Linear Algebra Research Articles

Towards a platform-portable linear algebra backend for OpenFOAM

AbstractGraphics processing unit accelerators have become a widespread technology on modern high performance computing clusters for increasing the performance of scientific computing algorithms. Despite early efforts to adopt linear solvers that utilize graphics processing units for OpenFOAM, to this date no widely accepted approach has gotten traction. In recent years, the number of different vendors providing graphics processing units accelerators has grown, and as of the writing of this paper, no commonly accepted, unified approach to leverage accelerators exists. This makes platform-portable solutions to increase the efficiency of graphics processing units offloading techniques desirable, and an important research topic. In this work, we investigate a platform-portable solution using the Ginkgo sparse linear algebra library.

Some Results on the Properties of Discrete-Time LTI State-Space Systems

Abstract This work brings forward several interesting facts about system properties of linear time-invariant discrete-time systems like output controllability, state controllability and state observability, trackability, and input and state observability. Particularly, the work brings about the inter-relationships between these properties while taking a state-space formulation-based approach to come up with simple facts that are backed up by proofs using preliminary linear algebra.

On linear algebra of r-Hankel and r-Toeplitz matrices with geometric sequence

AbstractMatrix theory plays a crucial role in solving practical problems and performing computational operations. In particular, specific types of matrices and their linear algebraic properties are of paramount significance for these processes. In this paper, we study the properties of r-Hankel and r-Toeplitz matrices whose entries are geometric sequences, and then the determinants, inverse matrix, generalized inverse matrix (the Moore-Penrose inverse), and spectral norms of such matrices are obtained.

The Truncated Moment Problem on Reducible Cubic Curves I: Parabolic and Circular Type Relations

In this article we study the bivariate truncated moment problem (TMP) of degree 2k on reducible cubic curves. First we show that every such TMP is equivalent after applying an affine linear transformation to one of 8 canonical forms of the curve. The case of the union of three parallel lines was solved in Zalar (Linear Algebra Appl 649:186–239, 2022. https://doi.org/10.1016/j.laa.2022.05.008), while the degree 6 cases in Yoo (Integral Equ Oper Theory 88:45–63, 2017). Second we characterize in terms of concrete numerical conditions the existence of the solution to the TMP on two of the remaining cases concretely, i.e., a union of a line and a circle y(ay+x2+y2)=0,a∈R\\{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y(ay+x^2+y^2)=0, a\\in {\\mathbb {R}}{\\setminus } \\{0\\}$$\\end{document}, and a union of a line and a parabola y(x-y2)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y(x-y^2)=0$$\\end{document}. In both cases we also determine the number of atoms in a minimal representing measure.

Applied Mathematics: Building Theory and Practice

The explanation, assessment, and resolution of real-world problems in a variety of fields, including the social sciences, engineering, biology, physics, and economics, depend heavily on applied mathematics. By applying mathematical models, algorithms, and computational techniques, experts may interpret complex events, improve workflows, and offer creative solutions. Moreover, by aiding in the formulation of hypotheses, the planning of experiments, and the interpretation of results, applied mathematics fosters a greater understanding of both natural and artificial systems.Applied mathematics is based on strong theoretical frameworks that facilitate modeling, analysis, and prediction. These foundations span several areas of mathematics, such as calculus, linear algebra, differential equations, probability theory, and optimization. With the use of these tools, scientists may develop mathematical representations of real-world events that accurately reflect their essential features and dynamics. Theoretical insights also enable practitioners to investigate complex systems and derive actionable knowledge through the development of analytical methodologies, numerical algorithms, and simulation tools.

Structured ramp secret sharing schemata over rings of real polynomials

Two new ramp secret sharing schemata based on polynomials are proposed. For both schemata, the secret is considered to be a polynomial created by the dealer. The participants are separated into ℓ⩾2, groups, that are specified by the dealer's levels Li for i=1,2,…,ℓ and each level Li, i⩾2, is separated into subsets. The shares of the secret are given to participants in the form of polynomials. For the first proposed scheme, the dealer creates ℓ polynomials one for each level. Specific participants from every subset of each level have to cooperate all together in order to construct the polynomial of their level. Next all the authorized participants cooperate for computing the greatest common divisor of the polynomials in order to retrieve the secret. In the second scheme, the authorized participants cooperate per two levels using a bottom-up procedure. In both schemata the greatest common divisor can be evaluated by implementing numerical linear algebra methods, and precisely factorization of matrices of special form such as Sylvester matrices. The triangularization of these matrices can be obtained by exploiting their special structure for the reduction of the required floating point operations. The innovative idea of the paper at hand is the use of real polynomials in secret sharing schemata. This is particularly useful since the greatest common divisor can always be computed with efficient accuracy using effective numerical methods. New theoretical results are proved and provided that support the error analysis of our approach.

