Linear Algebra Problems Research Articles

Thermodynamic linear algebra

AbstractLinear algebra is central to many algorithms in engineering, science, and machine learning; hence, accelerating it would have tremendous economic impact. Quantum computing has been proposed for this purpose, although the resource requirements are far beyond current technological capabilities. We consider an alternative physics-based computing paradigm based on classical thermodynamics, to provide a near-term approach to accelerating linear algebra. At first sight, thermodynamics and linear algebra seem to be unrelated fields. Here, we connect solving linear algebra problems to sampling from the thermodynamic equilibrium distribution of a system of coupled harmonic oscillators. We present simple thermodynamic algorithms for solving linear systems of equations, computing matrix inverses, and computing matrix determinants. Under reasonable assumptions, we rigorously establish asymptotic speedups for our algorithms, relative to digital methods, that scale linearly in matrix dimension. Our algorithms exploit thermodynamic principles like ergodicity, entropy, and equilibration, highlighting the deep connection between these two seemingly distinct fields, and opening up algebraic applications for thermodynamic computers.

Classifying Abelian Groups Through Acyclic Matchings

AbstractThe inquiry into identifying sets of monomials that can be eliminated from a generic homogeneous polynomial via a linear change of coordinates was initiated by E. K. Wakeford. This linear algebra problem prompted C. K. Fan and J. Losonczy to introduce the notion of acyclic matchings in the additive group $$\mathbb {Z}^n$$ Z n , subsequently extended to abelian groups by the latter author. Alon, Fan, Kleitman, and Losonczy established the acyclic matching property for $$\mathbb {Z}^n$$ Z n . This note aims to classify all abelian groups with respect to the acyclic matching property.

Total Positivity and Accurate Computations Related to q-Abel Polynomials

The attainment of accurate numerical solutions of ill-conditioned linear algebraic problems involving totally positive matrices has been gathering considerable attention among researchers over the last years. In parallel, the interest of q-calculus has been steadily growing in the literature. In this work the q-analogue of the Abel polynomial basis is studied. The total positivity of the matrix of change of basis between monomial and q-Abel bases is characterized, providing its bidiagonal factorization. Moreover, well-known high relative accuracy results of Vandermonde matrices corresponding to increasing positive nodes are extended to the decreasing negative case. This further allows to solve with high relative accuracy several algebraic problems concerning collocation, Wronskian and Gramian matrices of q-Abel polynomials. Finally, a series of numerical tests support the presented theoretical results and illustrate the goodness of the method where standard approaches fail to deliver accurate solutions.

On the number of solutions to a random instance of the permuted kernel problem

The Permuted Kernel Problem (PKP) is a problem in linear algebra that was first introduced by Shamir in 1989. Roughly speaking, given an ℓ×m matrix A and an m×1 vector b over a finite field of q elements Fq, the PKP asks to find an m×m permutation matrix π such that πb belongs to the kernel of A. In recent years, several post-quantum digital signature schemes whose security can be provably reduced to the hardness of solving random instances of the PKP have been proposed. In this regard, it is important to know the expected number of solutions to a random instance of the PKP in terms of the parameters q,ℓ,m. Previous works have heuristically estimated the expected number of solutions to be m!/qℓ.We provide, and rigorously prove, exact formulas for the expected number of solutions to a random instance of the PKP and the related Inhomogeneous Permuted Kernel Problem (IPKP), considering two natural ways of generating random instances.

Quantum Computing Techniques for Numerical Linear Algebra in Computational Mathematics

Quantum computing is a new and exciting area of computational mathematics that has the ability to solve very hard problems that traditional computing methods have not been able to solve for a long time. This abstract goes into detail about how quantum computing can be used in numerical linear algebra, which is an important part of computational mathematics that is used in many fields, such as science, engineering, and data analysis. The idea behind quantum computing is new. It uses quantum bits (qubits) and quantum gates to do calculations in a very different way, based on the rules of quantum physics. Because of this change in thinking, quantum computers might be able to solve some numerical linear algebra problems a lot faster than traditional computers. The outline then talks about some of the most important quantum computing methods and algorithms that are made for numerical linear algebra problems. There are quantum versions of standard algorithms like matrix multiplication, Gaussian elimination, and singular value decomposition. There are also new algorithms that are intended to use quantum parallelism and interference to speed up computations. For situations where full-scale quantum gear is not yet available, quantum-inspired methods are also talked about. These blend conventional and quantum techniques. The abstract talks about the difficulties and chances of using quantum algorithms for numerical linear algebra on quantum hardware systems that are available now and in the near future. It talks about things like error correction, qubit coherence times, and making efficient quantum circuit implementations. Lastly, the abstract talks about how quantum computing might change computational mathematics. It imagines that advances in quantum computing could help solve numerical linear algebra problems that were previously impossible to solve. This could have huge effects on many fields, from machine learning and optimization to quantum chemistry and cryptography. In the end, it stresses how important it is for mathematicians, computer scientists, physicists, and engineers to work together across disciplines in order to fully utilize the transformative power of quantum computing in numerical linear algebra and other areas.

