Band Matrix Research Articles

Complexity measure diagnostics of ergodic to many-body localization transition

We introduce diagnostics of the transition between the ergodic and many-body localization phases, which are based on complexity measures defined via the probability distribution function of the Lanczos coefficients of the tridiagonalized Hamiltonian. We use these complexity measures to analyze the power-law random banded matrix model as a function of the correlation strength and show that the moments and the entropy of the distribution diagnose the ergodic to many-body transition, as well as the distinctive feature of the phases concerning the memory of the initial conditions. Published by the American Physical Society 2024

Variational quantum algorithm for the Poisson equation based on the banded Toeplitz systems

Abstract For solving the Poisson equation it is usually possible to discretize it into solving the corresponding linear system $Ax=b$. Variational quantum algorithms (VQAs) for \textcolor{red}{the discreted} Poisson equation have been studied before. We give a VQA based on the banded Toeplitz systems for solving the Poisson equation with respect to the structural features of matrix $A$. In detail, we decompose the matrix $A$ and $A^2$ into a linear combination of the corresponding banded Toeplitz matrix and sparse matrices with only a few non-zero elements. For the one-dimensional Poisson equation with different boundary conditions and the $d$-dimensional Poisson equation with Dirichlet boundary conditions, \textcolor{red}{the number of decomposition terms is less than the work in [Phys. Rev. A108, 032418 (2023)]}. Based on the decomposition of the matrix, we design quantum circuits that \textcolor{red}{evaluate efficiently} the cost function. \textcolor{red}{Additionally, numerical simulation verifies the feasibility of the proposed algorithm. In the end,} the VQAs for linear systems of equations and matrix–vector multiplications with $K$-banded Teoplitz matrix $T_n^K$ \textcolor{red}{are} given, where $T_n^K\in R^{n\times n}$ and $K\in O(ploy\log n)$.

Non-Hermitian diluted banded random matrices: Scaling of eigenfunction and spectral properties

Non-Hermitian diluted banded random matrices: Scaling of eigenfunction and spectral properties

Magnetic instability and spin-glass order beyond the Anderson-Mott transition in interacting power-law random banded matrix fermions

Magnetic instability and spin-glass order beyond the Anderson-Mott transition in interacting power-law random banded matrix fermions

Orlicz Extension of New Sequence Spaces Engendered by the Composition of Binomial Matrix and Double Band Matrix

The present paper is emphasis on introducing Orlicz extension of new sequence spaces (i.e br,s0(M,G), br,sc(M,G) and br,s∞(M,G)) by way of the composition of binomial matrix and double band matrix, which are BK-spaces, moreover we prove that these spaces are linearly isomorphic to the spaces l∞, c0 and c. We also derive some inclusion relations. Additionally, we find the Schauder basis for these spaces and finally we also determine the α−, β− and γ− duals of these spaces.

Mapping the Slow Stabilization of End States with Length along a Laterally Extended Graphene Nanoribbon.

With a lateral bisnaphtho-extended chemical structure, finite 7-13 carbon atom wide armchair graphene nanoribbons (7-13-aGNRs) were on-surface synthesized. For all lengths up to N = 7 monomer units, low-temperature ultrahigh vacuum scanning tunneling spectroscopy and spatial dI/dV maps were recorded at each captured tunneling resonance. The degeneracy of the two central electronic end states (ESs) occurs in a slowly decaying regime with N converging toward zero for N = 6 long 7-13-aGNR (12 bonded anthracenes), while it is N = 2 (4 bonded anthracenes) for seven carbon atoms wide armchair GNRs (7-aGNRs). The two end dI/dV conductance maxima of ESs are also shifted away from strictly two ends of the 7-13-aGNR compared to the 7-aGNR. Using the quantum topology graph filiation between finite length polyacetylene and 7-13-aGNRs wires, we show that this slow decay of 7-13-aGNR ESs is coming from the property of the topological Hückel band matrix that expels the ESs into its eigenvalue spectrum gaps to keep harmony in the core spectrum.

Band Matrices, Markov Chains, and the Reliability of Engineering Systems

Band Matrices, Markov Chains, and the Reliability of Engineering Systems

Polynomial and Rational Measure Modifications of Orthogonal Polynomials via Infinite-Dimensional Banded Matrix Factorizations

AbstractWe describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via infinite-dimensional banded matrix factorizations and may be used to compute the modified Jacobi matrices all in linear complexity with respect to the truncation degree. A family of orthogonal polynomials with modified classical weights is constructed that support banded differentiation matrices, enabling sparse spectral methods with modified classical orthogonal polynomials. We present several applications and numerical experiments using an open source implementation which make direct use of these results.