Batched sparse and mixed-precision linear algebra interface for efficient use of GPU hardware accelerators in scientific applications

Batched Sparse Linear Algebra has become an emergent processing mode on modern hardware accelerators based on Graphics Processing Units (GPUs) developed over the years to serve as the main compute devices in the largest computing clusters and supercomputers. We propose a set solver interface designs for batched sparse numerical solvers on these hardware accelerators. We motivate our specific designs by both their use in scientific applications of national importance and also by the possibility of implementing them in an efficient and portable manner with multiple options for vendor-specific optimizations. We present the C language interface calls for the linker-agnostic interchange of functional entry points. We also show how using C++ for the batched solvers simplifies the interface design while giving the user much broader set of opportunities for customization, testing, and debugging. We also cover in our proposals the option of exploiting multiple floating-point arithmetic precisions to directly match the application needs in terms of accuracy. Finally, a selected sample of performance experiments show how our proposed interface can be efficiently implemented to outperform the available alternatives many times over. In the end, we plan for an ongoing evolution of our newly proposed interface standard to keep up with the updates in programming languages, accelerator hardware, and application needs.

Orthogonality and unitary conditions for solutions to some consistent linear matrix equations

The nonlinear matrix equation (A1 + B1X1D1)(A2 + B2X2D2) = A, presents a matrix identity, where A, Ai, Bi, and Di are known matrices of suitable sizes and Xi are unknown matrices for i = 1,2, over the field of complex numbers ℂ. Several known simple methods in linear algebra can handle the orthogonality problem. In this paper, we select some linear matrix equations as illustrative examples to discuss some properties of pairs of identically dimensional consistent linear matrix equations, then apply specific matrix analytic tools, among them is the matrix rank method, the rank of a matrix is one of the most basic quantities and useful methods and tools that are widely used in linear algebra specifically, in matrix theory and its applications. We consider certain forms of linear matrix equations such as T1X1 = N1 and the system A3X3 = C3, X3B3 = D3, to derive novel conditions dictating the orthogonality of all solutions to two consistent matrix equations, as well as the necessary and sufficient conditions for all solutions to be unitary, further we give an illustrative example.

Quantum simulation of discrete linear dynamical systems and simple iterative methods in linear algebra

Quantum simulation is capable of simulating certain dynamical systems in continuous time—Schrödinger’s equations being the most direct and well known—more efficiently than classical simulation. Any linear dynamical system can in fact be transformed into a system of Schrödinger’s equations via a method called Schrödingerisation (Jin et al. 2022. Quantum simulation of partial differential equations via Schrödingerisation. ( https://arxiv.org/abs/2212.13969 ) and Jin et al. 2023. Phys. Rev. A 108 , 032603. ( doi:10.1103/PhysRevA.108.032603 )). We show how Schrödingerisation allows quantum simulation to be directly used for the simulation of continuous-time versions of general (explicit) iterative schemes or discrete linear dynamical systems. In particular, we use this new method to solve linear systems of equations and for estimating the maximum eigenvector and eigenvalue of a matrix, respectively. This method is applicable using either discrete-variable quantum systems or on hybrid continuous-variable and discrete-variable quantum systems. This framework provides an interesting alternative to solve linear algebra problems using quantum simulation.

Performance of Linear Algebra Factorization in Multi-Accelerator Architectures

Hardware and software are required to effectively solve problems in many domains. The idea of creating a hybrid architecture based on graphics processors arose to meet the increasing demands of modern scientific problems. Most of these problems are reduced to solving linear algebra problems. A set of efficient linear solutions has been successfully used to solve important scientific problems for many years. Factorizations play a crucial role in solving linear algebra problems. This work presents implementations of LU, QR and Cholesky factorizations on two graphics processors using the MAGMA 2.6.0 library. Their performances are given for matrices with real and complex numbers in single and double precision.

