If a real-valued function is continuous on a real interval and takes on two different values, then it will also take every value between those two, by the Intermediate Value Theorem. It is not immediately clear what would be a natural generalization for functions whose domain and range are in higher-dimensional Euclidean spaces. In this article, we analyze this problem, by first arriving at what we think is the appropriate question to ask, and then restricting to linear transformations. It turns out that the matrices that will satisfy an appropriate version of the Intermediate Value Theorem are the so called monotone and weakly monotone matrices, which have applications in numerical approximation to the solutions to systems of linear equations.

This work focuses on the problem of exact model reduction of positive linear systems, by leveraging minimal realization theory. While determining the existence of a positive reachable realization remains in general an open problem, we are able to fully characterize the cases in which the new model is obtained with non-negative reduction matrices, and hence positivity of the reduced model is robust with respect to small perturbations of the original system. The characterization is obtained by specializing monotone matrix theory to positive matrices. In addition, we provide a systematic method to construct positive reductions also when minimal ones are not available, by exploiting algebraic techniques.

In this paper, we introduce an inexact Noda iteration method featuring inner and outer iterations for computing the smallest eigenvalue and corresponding eigenvector of an irreducible monotone matrix. The proposed method includes two primary relaxation steps designed to compute the smallest eigenvalue and its associated eigenvector. These steps are influenced by specific relaxation factors, and we examine how these factors impact the convergence of the outer iterations. By applying two distinct relaxation factors to solve the inner linear systems, we demonstrate that the convergence can be globally linear or superlinear, contingent upon the relaxation factor used. Additionally, the relaxation factor affects the rate of convergence. The inexact Noda iterations we propose are structure-preserving and ensure the positivity of the approximate eigenvectors. Numerical examples are provided to demonstrate the practicality of the proposed method, consistently preserving the positivity of approximate eigenvectors.

In this letter, we consider an optimization problem with box constraints coupled with polytope constraints. The existing deep learning methodologies for solving such constrained optimization problems are incapable to ensure feasible solution, require a resource-intensive projection step, or involve iterations. To this end, we propose a novel and efficient deep unsupervised learning (DUL)-based approach that minimizes the loss function while guaranteeing satisfaction of both box and polytope constraints involving a monotone matrix. In particular, we utilize properties of a monotone matrix to transform the original constrained optimization problem into an equivalent problem which can easily be optimized using DUL. We have conducted a performance comparison between our proposed approach and existing DUL-based benchmarks considering a device-to-device (D2D) wireless network. Simulation results have shown that our proposed scheme outperforms the benchmark schemes in terms of both constraint satisfaction probability and achievable average sum-rate.

This work is concerned with computing low-rank approximations of a matrix function f (A) for a large symmetric positive semi-definite matrix A, a task that arises in, e.g., statistical learning and inverse problems. The application of popular randomized methods, such as the randomized singular value decomposition or the Nyström approximation, to f (A) requires multiplying f (A) with a few random vectors. A significant disadvantage of such an approach, matrix-vector products with f (A) are considerably more expensive than matrix-vector products with A, even when carried out only approximately via, e.g., the Lanczos method. In this work, we present and analyze funNyström, a simple and inexpensive method that constructs a low-rank approximation of f (A) directly from a Nyström approximation of A, completely bypassing the need for matrixvector products with f (A). It is sensible to use funNyström whenever f is monotone and satisfies f (0) = 0. Under the stronger assumption that f is operator monotone, which includes the matrix square root A 1/2 and the matrix logarithm log(I + A), we derive probabilistic bounds for the error in the Frobenius, nuclear, and operator norms. These bounds confirm the numerical observation that funNyström tends to return an approximation that compares well with the best low-rank approximation of f (A). Furthermore, compared to existing methods, funNyström requires significantly fewer matrix-vector products with A to obtain a lowrank approximation of f (A), without sacrificing accuracy or reliability. Our method is also of interest when estimating quantities associated with f (A), such as the trace or the diagonal entries of f (A). In particular, we propose and analyze funNyström++, a combination of funNyström with the recently developed Hutch++ method for trace estimation.