A high-dimensional additive nonparametric model

Nonparametric additive models are garnering increasing attention in applied research across fields like statistics and economics, attributed to their distinct interpretability, versatility, and their adeptness at addressing the curse of dimensionality. This paper introduces a novel and efficient fully Bayesian method for estimating nonparametric additive models, employing a band matrix smoothness prior. Our methodology leverages unobserved binary indicator parameters, promoting linearity in each additive component while allowing for deviations from it. We validate the efficacy of our approach through experiments on synthetic data derived from ten-component additive models, encompassing diverse configurations of linear, nonlinear, and zero function components. Additionally, the robustness of our algorithm is tested on high-dimensional models featuring up to one hundred components, and models correlated components. The practical utility and computational efficiency of our technique are further underscored by its application to two real-world datasets, showcasing its broad applicability and effectiveness in various scenarios.

Numerical analysis of fractional order Black–Scholes option pricing model with band equation method

In recent years, there has been a surge of interest in fractional calculus, primarily due to its enhanced capability in accurately modeling complex systems compared to classical methods. This study delves into the fractional Black–Scholes option pricing model, which adeptly captures the intricate dynamics and memory effects inherent in financial markets. The novelty of this research lies in introducing a novel approach employing band matrix equations for numerically solving the fractional order Black–Scholes option pricing model. We explore the numerical solutions for fractional European options using three distinct methods: Laplace Transform, Hilbert Transform, and Monte Carlo simulation. Each method’s convergence is rigorously proven, and their respective convergence orders are derived. Furthermore, we conduct a comprehensive analysis to discern the factors influencing the convergence rates for each method. Additionally, we observe the effectiveness and efficiency of the proposed methods in solving fractional European options through numerical examples. Our findings offer an in-depth insight into the comparative performance of Laplace transform, Hilbert transform, and Monte Carlo simulation in terms of accuracy, computational complexity, and stability. This research contributes significantly to the comprehension of fractional European option pricing and provides valuable guidance for both practitioners and researchers in selecting the most suitable method for pricing these intricate financial instruments.

An attention mechanism module with spatial perception and channel information interaction

In the field of deep learning, the attention mechanism, as a technology that mimics human perception and attention processes, has made remarkable achievements. The current methods combine a channel attention mechanism and a spatial attention mechanism in a parallel or cascaded manner to enhance the model representational competence, but they do not fully consider the interaction between spatial and channel information. This paper proposes a method in which a space embedded channel module and a channel embedded space module are cascaded to enhance the model’s representational competence. First, in the space embedded channel module, to enhance the representational competence of the region of interest in different spatial dimensions, the input tensor is split into horizontal and vertical branches according to spatial dimensions to alleviate the loss of position information when performing 2D pooling. To smoothly process the features and highlight the local features, four branches are obtained through global maximum and average pooling, and the features are aggregated by different pooling methods to obtain two feature tensors with different pooling methods. To enable the output horizontal and vertical feature tensors to focus on different pooling features simultaneously, the two feature tensors are segmented and dimensionally transposed according to spatial dimensions, and the features are later aggregated along the spatial direction. Then, in the channel embedded space module, for the problem of no cross-channel connection between groups in grouped convolution and for which the parameters are large, this paper uses adaptive grouped banded matrices. Based on the banded matrices utilizing the mapping relationship that exists between the number of channels and the size of the convolution kernels, the convolution kernel size is adaptively computed to achieve adaptive cross-channel interaction, enhancing the correlation between the channel dimensions while ensuring that the spatial dimensions remain unchanged. Finally, the output horizontal and vertical weights are used as attention weights. In the experiment, the attention mechanism module proposed in this paper is embedded into the MobileNetV2 and ResNet networks at different depths, and extensive experiments are conducted on the CIFAR-10, CIFAR-100 and STL-10 datasets. The results show that the method in this paper captures and utilizes the features of the input data more effectively than the other methods, significantly improving the classification accuracy. Despite the introduction of an additional computational burden (0.5 M), however, the overall performance of the model still achieves the best results when the computational overhead is comprehensively considered.

Computation of quasiseparable representations of Green matrices

The well-known Asplund theorem states that the inverse of a (possibly one-sided) band matrix A is a Green matrix. In accordance with quasiseparable theory, such a matrix admits a quasiseparable representation in its rank-structured part. Based on this idea, we derive algorithms that compute a quasiseparable representation of A−1 with linear complexity.Many inversion algorithms for band matrices exist in the literature. However, algorithms based on a computation of the rank structure performed theoretically via the Asplund theorem appear for the first time in this paper. Numerical experiments confirm complexity estimates and offer insight into stability properties.

