Pentadiagonal Toeplitz Research Articles

A novel numerical algorithm for solving linear systems with periodic pentadiagonal Toeplitz coefficient matrices

A novel numerical algorithm for solving linear systems with periodic pentadiagonal Toeplitz coefficient matrices

A complexity efficient penta-diagonal quantum smoothing filter for bio-medical signal denoising: a study on ECG

Extracting information-bearing signal from a noisy environment has been a practical challenge in both classical and quantum computing formalism, especially in critical signal processing applications. To filter out the effect of noise, we propose a quantum smoothing filter built upon quantum formalism-based circuits applied for electrocardiogram signal denoising. The proposed quantum filter is a conceptually novel framework with an advantage in computational complexity as compared to the existing classical filters, such as discrete wavelet transform and empirical mode decomposition, whereas it achieves similar performance metrics for the accuracy of the filter. Further, we exploit the penta-diagonal Toeplitz structure of the smoothing filter, which gives approximately 48%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$48\\%$$\\end{document} gate cost reduction for 10 qubit circuit compared to the standard Hamiltonian simulation without structure. The run-time complexity using the quantum matrix inversion technique for the structured matrix is given by O~κ2poly(logN)εP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ilde{{\\mathscr {O}}}\\left( \\frac{\\kappa ^2 \ ext {poly}(\\log {N})}{\\varepsilon _P}\\right)$$\\end{document} for condition number κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document} of the N×N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\ imes N$$\\end{document} filter matrix within precision εP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon _P$$\\end{document}. Embedding fixed sparsity of the banded matrix, the quantum filter shows potentially better run-time complexity than classical filtering techniques. For the quantifiable research results of our work, we have shown several performance metrics, such as mean-square error and peak signal-to-noise ratio analysis, with a bound of error due to observation noise, simulation error and quantum measurement uncertainty.

Exact Inversion of Pentadiagonal Matrix for Semi-Analytic Solution of 2D Poisson Equation

This work essentially consists in inverting in an exact, explicit, and original way the pentadiagonal Toeplitz matrix or tridiagonal block matrix resulting from the discretization of the two-dimensional Laplace operator. This method is an algorithm facilitating the resolution of a large number of problems governed by PDEs involving the Laplacian in two dimensions. It guarantees high precision and high efficiency in solving various differential equations.

The Explicit Formulae of the Determinants of Specific Pentadiagonal Toeplitz Hessenberg Matrices

The pentadiagonal Toeplitz matrix is a special kind of sparse matrix widely used in linear algebra, combinatorics, computational mathematics, and has been attracted much attention. We use the determinants of two specific Hessenberg matrices to represent the recurrence relations to prove two explicit formulae to evaluate the determinants of specific pentadiagonal Toeplitz matrices proposed in a recent paper [3]. Further, four new results are established.

Asymptotics of eigenvalues for Toeplitz matrices with rational symbols that have a minimum of the 4th order

In Barrera M, Grudsky SM. Asymptotics of eigenvalues for pentadiagonal symmetric Toeplitz matrices. In: Large truncated Toeplitz matrices, toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhäuser, Cham.; 2017; p. 51–77. we have considered the problem about asymptotic formulas for all eigenvalues of , as n goes to infinity, assuming that a is a specific model symbol with a unique zero of order 4. In this paper, we continue our investigation and we explore the case where a is a more general real-valued rational symbol with a unique zero of order 4. It should be noted that we apply a different method than the one used in Barrera M, Grudsky SM. Asymptotics of eigenvalues for pentadiagonal symmetric Toeplitz matrices. In: Large truncated Toeplitz matrices, Toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhäuser, Cham.; 2017; p. 51–77. This method coming from works Bogoya JM, Böttcher A, Grudsky SM, et al. Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols. J Math Anal Appl. 2015;422(2):1308–1334 and Bogoya JM, Böttcher A, Grudsky SM, et al. Eigenvalues of Hermitian Toeplitz matrices generated by simple-loop symbols with relaxed smoothness. In: Large truncated Toeplitz matrices, Toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhäuser, Cham.; 2017. p. 179–212, where it is considered the class of all symbols having zeros of second order and one can reduce the problem to asymptotic analysis of a nonlinear equation. As well, we construct uniform asymptotic expansions for all eigenvalues, which allow us to precise the classical results of Widom and Parter for first and very last eigenvalues.