Abstract Monotone matrices are stochastic matrices that satisfy the monotonicity conditions as introduced by Daley in 1968. Monotone Markov chains are useful in modeling phenomena in several areas. Most previous work examines the embedding problem for Markov chains within the entire set of stochastic transition matrices, and only a few studies focus on the embeddability within a specific subset of stochastic matrices. This article examines the embedding in a discrete-time monotone Markov chain, i.e., the existence of monotone matrix roots. Monotone matrix roots of ( 2 × 2 ) \left(2\times 2) monotone matrices are investigated in previous work. For ( 3 × 3 ) \left(3\times 3) monotone matrices, this article proves properties that are useful in studying the existence of monotone roots. Furthermore, we demonstrate that all ( 3 × 3 ) \left(3\times 3) monotone matrices with positive eigenvalues have an m m th root that satisfies the monotonicity conditions (for all values m ∈ N , m ≥ 2 m\in {\mathbb{N}},m\ge 2 ). For monotone matrices of order n > 3 n\gt 3 , diverse scenarios regarding the matrix roots are pointed out, and interesting properties are discussed for block diagonal and diagonalizable monotone matrices.

Projection and contraction method for updating simultaneously mass and stiffness matrices

Monotonicity correction for second order element finite volume methods of anisotropic diffusion problems

Convergence of two-stage iterative scheme for [formula omitted]-weak regular splittings of type II

Divisibility of LCM matrices by totally nonnegative GCD matrices

<abstract> The comparison results for $ K $-double splittings of one $ K $-monotone matrix are established in the literatures. As comparison theorems between the spectral radii of different matrices are a useful tool for judging the efficiency of preconditioners, we propose some comparison results for $ K $-nonnegative double splittings of different $ K $-monotone matrices in this note. The obtained results generalize the previous ones. </abstract>

SummaryMotivated by the problem of estimating bacterial growth rates for genome assemblies from shotgun metagenomic data, we consider the permuted monotone matrix model $Y=\Theta\Pi+Z$ where $Y\in \mathbb{R}^{n\times p}$ is observed, $\Theta\in \mathbb{R}^{n\times p}$ is an unknown approximately rank-one signal matrix with monotone rows, $\Pi \in \mathbb{R}^{p\times p}$ is an unknown permutation matrix, and $Z\in \mathbb{R}^{n\times p}$ is the noise matrix. In this article we study estimation of the extreme values associated with the signal matrix $\Theta$, including its first and last columns and their difference. Treating these estimation problems as compound decision problems, minimax rate-optimal estimators are constructed using the spectral column-sorting method. Numerical experiments on simulated and synthetic microbiome metagenomic data are conducted, demonstrating the superiority of the proposed methods over existing alternatives. The methods are illustrated by comparing the growth rates of gut bacteria in inflammatory bowel disease patients and control subjects.

ABSTRACT Iterative methods based on matrix splittings are useful tools in solving real large sparse linear systems. In this aspect, the type I double splitting approaches are straight forward from the formulation of the iteration scheme and its convergence theory is well established in the literature. However, if a double splitting is of type II, then the convergence of the iteration scheme seems not to be straight forward. In this paper, we develop convergence theory for type II double splittings to make the implementation quite simple. In this direction, we first introduce two new subclasses of double splittings and establish their convergence theory. Using this theory, we prove a new characterization of a monotone matrix. Finally, we apply our theoretical findings to the double splitting of an M-matrix in the Gauss–Seidel double SOR method to obtain a comparison result.