Bulk universality and quantum unique ergodicity for random band matrices in high dimensions

Bulk universality and quantum unique ergodicity for random band matrices in high dimensions

Chiral random band matrices at zero energy

Chiral random band matrices at zero energy

Toda and Laguerre–Freud equations and tau functions for hypergeometric discrete multiple orthogonal polynomials

In this paper, the authors investigate the case of discrete multiple orthogonal polynomials with two weights on the step line, which satisfy Pearson equations. The discrete multiple orthogonal polynomials in question are expressed in terms of τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-functions, which are double Wronskians of generalized hypergeometric series. The shifts in the spectral parameter for type II and type I multiple orthogonal polynomials are described using banded matrices. It is demonstrated that these polynomials offer solutions to multicomponent integrable extensions of the nonlinear Toda equations. Additionally, the paper characterizes extensions of the Nijhoff–Capel totally discrete Toda equations. The hypergeometric τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-functions are shown to provide solutions to these integrable nonlinear equations. Furthermore, the authors explore Laguerre–Freud equations, nonlinear equations for the recursion coefficients, with a particular focus on the multiple Charlier, generalized multiple Charlier, multiple Meixner II, and generalized multiple Meixner II cases.

The variable two-step BDF method for parabolic equations

The two-step backward difference formula (BDF) method on variable grids for parabolic equations with self-adjoint elliptic part is considered. Standard stability estimates for adjacent time-step ratios rj:=kj/kj-1⩽1.8685\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_j:=k_j/k_{j-1}\\leqslant 1.8685$$\\end{document} and 1.9104, respectively, have been proved by Becker (BIT 38:644–662, 1998) and Emmrich (J Appl Math Comput 19:33–55, 2005) by the energy technique with a single multiplier. Even slightly improving the ratio is cumbersome. In this paper, we present a novel technique to examine the positive definiteness of banded matrices that are neither Toeplitz nor weakly diagonally dominant; this result can be viewed as a variant of the Grenander–Szegő theorem. Then, utilizing the energy technique with two multipliers, we establish stability for adjacent time-step ratios up to 1.9398.

Hybrid cubic and hyperbolic b-spline collocation methods for solving fractional Painlevé and Bagley-Torvik equations in the Conformable, Caputo and Caputo-Fabrizio fractional derivatives

In this paper, we approximate the solution of fractional Painlevé and Bagley-Torvik equations in the Conformable (Co), Caputo (C), and Caputo-Fabrizio (CF) fractional derivatives using hybrid hyperbolic and cubic B-spline collocation methods, which is an extension of the third-degree B-spline function with more smoothness. The hybrid B-spline function is flexible and produces a system of band matrices that can be solved with little computational effort. In this method, three parameters m, η, and λ play an important role in producing accurate results. The proposed methods reduce to the system of linear or nonlinear algebraic equations. The stability and convergence analysis of the methods have been discussed. The numerical examples are presented to illustrate the applications of the methods and compare the computed results with those obtained using other methods.

Adaptive Tests for Bandedness of High-dimensional Covariance Matrices

Estimation of the high-dimensional banded covariance matrix is widely used in multivariate statistical analysis. To ensure the validity of estimation, we aim to test the hypothesis that the covariance matrix is banded with a certain bandwidth under the high-dimensional framework. Though several testing methods have been proposed in the literature, the existing tests are only powerful for some alternatives with certain sparsity levels, whereas they may not be powerful for alternatives with other sparsity structures. The goal of this paper is to propose a new test for the bandedness of high-dimensional covariance matrix, which is powerful for alternatives with various sparsity levels. The proposed new test also be used for testing the banded structure of covariance matrices of error vectors in high-dimensional factor models. Based on these statistics, a consistent bandwidth estimator is also introduced for a banded high dimensional covariance matrix. Extensive simulation studies and an application to a prostate cancer dataset from protein mass spectroscopy are conducted for evaluating the effectiveness of the proposed adaptive tests blue and bandwidth estimator for the banded covariance matrix.

A smoothing spline algorithm to interpolate and predict the eigenvalues of matrices extracted from the sequence of preconditioned banded symmetric Toeplitz matrices

<abstract><p>Understanding the eigenvalue distribution of sequence Toeplitz matrices has advanced significantly in recent years. Notable contributors include Bogoya, Grudsky, Böttcher, and Maximenko, who have derived precise asymptotic expansions for these eigenvalues under certain conditions related to the generating function as the matrix size increases. Building on this foundation, the Stefano Serra-Capizzano conjectured that, under certain assumptions about $ \Omega $ and $ \Phi $, a similar expansion may hold for the eigenvalues of a sequence of preconditioned Toeplitz matrices $ T_{n}^{-1}(\Phi) T_n(\Omega) $, given a monotonic ratio $ r = \Omega/\Phi $. In contrast to current eigenvalue solvers, this work presents a novel method for efficiently calculating the eigenvalues of a sequence of large preconditioned banded symmetric Toeplitz matrices (PBST). Our algorithm uses a higher-order spline fitting extrapolation technique to gather spectral data from a smaller sequence of PBST matrices in order to forecast the spectrum of bigger matrices.</p></abstract>

Random band and block matrices with correlated entries

Random band and block matrices with correlated entries