Quantum hitting time according to a given distribution

In this work we focus on the notion of quantum hitting time for discrete-time Szegedy quantum walks, compared to its classical counterpart. Under suitable hypotheses, quantum hitting time is known to be of the order of the square root of classical hitting time: this quadratic speedup is a remarkable example of the computational advantages associated with quantum approaches. Our purpose here is twofold. On one hand, we provide a detailed proof of quadratic speedup for time-reversible walks within the Szegedy framework, in a language that should be familiar to the linear algebra community. Moreover, we explore the use of a general distribution in place of the stationary distribution in the definition of quantum hitting time, through theoretical considerations and numerical experiments.

Systems of linear equations and the theory of divisibility in the structure of mathematical competence of a chemistry teacher

This article is dedicated to the analysis of the role of linear algebraic models and related divisibility theory issues in the mathematical activity of a chemistry teacher. The work has investigated the place and informational-logical connections of linear algebraic models in the system of mathematical competence of a chemistry teacher and has justified its significance. Examples from the school chemistry course have been given, where systems of linear algebraic equations have been actively used both over the field of real numbers and over the ring of integers. Various scenarios for solving problems, in fact reduced to linear algebraic models, have been illustrated. Such tasks have included classic problems with many reagent components as well as setting coefficients in complex chemical equations. The advantages and disadvantages of synthetic methods using chemical reasoning and formal algebraic methods have been analyzed. It has been substantiated that the teacher needed to understand the mathematical essence of the corresponding models to generate any solution and guide the students' corresponding work. The possibility of applying the basics of number theory in teaching chemistry has been shown. An analysis of the possession of relevant mathematical skills of working chemistry teachers has been conducted. Approaches were developed to improve the level of chemistry teachers’ skills of solving and analyzing systems of linear equations with real and integer variables in the conditions of postgraduate pedagogical education of teachers. The results, obtained in the work, have shown the need for an analysis of mathematics curricula in pedagogical and classical universities, sections related to linear algebra, as well as the introduction of the basics of number theory into mathematics curricula. The need to introduce mathematical training into the system of advanced training of chemistry teachers has been proven. Possible further research on this topic is related to the active implementation of blocks of linear algebraic models into the courses of advanced training of chemistry teachers and further analysis of their effectiveness

Decay and Fission of Magnetic Quivers.

In exploring supersymmetric theories with eight supercharges, the Higgs branches present an intriguing window into strong coupling dynamics. Magnetic quivers serve as crucial tools for understanding these branches. Here, we introduce the decay and fission algorithm for unitary magnetic quivers. It efficiently derives complete phase diagrams (Hasse diagrams) through convex linear algebra. It allows magnetic quivers to undergo decay or fission, reflecting Higgs branch RG flows in the theory. Importantly, the algorithm generates magnetic quivers for the RG fixed points and simplifies the understanding of transverse slice geometry with no need for a list of minimal transitions. In contrast, the algorithm hints to the existence of a new minimal transition, whose geometry and physics need to be explored.

Quantum Intelligence in Beyond 5G Networks: Current Progress, and Open Avenues

The emergence of fifth-generation (5G) wireless technologies leads to enlarging network complexity as a result of massive data generation, exhaustive operating costs, time, energy, and the burdens of planning and management. Artificial intelligence (AI) has been demonstrated to have a vital role in improving data analytics and decision-making in massive 5G networks. On the other hand, Quantum Computing is an evolving technology for handling exponential expansion in the data dimensions and calculating linear algebra quicker and more proficiently than traditional computers implying reduced computational costs and energy consumption. The unification between these disciplines engenders the concept of “Quantum Intelligence", which is an innovative and quite promising field with the possibility of unbounded capabilities for the 5G network. Beyond centralized learning, our discussions extend to debate the potentials of quantum intelligence to improve the distributed (federated) learning scenarios over several quantum computers, aiming to drastically enhance computational efficiency and energy consumption. Multiple simulation experiments are performed to evaluate and compare the performance of quantum intelligence on classical and quantum datasets. Finally, this article outlines the major technical and research challenges and open problems for future research on quantum intelligence in 5G wireless networks.

