arXiv subjects

John Urschel

Publications and source records attributed to John Urschel.

At least 19 recordsLinked to original sources

arXiv2026

Entry growth in Gaussian elimination

Gaussian elimination is one of the oldest algorithms in mathematics, and the most popular method for solving an unstructured linear system. Its stability in finite precision is controlled by its growth factor, which measures how large the entries produced during elimination can become. Understanding the worst-case behavior of this quantity has been a central problem in numerical analysis since the 1940s. Here we make a significant leap in that understanding, settling several open problems. In particular, we determine the asymptotic behavior of the maximum growth factor under complete and rook pivoting, proving that both are quasi-polynomial in dimension. We also show that the exponential growth under partial pivoting persists for sparse matrices and that randomized partial pivoting suffers the same instability. In contrast, we show that every non-singular matrix has a row permutation with polynomial growth, though finding the optimal row permutation is NP-hard.

math.NA↗

arXiv2026

LU Factorization of Discrete Random Matrices

We consider the probability that a discrete random matrix $M_n(ξ)$ is \emph{strongly non-singular}, meaning all its leading principal submatrices are non-singular. This property is equivalent to the existence of an LU factorization. We show that for any discrete random variable $ξ$ with finite support and $|ξ|_\infty < 1$, there is a constant probability that $M_n(ξ)$ is strongly non-singular with a growth factor bounded by $n^{5/2+δ}$. Furthermore, we provide a tight asymptotic lower bound for this probability as $|ξ|_\infty \to 0$. Finally, we provide exact counts for strongly non-singular binary matrices up to $n=9$ and use these to derive improved upper bounds for the Bernoulli case.

math.NA↗

arXiv2026

What is Jackson's constant?

We prove a refinement of Jackson's theorem on the approximation of Lipschitz functions by trigonometric polynomials. Our result precisely characterizes the leading error term associated with Jackson's construction. We do the same for a related construction commonly used in the kernel polynomial method for spectral density estimation, which is slightly better than Jackson's construction in this respect.

math.NA↗

arXiv2026

Spectral density estimation for normal matrices

The spectral density estimation problem asks for an algorithm that, given an $n\times n$ matrix $A$, outputs a probability measure that is a good approximation to the uniform distribution on the eigenvalues of $A$, called the spectral density of $A$. This paper considers the setting where $A$ is a large normal matrix that is accessible only through matrix-vector product queries. We provide an algorithm that makes just $m$ matrix-vector queries to $A$ and returns, with high probability, a measure within earth mover's distance $O(1/m+\log m/{\sqrt n})$ of the true spectral density of $A$. We provide a complementary lower bound that any algorithm producing an $\varepsilon$-approximation to the true spectral density for large matrices must make $Ω(1/\varepsilon)$ matrix-vector queries. The lower bound holds even for the more restricted case of real symmetric input matrices. In combination with our upper bound, it shows that spectral density estimation is essentially no harder for complex normal matrices than for real symmetric matrices.

math.NA↗

arXiv2026

On the exponential rate of the condition number of Fourier submatrices and Vandermonde matrices

The discrete Fourier transform matrix is one of the most important matrices in linear algebra, and submatrices of it arise in a variety of applications. Though the discrete Fourier transform matrix is unitary, its submatrices can be exponentially ill-conditioned, an obstacle to accurate computation. This work resolves the exact rate of the exponential ill-conditioning for square submatrices with contiguous rows and columns. As a consequence, we obtain a tight upper bound of $2 G/π$ on the exponential rate for all submatrices with contiguous columns, or, equivalently, all Vandermonde submatrices with distinct support points, where $G$ is Catalan's constant. These results follow from a more general analysis of Vandermonde and Vandermonde-like matrices for which exact estimates for exponential ill-conditioning are developed in terms of logarithmic potentials.

math.NA↗

arXiv2026

The largest 5th pivot may be the root of a 61st degree polynomial

This paper introduces a number of new techniques in the study of the famous question from numerical linear algebra: what is the largest possible growth factor when performing Gaussian elimination with complete pivoting? This question is highly complex, due to a complicated set of polynomial inequalities that need to be simultaneously satisfied. This paper introduces the JuMP + Groebner basis + discriminant polynomial approach as well as the use of interval arithmetic computations. Thus, we are introducing a marriage of numerical and exact mathematical computations. In 1988, Day and Peterson performed numerical optimization on $n=5$ with NPSOL and obtained a largest seen value of $4.1325...$. This same best value was reproduced by Gould with LANCELOT in 1991. We ran extensive comparable experiments with the modern software tool JuMP and also saw the same value $4.1325...$. While the combinatorial explosion of possibilities prevents us from knowing whether there may not be a larger maximum, we succeed in obtaining the exact mathematical value: the number $4.1325...$ is exactly the root of a 61st degree polynomial provided in this work, and is a maximum given the equality constraints seen by JuMP. In light of the numerics, we pose the conjecture that this lower bound is indeed the maximum. We also apply this technique to $n = 6$, $7$, and $8$. Furthermore, in 1969, an upper bound of $4\frac{17}{18}\approx 4.94$ was produced for the maximum possible growth for $n = 5$. We slightly lower this upper bound to $4.84$.