Metal matrix composites (MMCs) generally possess superior properties than the monotonic matrix alloys, and thus, they have become excellent candidate materials in various applications. Also, the ability of property tailoring at an affordable cost is of particular importance to industries. Among the many manufacturing techniques for MMCs, laser-assisted additive manufacturing (AM) techniques have emerged and drawn increasing attention in the past decade. In the literature, a wealth of studies have been carried out on the synthesis of MMCs via laser-assisted AM techniques, as well as the property evaluation of the obtained MMCs. In this paper, we review and analyze the relevant literature and summarize the material preparation, optimization of process parameters, resultant improvements, and corresponding strengthening mechanisms for each major category of MMCs. Moreover, the limitations and challenges related to MMC synthesis using the laser-assisted AM techniques are discussed, and the future research directions are suggested to address those issues.

Motivated by recent research on quantifying bacterial growth dynamics based on genome assemblies, we consider a permuted monotone matrix model , where the rows represent different samples, the columns represent contigs in genome assemblies and the elements represent log-read counts after preprocessing steps and Guanine-Cytosine (GC) adjustment. In this model, Θ is an unknown mean matrix with monotone entries for each row, Π is a permutation matrix that permutes the columns of Θ, and Z is a noise matrix. This article studies the problem of estimation/recovery of Π given the observed noisy matrix Y. We propose an estimator based on the best linear projection, which is shown to be minimax rate-optimal for both exact recovery, as measured by the 0-1 loss, and partial recovery, as quantified by the normalized Kendall’s tau distance. Simulation studies demonstrate the superior empirical performance of the proposed estimator over alternative methods. We demonstrate the methods using a synthetic metagenomics dataset of 45 closely related bacterial species and a real metagenomic dataset to compare the bacterial growth dynamics between the responders and the nonresponders of the IBD patients after 8 weeks of treatment. Supplementary materials for this article are available online.

Book Review: Loewner’s theorem on monotone matrix functions

In previous work, the embedding problem is examined within the entire set of discrete-time Markov chains. However, for several phenomena, the states of a Markov model are ordered categories and the transition matrix is state-wise monotone. The present paper investigates the embedding problem for the specific subset of state-wise monotone Markov chains. We prove necessary conditions on the transition matrix of a discrete-time Markov chain with ordered states to be embeddable in a state-wise monotone Markov chain regarding time-intervals with length 0.5: A transition matrix with a square root within the set of state-wise monotone matrices has a trace at least equal to 1.

The purpose of this note is two-fold: (1) to study when quasi-Euclidean rings, regular rings and regular separative rings have the property (∗) that each right (left) singular element is a product of idempotents, and (2) to consider the question: “when is a singular nonnegative square matrix a product of nonnegative idempotent matrices?” The importance of the class of quasi- Euclidean rings in connection with the property (∗) is given by the first three authors and T.Y. Lam [Journal of Algebra, 406:154–170, 2014], where it is shown that every singular matrix over a right and left quasi-Euclidean domain is a product of idempotents, generalizing the results of J. A Erdos [Glasgow Mathematical Journal, 8: 118–122, 1967] for matrices over fields and that of T. J. Laffey [Linear and Multilinear Algebra, 14:309–314, 1983] for matrices over commutative Euclidean domains. We have shown in this paper that quasi-Euclidean rings appear among many interesting classes of rings and hence they are in abundance. We analyze the properties of triangular matrix rings and upper triangular matrices with respect to the decomposition into product of idempotents and show, in particular, that nonnegative nilpotent matrices are products of nonnegative idempotent matrices. We study as to when each singular matrix is a product of idempotents in special classes of rings. Regarding the second question for nonnegative matrices, bounds are obtained for a rank one nonnegative matrix to be a product of two idempotent matrices. It is shown that every nonnegative matrix of rank one is a product of three nonnegative idempotent matrices. For matrices of higher orders, we show that some power of a group monotone matrix is a product of idempotent matrices.

Comparison theorems for double splittings of K-monotone matrices

Since lattice matrices are useful tools in various domains like automata theory, design of switching circuits, logic of binary relations, medical diagnosis, markov chains, computer network, traffic control and so on, the study of the properties of lattice matrices is valuable. A lattice matrix A is called monotone if A is transitive or A is monotone increasing. In this paper, the convergence of monotone matrices is studied. The results obtained here develop the corresponding ones on lattice matrices shown in the references.