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.

The Application of the Bidiagonal Factorization of Totally Positive Matrices in Numerical Linear Algebra

The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered in this review article, the starting point being the classical Björck–Pereyra algorithms for Vandermonde systems, published in 1970 and carefully analyzed by Higham in 1987. The work of Higham briefly considered the role of total positivity in obtaining accurate results, which led to the generalization of this approach to totally positive Cauchy, Cauchy–Vandermonde and generalized Vandermonde matrices. Then, the solution of other linear algebra problems (eigenvalue and singular value computation, least squares problems) is addressed, a fundamental tool being the bidiagonal decomposition of the corresponding matrices. This bidiagonal decomposition is related to the theory of Neville elimination, although for achieving high relative accuracy the algorithm of Neville elimination is not used. Numerical experiments showing the good behavior of these algorithms when compared with algorithms that ignore the matrix structure are also included.

Quantum Communication Complexity of Linear Regression

Quantum computers may achieve speedups over their classical counterparts for solving linear algebra problems. However, in some cases—such as for low-rank matrices—dequantized algorithms demonstrate that there cannot be an exponential quantum speedup. In this work, we show that quantum computers have provable polynomial and exponential speedups in terms of communication complexity for some fundamental linear algebra problems if there is no restriction on the rank. We mainly focus on solving linear regression and Hamiltonian simulation. In the quantum case, the task is to prepare the quantum state of the result. To allow for a fair comparison, in the classical case, the task is to sample from the result. We investigate these two problems in two-party and multiparty models, propose near-optimal quantum protocols, and prove quantum/classical lower bounds. In this process, we propose an efficient quantum protocol for quantum singular value transformation, which is a powerful technique for designing quantum algorithms. We feel this will be helpful in developing efficient quantum protocols for many other problems.

Total positivity and high relative accuracy for several classes of Hankel matrices

SummaryGramian matrices with respect to inner products defined for Hilbert spaces supported on bounded and unbounded intervals are represented through a bidiagonal factorization. It is proved that the considered matrices are strictly totally positive Hankel matrices and their catalecticant determinants are also calculated. Using the proposed representation, the numerical resolution of linear algebra problems with these matrices can be achieved to high relative accuracy. Numerical experiments are provided, and they illustrate the excellent results obtained when applying the theoretical results.

Evaluating ChatGPT-Generated Linear Algebra Formative Assessments

This research explored Large Language Models potential uses on formative assessment for mathematical problem-solving process. The study provides a conceptual analysis of feedback and how the use of these models is related in the context of formative assessment for Linear Algebra problems. Particularly, the performance of a popular model known as ChatGPT in mathematical problems fails on reasoning, proofs, model construction, among others. Formative assessment is a process used by teachers and students during instruction that provides feedback to adjust ongoing teaching and learning to improve student’s achievement of intended instructional outcomes. The study analyzed and evaluated feedback provided to engineering students in their solutions, from both, instructors and ChatGPT, against fine-grained criteria of a formative feedback model that includes affective aspects. Considering preliminary outputs, and to improve performance of feedback from both agents’ instructors and ChatGPT, we developed a framework for formative assessment in mathematical problemsolving using a Large Language Model (LLM). We designed a framework to generate prompts, supported by common Linear Algebra mistakes within the context of concept development and problem-solving strategies. In this framework, the instructor acts as an agent to verify tasks in a math problem assigned to students, establishing a virtuous cycle of learning of queries supported by ChatGPT. Results revealed potentialities and challenges on how to improve feedback on graduate-level math problems, by which both educators and students adapt teaching and learning strategies.