math.NA↗

arXiv2026

A Class of Optimal Directed Graphs for Network Synchronization

In a paper by Nishikawa and Motter, a quantity called the normalized spread of the Laplacian eigenvalues is used to measure the synchronizability of certain network dynamics. Through simulations, and without theoretical validation, it is conjectured that among all simple directed graphs with a fixed number of vertices and arcs, the optimal value of this quantity is achieved if the Laplacian spectrum satisfies a specific pattern. This paper proves this conjecture and further shows that the conjectured spectral condition is not only sufficient but also necessary. Moreover, the paper proves that the optimal Laplacian spectrum is always achievable by a class of almost regular directed graphs, which can be constructed through an inductive algorithm.

math.OC↗

arXiv2025

On the Maximum Spread of Non-Negative Matrices

Given a directed graph $G$, the spread of $G$ is the largest distance between any two eigenvalues of its adjacency matrix. In 2022, Breen, Riasanovsky, Tait, and Urschel asked what $n$-vertex directed graph maximizes spread, and whether this graph is undirected. We prove the more general result that the spread of any $n \times n$ non-negative matrix $A$ with $\|A\|_{\max} \le 1$ is at most $2n/\sqrt{3}$, which is tight up to an additive factor and exact when $n$ is a multiple of three. Furthermore, our results show that the matrix with maximum spread is always symmetric.

math.CO↗

arXiv2025

Nodal Count for Orthogonally Invariant Ensembles

We investigate the nodal count of eigenvectors of random matrices interpreted as operators on signed complete graphs. Our focus is on orthogonally invariant ensembles, with particular attention to the Gaussian Orthogonal Ensemble (GOE). We establish that, as the matrix size tends to infinity, the distribution of nodal counts converges to the same limiting law as the eigenvalue distribution. In the GOE case, this limit is the semicircle law. This result refutes a conjecture, motivated by quantum chaos and quantum graphs, which predicted Gaussian behavior of the nodal count.

math-ph↗

arXiv2025

On a perturbation analysis of Higham squared maximum Gaussian elimination growth matrices

Gaussian elimination is the most popular technique for solving a dense linear system. Large errors in this procedure can occur in floating point arithmetic when the matrix's growth factor is large. In the study of numerical linear algebra, it is often valuable to study and characterize the worst case examples. To this end, in their 1989 paper, Higham and Higham characterized the complete set of real n by n matrices that achieves the maximum growth factor under partial pivoting. Left undone is a sensitivity analysis for these matrices under perturbations. The growth factor of these and nearby matrices is the subject of this work. Through theoretical insights and empirical results, we illustrate the high sensitivity of the growth factor of these matrices to perturbations and show how subtle changes can be strategically applied to matrix entries to significantly reduce the growth.

math.NA↗

arXiv2025

A New Upper Bound For the Growth Factor in Gaussian Elimination with Complete Pivoting

The growth factor in Gaussian elimination measures how large the entries of an LU factorization can be relative to the entries of the original matrix. It is a key parameter in error estimates, and one of the most fundamental topics in numerical analysis. We produce an upper bound of $n^{0.2079 \ln n +0.91}$ for the growth factor in Gaussian elimination with complete pivoting -- the first improvement upon Wilkinson's original 1961 bound of $2 \, n ^{0.25\ln n +0.5}$.

math.NA↗

arXiv2024

Estimating the numerical range with a Krylov subspace

Krylov subspace methods are a powerful tool for efficiently solving high-dimensional linear algebra problems. In this work, we study the approximation quality that a Krylov subspace provides for estimating the numerical range of a matrix. In contrast to prior results, which often depend on the gaps between eigenvalues, our estimates depend only on the dimensions of the matrix and Krylov subspace, and the conditioning of the eigenbasis of the matrix. In addition, we provide nearly matching lower bounds for our estimates, illustrating the tightness of our arguments.

math.NA↗

arXiv2024

Recovering a Magnitude-Symmetric Matrix from its Principal Minors

We consider the inverse problem of finding a magnitude-symmetric matrix (matrix with opposing off-diagonal entries equal in magnitude) with a prescribed set of principal minors. This problem is closely related to the theory of recognizing and learning signed determinantal point processes in machine learning, as kernels of these point processes are magnitude-symmetric matrices. In this work, we prove a number of properties regarding sparse and generic magnitude-symmetric matrices. We show that principal minors of order at most $\ell$, for some invariant $\ell$ depending only on principal minors of order at most two, uniquely determines principal minors of all orders. In addition, we produce a polynomial-time algorithm that, given access to principal minors, recovers a matrix with those principal minors using only a quadratic number of queries. Furthermore, when principal minors are known only approximately, we present an algorithm that approximately recovers a matrix, and show that the approximation guarantee of this algorithm cannot be improved in general.