Solutions to the Landau–Lifshitz–Gilbert equation in the frequency space: Discretization schemes for the dynamic-matrix approach

The dynamic matrix method addresses the Landau–Lifshitz–Gilbert (LLG) equation in the frequency domain by transforming it into an eigenproblem. Subsequent numerical solutions are derived from the eigenvalues and eigenvectors of the dynamic matrix. In this work we explore discretization methods needed to obtain a matrix representation of the dynamic operator, a fundamental counterpart of the dynamic matrix. Our approach opens a new set of linear algebra tools for the dynamic matrix method and expose the approximations and limitations intrinsic to it. Moreover, our discretization algorithms can be applied to various discretization schemes, extending beyond micromagnetism problems. We present some application examples, including a technique to obtain the dynamic matrix directly from the magnetic free energy function of an ensemble of macrospins, and an algorithmic method to calculate numerical micromagnetic kernels, including plane wave kernels. We also show how to exploit symmetries and reduce the numerical size of micromagnetic dynamic-matrix problems by a change of basis. This procedure significantly reduces the size of the dynamic matrix by several orders of magnitude while maintaining high numerical precision. Additionally, we calculate analytical approximations for the dispersion relations in magnonic crystals. This work contributes to the understanding of the current magnetization dynamics methods, and could help the development and formulations of novel analytical and numerical methods for solving the LLG equation within the frequency domain.

Linear Algebra and Bayes’ Theorem in Probability

Linear Algebra and Bayes’ Theorem in Probability

MISTER-T: An open-source software package for quantum optimal control of multi-electron systems on arbitrary geometries

We present an open-source software package, MISTER-T (Manipulating an Interacting System of Total Electrons in Real-Time), for the quantum optimal control of interacting electrons within a time-dependent Kohn-Sham formalism. In contrast to other implementations restricted to simple models on rectangular domains, our method enables quantum optimal control calculations for multi-electron systems (in the effective mass formulation) on nonuniform meshes with arbitrary two-dimensional cross-sectional geometries. Our approach is enabled by forward and backward propagator integration methods to evolve the Kohn-Sham equations with a pseudoskeleton decomposition algorithm for enhanced computational efficiency. We provide several examples of the versatility and efficiency of the MISTER-T code in handling complex geometries and quantum control mechanisms. The capabilities of the MISTER-T code provide insight into the implications of varying propagation times and local control mechanisms to understand a variety of strategies for manipulating electron dynamics in these complex systems. Program summaryProgram Title: MISTER-TCPC Library link to program files:https://doi.org/10.17632/psymy4ddnw.1Licensing provisions: GNU General Public License 3Programming language: MATLABSupplementary material: animated movies of total electron densities under the influence of optimal control fields for (1) an asymmetric double-well potential for long propagation times, (2) an asymmetric double-well potential for short propagation times, and (3) a triple-well potential with a position-dependent effective mass.Nature of problem: The MISTER-T code solves quantum optimal control problems for interacting electrons within a time-dependent Kohn-Sham formalism. It can handle two-dimensional systems with arbitrary cross-sectional geometries within the effective mass formulation. The user-friendly code uses forward and backward propagator integration methods to evolve the Kohn-Sham equations with a pseudoskeleton decomposition algorithm for enhanced computational efficiency.Solution method: iterative solution of the quantum optimal control equations using finite element methods, effective mass formulation, pseudoskeleton decomposition, sparse matrix linear algebra, and nonuniform fast Fourier transforms.

Software Implementation of the Epps-Pulley Criterion in Matlab Modeling Environment