A proof of Anđelić-Fonseca conjectures on the determinant of some Toeplitz matrices and their generalization

In this work, we present and prove the explicit formula for the determinant of a class of nonsymmetric Toeplitz matrices. Setting up one of the nonzero subdiagonals to zero results in special pentadiagonal Toeplitz matrices, whose determinant formulas are conjectured by Anđelić and Fonseca in Anđelić [Some determinantal considerations for pentadiagonal matrices. Linear Multilinear Algebra. 2020. DOI:10.1080/03081087.2019.1708845]. By using the explicit formula with this setting, the conjectures are therefore proved.

Covariance regularization for metabolomic data on the drought resistance of barley

Summary Modern chromatography largely uses the technique of gas chromatography coupled with mass spectrometry (GC–MS). For a set of data concerning the drought resistance of barley, the problem of the characterization of a covariance structure is investigated with the use of two methods. The first is based on the Frobenius norm and the second on the entropy loss function. For the four considered covariance structures – compound symmetry, three-diagonal and penta-diagonal Toeplitz and autoregression of order one – the Frobenius norm indicates the compound symmetry matrix and autoregression of order one as the most relevant, whilst the entropy loss function gives a slight indication in favor of the compound symmetry structure.

Computational properties of pentadiagonal and anti-pentadiagonal block band matrices with perturbed corners

In this paper, we derive an orthogonal block diagonalization and a number of formulas for pentadiagonal block band symmetric Toeplitz matrices and anti-pentadiagonal block band persymmetric Hankel matrices, both with perturbed corners. Namely, our formulas include block diagonalization, inverse, eigenvalues of these matrices. Our approach uses a suitable modification technique for pentadiagonal block band symmetric Toeplitz matrices and anti-pentadiagonal block band persymmetric Hankel matrices. Some numerical experiments are also presented to demonstrate the performance and effectiveness of the proposed theorems.

On a structure-preserving matrix factorization for the determinants of cyclic pentadiagonal Toeplitz matrices

In recent years, a number of numerical algorithms of O(n) for computing the determinants of cyclic pentadiagonal matrices have been developed. In this paper, a cost-efficient numerical algorithm for the determinant of an n-by-n cyclic pentadiagonal Toeplitz matrix is proposed whose computational cost is estimated at $$O(\log n)$$ . The algorithm is based on a structure-preserving matrix factorization and a three-term recurrence relation. We provide some numerical results with simulations in Matlab implementation in order to demonstrate the accuracy and effectiveness of the proposed algorithm, and its competitiveness with other existing algorithms.

On formulae for the determinant of symmetric pentadiagonal Toeplitz matrices

We show that the characteristic polynomial of a symmetric pentadiagonal Toeplitz matrix is the product of two polynomials given explicitly in terms of the Chebyshev polynomials.

An efficient numerical algorithm for the determinant of a cyclic pentadiagonal Toeplitz matrix

In the past few years, a number of numerical and symbolic algorithms for evaluating the determinants of cyclic pentadiagonal matrices have been developed. In this paper, we present a fast numerical algorithm for the determinant of an n-by-n cyclic pentadiagonal matrix with Toeplitz structure. The algorithm is based upon a certain type of matrix reordering and partitioning, and linear transformation. Some numerical examples are provided, and the results are compared with the ones obtained via Matlab built-in function and two existing algorithms. All of the numerical experiments are performed on a computer with the aid of programs written in Matlab.

On determinants of cyclic pentadiagonal matrices with Toeplitz structure

Cyclic pentadiagonal matrices with Toeplitz structure have received tremendous attention in recent years. In the current paper, we present a block upper triangular transformation of the cyclic pentadiagonal Toeplitz matrices. By using the transformation, the determinant of an n-by-n cyclic pentadiagonal Toeplitz matrix can be readily evaluated since one just needs to compute the determinant of a 4-by-4 matrix obtained from the transformation. In addition, an efficient numerical algorithm of O(n) is derived for computing nth order cyclic pentadiagonal Toeplitz determinants. Some numerical experiments are given to show the performance of the proposed algorithm.