math.CO↗

arXiv2024

On the Frobenius norm of the inverse of a non-negative matrix

We prove a new lower bound for the Frobenius norm of the inverse of an non-negative matrix. This bound is only a modest improvement over previous results, but is sufficient for fully resolving a conjecture of Harwitz and Sloane, commonly referred to as the S-matrix conjecture, for all dimensions larger than a small constant.

math.CO↗

arXiv2024

Hamilton Powers of Eulerian Digraphs

In this note, we prove that the $\lceil \tfrac{1}{2} \sqrt{n} \log_2^2 n \rceil^{th}$ power of a connected $n$-vertex Eulerian digraph is Hamiltonian, and provide an infinite family of digraphs for which the $\lfloor \sqrt{n}/2 \rfloor^{th}$ power is not.

math.CO↗

arXiv2024

Some New Results on the Maximum Growth Factor in Gaussian Elimination

This paper combines modern numerical computation with theoretical results to improve our understanding of the growth factor problem for Gaussian elimination. On the computational side we obtain lower bounds for the maximum growth for complete pivoting for $n=1:75$ and $n=100$ using the Julia JuMP optimization package. At $n=100$ we obtain a growth factor bigger than $3n$. The numerical evidence suggests that the maximum growth factor is bigger than $n$ if and only if $n \ge 11$. We also present a number of theoretical results. We show that the maximum growth factor over matrices with entries restricted to a subset of the reals is nearly equal to the maximum growth factor over all real matrices. We also show that the growth factors under floating point arithmetic and exact arithmetic are nearly identical. Finally, through numerical search, and stability and extrapolation results, we provide improved lower bounds for the maximum growth factor. Specifically, we find that the largest growth factor is bigger than $1.0045n$ for $n>10$, and the lim sup of the ratio with $n$ is greater than or equal to $3.317$. In contrast to the old conjecture that growth might never be bigger than $n$, it seems likely that the maximum growth divided by $n$ goes to infinity as $n \rightarrow \infty$.

math.NA↗

arXiv2024

Average Nodal Count and the Nodal Count Condition for Graphs

The nodal edge count of an eigenvector of the Laplacian of a graph is the number of edges on which it changes sign. This quantity extends to any real symmetric $n\times n$ matrix supported on a graph $G$ with $n$ vertices. The average nodal count, averaged over all eigenvectors of a given matrix, is known to be bounded between $\frac{n-1}{2}$ and $\frac{n-1}{2}+β(G)$, where $β(G)$ is the first Betti number of $G$ (a topological quantity), and it was believed that generically the average should be around $\frac{n-1}{2}+β(G)/2$. We prove that this is not the case: the average is bounded between $\frac{n-1}{2}+β(G)/n$ and $\frac{n-1}{2}+β(G)-β(G)/n$, and we provide graphs and matrices that attain the upper and lower bounds for any possible choice of $n$ and $β$. A natural condition on a matrix for defining the nodal count is that it has simple eigenvalues and non-vanishing eigenvectors. For any connected graph $G$, a generic real symmetric matrix supported on $G$ satisfies this nodal count condition. However, the situation for constant diagonal matrices is far more subtle. We completely characterize the graphs $G$ for which this condition is generically true, and show that if this is not the case, then any real symmetric matrix supported on $G$ with constant diagonal has a multiple eigenvalue or an eigenvector that vanishes somewhere. Finally, we discuss what can be said when this nodal count condition fails, and provide examples.

math-ph↗

arXiv2023

Estimating the matrix $p \rightarrow q$ norm

The matrix $p \rightarrow q$ norm is a fundamental quantity appearing in a variety of areas of mathematics. This quantity is known to be efficiently computable in only a few special cases. The best known algorithms for approximately computing this quantity with theoretical guarantees essentially consist of computing the $p\to q$ norm for $p,q$ where this quantity can be computed exactly or up to a constant, and applying interpolation. We analyze the matrix $2 \to q$ norm problem and provide an improved approximation algorithm via a simple argument involving the rows of a given matrix. For example, we improve the best-known $2\to 4$ norm approximation from $m^{1/8}$ to $m^{1/12}$. This insight for the $2\to q$ norm improves the best known $p \to q$ approximation algorithm for the region $p \le 2 \le q$, and leads to an overall improvement in the best-known approximation for $p \to q$ norms from $m^{25/128}$ to $m^{3 - 2 \sqrt{2}}$.

cs.DS↗