Profile of Student Metacognition in Solving Elementary Linear Algebra Problems Viewed from Tempo Conceptual Cognitive Style

Metacognition is one of the factors that influences students' success in solving mathematical problems. On the other hand, the reality in the field is that students experience difficulties in solving mathematical problems related to Elementary Linear Algebra because they contain a lot of repetitive calculations, so they require high levels of perseverance and accuracy. This research aims to describe students' metacognitive profiles in solving Elementary Linear Algebra problems in terms of cognitive style. The cognitive style chosen is the conceptual cognitive style of pacing through Matching Familiar Figures Tests with research subjects being mathematics students at Nahdlatul Ulama University, Purwokerto. The resulting metacognition profile was obtained from Elementary Linear Algebra problem-solving test data and interviews, which were analyzed and concluded through a series of method triangulation processes. The research results show that students are reflective and impulsive in carrying out all activities that are indicators of metacognitive awareness and regulation, as well as several indicators from metacognitive evaluation. Fast, accurate students carry out all activities that become indicators of metacognitive awareness, regulation, and evaluation. Meanwhile, slowly, inaccurate students carry out all activities that are indicators of metacognitive awareness and several indicators of metacognitive regulation but still need to be able to carry out activities of metacognitive evaluation.

FEATURES OF THE APPLICATION OF THE SCILAB MATHEMATICAL PACKAGE IN THE PROCESS OF TEACHING THE DISCIPLINE «HIGHER MATHEMATICS»

The article notes that the choice of software is of great importance in teaching the discipline «Higher Mathematics». The complexity and type of tasks to be solved depend on the functionality of the selected program. It is emphasised that the methods of approximate, primarily numerical, solution are used. The methods of numerical solution of mathematical problems have always been an integral part of mathematics and have always been part of the content of natural-mathematical and engineering education. This article uses the Skilab math package as software. It is a multi-functional package designed for performing engineering and scientific calculations, which allows you to perform complex algebraic calculations, solve problems of differentiation and integration. All this contributes to the successful use of the Scilab mathematical package in the process of teaching the discipline «Higher Mathematic». The author draws attention to the fact that Skilab is a computer mathematics system designed to perform engineering and scientific calculations, such as: solving nonlinear equations and systems; solving linear algebra problems; solving optimization tasks; differentiation and integration; processing of experimental data (interpolation and approximation, method of least squares); solution of ordinary differential equations and systems. Skilab provides wide opportunities for creating and editing various types of graphs and surfaces. The Skilab system contains a sufficient number of built-in commands, operators and functions, and its distinctive feature is flexibility. The user can create any new command or function and then use it alongside the built-in ones. The article states that the use of modern mathematical packages significantly increases the intensity of cognitive activity, raises the level of mathematical preparation, improves the knowledge control system, and promotes learning motivation. The perspective of the development of the study of the use of mathematical packages is the development of a database on the issues of their use during the educational process, which should take into account the professional orientation of future activities. This approach provides a more effective application of interdisciplinary connections, which contributes to the in-depth study of the material and the expansion of opportunities for independent learning.

Robust linear algebra

We propose a robust optimization (RO) framework that immunizes some of the central linear algebra problems in the presence of data uncertainty. Namely, we formulate linear systems, matrix inversion, eigenvalues–eigenvectors and matrix factorization under uncertainty, as robust optimization problems using appropriate descriptions of uncertainty. The resulting optimization problems are computationally tractable and scalable. We show in theory that RO improves the relative error of the linear system by reducing the condition number of the underlying matrix. Moreover, we provide empirical evidence showing that the proposed approach outperforms state of the art methods for linear systems and matrix inversion, when applied on ill-conditioned matrices. We show that computing eigenvalues–eigenvectors under RO, corresponds to solving linear systems that are better conditioned than the nominal and illustrate with numerical experiments that the proposed approach is more accurate than the nominal, when perturbing ill-conditioned matrices. Finally, we demonstrate empirically the benefit of the robust Cholesky factorization over the nominal.

Alternative to mean and least squares methods used in processing the results of scientific and technical experiments

Increasing the complexity and size of systems of various nature requires constant improvement of modeling and verification of the obtained results by experiment. It is possible to clearly conduct each experiment, objectively evaluate the summaries of the researched process, and spread the material obtained in one study to a series of other studies only if they are correctly set up and processed. On the basis of experimental data, algebraic expressions are selected, which are called empirical formulas, which are used if the analytical expression of some function is complex or does not exist at this stage of the description of the object, system or phenomenon. When selecting empirical formulas, polynomials of the form: у = А0 + А1х+ А2х2+ А3х3+…+ Аnхn are widely used, which can be used to approximate any measurement results if they are expressed as continuous functions. It is especially valuable that even if the exact expression of the solution (polynomial) is unknown, it is possible to determine the value of the coefficients An using the methods of mean and least squares. But in the method of least squares, there is a shift in estimates when the noise in the data is increased, as it is affected by the noise of the previous stages of information processing. Therefore, for real-time information processing procedures, a pseudo-reverse operation is proposed, which is performed using recurrent formulas. This procedure is a procedure of successive updating (with a shift) along the columns of the matrix of given sizes and pseudo-reversal at each step of information change. This approach is straightforward and takes advantage of the bounding method. With pseudo-inversion, it is possible to control the correctness of calculations at each step, using Penrose conditions. The need for pseudo-inversion may arise during optimization, forecasting of certain parameters and characteristics of systems of various purposes, in various problems of linear algebra, statistics, presentation of the structure of the obtained solutions, to understand the content of the incorrectness of the resulting solution, in the sense of Adomar-Tikhonov, and to see the ways of regularization of such solutions.