On a homogeneous recurrence relation for the determinants of general pentadiagonal Toeplitz matrices

Pentadiagonal Toeplitz matrices frequently arise in many application areas and have been attracted much attention in recent years. In this paper, we present a numerical algorithm of O(logn) for computing the determinants of general pentadiagonal Toeplitz matrices without imposing any restrictive conditions. In addition, we investigate some special pentadiagonal Toeplitz determinants and their relations to well-known number sequences, and give a few identity formulas for the ordinary Fibonacci sequences and the generalized k-Fibonacci sequences.

Numerical algorithm for the determinant evaluation of cyclic pentadiagonal matrices with Toeplitz structure

Very recently, an efficient computational algorithm (DETQPT algorithm) for the determinant evaluation of general cyclic pentadiagonal Toeplitz matrices has been proposed by Y.L. Jiang and J.T. Jia (J. Math. Chem. 51: 2503-2513, 2013). In this paper, an explicit formula for the determinant of a cyclic pentadiagonal Toeplitz matrix is derived at first. Then, we present a more efficient numerical algorithm with the cost of 7n+O(logn)$7n+O(\log \ n)$ for evaluating n-th order cyclic pentadiagonal Toeplitz determinants. The algorithm is based on the use of a certain type of matrix reordering and matrix partition, and equalities involving the products of special kind of relevant matrices. Three numerical examples demonstrate the performance and effectiveness of the proposed algorithm and its competitiveness with other already existing algorithms.

An explicit formula for the inverse of a pentadiagonal Toeplitz matrix

In this paper, we mainly consider finding an explicit formula for the inverse of a pentadiagonal Toeplitz matrix. For that purpose, we first factorize the modified form of a pentadiagonal Toeplitz matrix by two tridiagonal Toeplitz matrices, and then use the Sherman–Morrison–Woodbury inversion formula. As a result, an explicit inverse of a pentadiagonal Toeplitz matrix is obtained under certain assumptions. And numerical experiments are given to show the effectiveness of our results.

An algorithm for solving nonsymmetric penta-diagonal Toeplitz linear systems

Fast algorithms for solving symmetric penta-diagonal Toeplitz linear systems have been proposed in McNally (2010) [7], Nemani (2010) [8] and Xu (2010) [11]. In this paper, we discuss the general nonsymmetric problem and propose an algorithm for solving nonsymmetric penta-diagonal Toeplitz linear systems. Numerical examples are given to illustrate the efficiency of our algorithm.

A Note on Computing the Inverse and the Determinant of a Pentadiagonal Toeplitz Matrix in Signal Processing

A Note on Computing the Inverse and the Determinant of a Pentadiagonal Toeplitz Matrix in Signal Processing

A structure preserving matrix factorization for solving general periodic pentadiagonal Toeplitz linear systems

In the current article, a new algorithm for solving linear systems with the periodic pentadiagonal Toeplitz (PPT) coefficient matrix is presented. This algorithm is based on a structure preserving matrix factorization and Sherman–Morrison–Woodbury formula. Some numerical examples are given in order to illustrate the effectiveness of the proposed algorithms. All of the experiments are performed on a computer with the aid of programs written in MATLAB.

On the determinant evaluation of quasi penta-diagonal matrices and quasi penta-diagonal Toeplitz matrices

Recently, three computational algorithms for evaluating the determinant of quasi penta-diagonal matrices have been proposed by El-Mikkawy and Rahmo (Comput Math Appl 59:1386–1396, 2010), by Neossi Nguetchue and Abelman (Appl Math Comput 203:629–634, 2008), and by Jia et al. (Int J Comput Math 89:851–860, 2013), respectively. In the current paper, two novel algorithms with less computational costs are proposed for the determinant evaluation of general quasi penta-diagonal matrices and quasi penta-diagonal Toeplitz matrices. Furthermore, three numerical experiments are given to show the performance of our algorithms. All of the numerical computations were performed on a computer with aid of programs written in MATLAB.

A New Algorithm for Computing the Determinant and the Inverse of a Pentadiagonal Toeplitz Matrix

An effective numerical algorithm for computing the determinant of a pentadiagonal Toeplitz matrix has been proposed by Xiao-Guang Lv and others [1]. The complexity of the algorithm is (9n + 3). In this paper, a new algorithm with the cost of (4n + 6) is presented to compute the determinant of a pentadiagonal Toeplitz matrix. The inverse of a pentadiagonal Toeplitz matrix is also considered.