Purpose. Modeling systems and programming platforms provide ample opportunities for the use of statistical tools in research activities. Since the normal distribution is one of the most common distribution laws, the criterion for checking the sample for normality is in high demand among statistical assessment tools, among which the Epps-Pulley test has the status as one of the most powerful tests to check the deviation of the distribution from the normal one. There are a number of implementations of this test in the R and Python languages. However, this test is not implemented in one of the most popular Matlab modeling software. Thus, the purpose of this study is to develop a software implementation of the Epps-Pulley criterion in the Matlab environment and verify the correctness of the performed calculations.Materials and Methods. We implemented the calculation of Epps-Pulley statistics by two methods – classical, using cycles, and matrix-vector, using linear algebra operations. The classical method requires calculating the intermediate values necessary to obtain the criterion statistics using two independent cycles, the second cycle being a double one, in which one cycle is nested into the other. The matrix-vector method requires fewer code by performing calculations using linear algebra operations on matrices and vectors. We obtained critical statistical values for the sample size from 8 to 1000 observations with two-dimensional linear interpolation of tabular values. We used an approximation by a beta function of the third kind for a sample of over 1000 elements.Results. An assessment of the computational efficiency of the methods showed that the cyclic approach is about three times higher than the matrix-vector approach in terms of consumed time, which is presumably due to the processing of insignificant elements in triangular matrices when performing component-by-component operations. The correctness of the software implementation of the Epps-Pulley criterion was tested on several examples, which confirmed the compliance of the calculated values of the criterion statistics, as well as the critical values of statistics, with known data. We carried out a criterion statistical evaluation based on the empirical values of the error of the first kind. We obtained the error values correspondence to the specified significance levels. We performed comparative estimates of the Epps-Pulley test with the Anders-Darling and Shapiro-Wilk tests in terms of the criterion empirical power and tabulated the evaluation results. We published the software implementation of the Epps-Pulley test on the MATLAB Central Internet resource and for free use.Conclusion. We developed software implementation of the Epps-Pulley criterion as a new research tool that was previously unavailable in the Matlab modeling environment. We used the time spent on calculations to make a reasonable choice of the calculation algorithm for the criterion statistics. We confirmed correctness of the calculation algorithms by a set of selective checks and statistical estimates that showed the compliance with well-known theoretical provisions.

On linear combinations of binomial OWA functions

We consider the binomial decomposition of ordered weighted averaging (OWA) functions proposed by Calvo and De Baets (1998) in the framework of Choquet integration. Our aim in the paper is to further investigate the equivalence between the two representations of OWA functions involved in the binomial decomposition: the usual canonical representation in terms of the order statistics, and the binomial representation in terms of the binomial OWA functions. We describe and discuss in detail the linear transformations that relate the coefficients of these two equivalent representations: the original expression of the weights in terms of the coefficients of the binomial representation, and its inverse, the expression of those coefficients in terms of the weights. In both cases simple and direct proofs are presented. Moreover, we consider the linear transformations between the two representations in the general linear algebra framework of unconstrained linear combinations of binomial OWA functions. In this perspective we obtain compact matrix expressions for the linear transformations, which also offer new insight on the geometry of the coefficient constraints in the binomial representation of OWA functions.

Novel progressive deep learning algorithm for uncovering multiple single nucleotide polymorphism interactions to predict paclitaxel clearance in patients with nonsmall cell lung cancer.

The rate at which the anticancer drug paclitaxel is cleared from the body markedly impacts its dosage and chemotherapy effectiveness. Importantly, paclitaxel clearance varies among individuals, primarily because of genetic polymorphisms. This metabolic variability arises from a nonlinear process that is influenced by multiple single nucleotide polymorphisms (SNPs). Conventional bioinformatics methods struggle to accurately analyze this complex process and, currently, there is no established efficient algorithm for investigating SNP interactions. We developed a novel machine-learning approach called GEP-CSIs data mining algorithm. This algorithm, an advanced version of GEP, uses linear algebra computations to handle discrete variables. The GEP-CSI algorithm calculates a fitness function score based on paclitaxel clearance data and genetic polymorphisms in patients with nonsmall cell lung cancer. The data were divided into a primary set and a validation set for the analysis. We identified and validated 1184 three-SNP combinations that had the highest fitness function values. Notably, SERPINA1, ATF3 and EGF were found to indirectly influence paclitaxel clearance by coordinating the activity of genes previously reported to be significant in paclitaxel clearance. Particularly intriguing was the discovery of a combination of three SNPs in genes FLT1, EGF and MUC16. These SNPs-related proteins were confirmed to interact with each other in the protein-protein interaction network, which formed the basis for further exploration of their functional roles and mechanisms. We successfully developed an effective deep-learning algorithm tailored for the nuanced mining of SNP interactions, leveraging data on paclitaxel clearance and individual genetic polymorphisms.