Some Comments about Zero and Non-Zero Eigenvalues from Connected Undirected Planar Graph Adjacency Matrices

Two linear algebra problems implore a solution to them, creating the themes pursued in this paper. The first problem interfaces with graph theory via binary 0-1 adjacency matrices and their Laplacian counterparts. More contemporary spatial statistics/econometrics applications motivate the second problem, which embodies approximating the eigenvalues of massively large versions of these two aforementioned matrices. The proposed solutions outlined in this paper essentially are a reformulated multiple linear regression analysis for the first problem and a matrix inertia refinement adapted to existing work for the second problem.

PANSATZ: pulse-based ansatz for variational quantum algorithms

Quantum computers promise a great computational advantage over classical computers, which might help solve various computational challenges such as the simulation of complicated quantum systems, finding optimum in large optimization problems, and solving large-scale linear algebra problems. Current available quantum devices have only a limited amount of qubits and a high level of noise, limiting the size of problems that can be solved accurately with those devices. Variational quantum algorithms (VQAs) have emerged as a leading strategy to address these limitations by optimizing cost function based on measurement results of shallow depth circuits. Recently, various pulse engineering methods were suggested in order to improve VQA results, including optimizing pulse parameters instead of gate angles as part of the VQA optimization process. In this paper, we suggest a novel pulse-based ansatz, which is parameterized mainly by pulses’ duration of pre-defined pulse structures. This ansatz structure provides relatively low amounts of optimization parameters while maintaining high expressibility, allowing fast convergence. In addition, the ansatz has structured adaptivity to the entanglement level required by the problem, allowing low noise and accurate results. We tested this ansatz against quantum chemistry problems. Specifically, finding the ground-state energy associated with the electron configuration problem, using the variational quantum eigensolver (VQE) algorithm for several different molecules. We manage to achieve chemical accuracy both in simulation for several molecules and on one of IBM’s NISQ devices for the H2 molecule in the STO-3G basis, without the need for extensive error mitigation. Our results are compared to a common gate-based ansatz and show better accuracy and significant latency reduction—up to 7× shorter ansatz schedules.

On simultaneous similarity of families of commuting operators

Characterization of simultaneous similarity for commuting m m - tuples of operators is an open problem even in finite-dimensional spaces; known as “A wild problem in linear algebra”. In this paper we offer a criterion for simultaneous similarity of m m -tuples of k k -cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found by Shekhtman [Math. Stat. 1 (2013), pp. 157–161] for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for m m -tuples of commuting square matrices.

Numerical Solution of Nonlinear Backward Stochastic Volterra Integral Equations

This work uses the collocation approximation method to solve a specific type of backward stochastic Volterra integral equations (BSVIEs). Using Newton’s method, BSVIEs can be solved using block pulse functions and the corresponding stochastic operational matrix of integration. We present examples to illustrate the estimate analysis and to demonstrate the convergence of the two approximating sequences separately. To measure their accuracy, we compare the solutions with values of exact and approximative solutions at a few selected locations using a specified absolute error. We also propose an efficient method for solving a triangular linear algebraic problem using a single integral equation. To confirm the effectiveness of our method, we conduct numerical experiments with issues from real-world applications.

Thick-restarted joint Lanczos bidiagonalization for the GSVD

The computation of the partial generalized singular value decomposition (GSVD) of large-scale matrix pairs can be approached by means of iterative methods based on expanding subspaces, particularly Krylov subspaces. We consider the joint Lanczos bidiagonalization method, and analyze the feasibility of adapting the thick restart technique that is being used successfully in the context of other linear algebra problems. Numerical experiments illustrate the effectiveness of the proposed method. We also compare the new method with an alternative solution via equivalent eigenvalue problems, considering accuracy as well as computational performance. The analysis is done using a parallel implementation in the SLEPc library.