heuristics by practitioners. In this manuscript, we perform a detailed analysis of stochastic probing methods and, in particular, expose conditions under which the expected approximation error in the stochastic probing method scales more favorably with the dimension of the matrix than the error in non-stochastic probing. Extending results from [E. Aune, D. P. Simpson, J. Eidsvik, Parameter estimation in high dimensional Gaussian distributions, Stat. Comput., 24, pp. 247--263, 2014], we also characterize situations in which using just one stochastic vector is always -- not only in expectation -- better than the deterministic probing method. Several numerical experiments illustrate our theory and compare with existing methods.
We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix $A$, defined as $\operatorname{tr}(f(A))$ where $f(x)=-x\log x$. After establishing some useful properties of this matrix function, we consider the use of both polynomial and rational Krylov subspace algorithms within two types of approximations methods, namely, randomized trace estimators and probing techniques based on graph colorings. We develop error bounds and heuristics which are employed in the implementation of the algorithms. Numerical experiments on density matrices of different types of networks illustrate the performance of the methods.
Spectral projectors of Hermitian matrices play a key role in many applications, and especially in electronic structure computations. Linear scaling methods for gapped systems are based on the fact that these special matrix functions are localized, which means that the entries decay exponentially away from the main diagonal or with respect to more general sparsity patterns. The relation with the sign function together with an integral representation is used to obtain new decay bounds, which turn out to be optimal in an asymptotic sense. The influence of isolated eigenvalues in the spectrum on the decay properties is also investigated and a superexponential behaviour is predicted.