Related papers: Scalable computation of Jordan chains

Given some binary matrix $M$, suppose we are presented with the collection of its rows and columns in independent arbitrary orderings. From this information, are we able to recover the unique original orderings and matrix? We present an…

We take matrix decompositions that are usually applied to matrices over the real numbers or complex numbers, and extend them to matrices over an algebra called the double numbers. In doing so, we unify some matrix decompositions: For…

A factorization of the inverse of a Hermetian positive definite matrix based on a diagonal by diagonal recurrence formulae permits the inversion of Toeplitz Block Toeplitz matrices using minimized matrix-vector products, with a complexity…

An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as…

In this paper we show how to recover a spectral approximations to broad classes of structured matrices using only a polylogarithmic number of adaptive linear measurements to either the matrix or its inverse. Leveraging this result we obtain…

In this paper, we propose iterative inner/outer approximations based on a recent notion of block factor-width-two matrices for solving semidefinite programs (SDPs). Our inner/outer approximating algorithms generate a sequence of upper/lower…

We develop a hierarchical matrix construction algorithm using matrix-vector multiplications, based on the randomized singular value decomposition of low-rank matrices. The algorithm uses $\mathcal{O}(\log n)$ applications of the matrix on…

We exhibit a randomized algorithm which given a matrix $A\in \mathbb{C}^{n\times n}$ with $\|A\|\le 1$ and $\delta>0$, computes with high probability an invertible $V$ and diagonal $D$ such that $\|A-VDV^{-1}\|\le \delta$ using…

The calculation of exclusive observables beyond the one-loop level requires elaborate techniques for the computation of multi-leg two-loop integrals. We discuss how the large number of different integrals appearing in actual two-loop…

Variational inequalities represent a broad class of problems, including minimization and min-max problems, commonly found in machine learning. Existing second-order and high-order methods for variational inequalities require precise…

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…

We consider variants of a recently-developed Newton-CG algorithm for nonconvex problems \citep{royer2018newton} in which inexact estimates of the gradient and the Hessian information are used for various steps. Under certain conditions on…

We consider the problem of quantifying uncertainty for the estimation error of the leading eigenvector from Oja's algorithm for streaming principal component analysis, where the data are generated IID from some unknown distribution. By…

We discuss the eigenvalue problem for 3x3 octonionic Hermitian matrices which is relevant to the Jordan formulation of quantum mechanics. In contrast to the eigenvalue problems considered in our previous work, all eigenvalues are real and…

In this paper a vectorized algorithm for simultaneously computing up to eight singular value decompositions (SVDs, each of the form $A=U\Sigma V^{\ast}$) of real or complex matrices of order two is proposed. The algorithm extends to a batch…

Let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field $F$ of characteristic $p$. For positive integers $r$ and $s$ with $r \leq s$, the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes…

We propose a more accurate variant of an algorithm for multiplying 4x4 matrices using 48 multiplications over any ring containing an inverse of 2. This algorithm has an error bound exponent of only log 4 $\gamma$$\infty$,2 $\approx$ 2.386.…

We introduce some basic notions and results for quaternionic linear operators analogous to those for complex linear operators. Our main result is to prove the additive and multiplicative Jordan-Chevalley decompositions for quaternionic…

Recently, a class of algorithms combining classical fixed point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as $10^{108} \times…

A novel algorithm for computing the action of a matrix exponential over a vector is proposed. The algorithm is based on a multilevel Monte Carlo method, and the vector solution is computed probabilistically generating suitable random paths…