Strict Diagonal Dominance Research Articles

INTERNET OF THINGS-BASED WIRELESS SENSOR NETWORK SYSTEM FOR EARLY DETECTION AND PREVENTION OF VANDALISM/LEAKAGE ON PIPELINE INSTALLATIONS IN THE OIL AND GAS INDUSTRY IN NIGERIA

Solving linear systems of equations stands as one of the fundamental challenges in linear algebra, given their prevalence across various fields. The demand for an efficient and rapid method capable of addressing diverse linear systems remains evident. In scenarios involving large and sparse systems, iterative techniques come into play to deliver solutions. This research paper contributes by introducing a refinement to the existing Jacobi method, referred to as the "Third Refinement of Jacobi Method." This novel iterative approach exhibits its validity when applied to coefficient matrices exhibiting characteristics such as symmetry, positive definiteness, strict diagonal dominance, and -matrix properties. Importantly, the proposed method significantly reduces the spectral radius, thereby curtailing the number of iterations and substantially enhancing the rate of convergence. Numerical experiments were conducted to assess its performance against the original Jacobi method, the second refinement of Jacobi, and the Gauss-Seidel method. The outcomes underscore the "Third Refinement of Jacobi" method's potential to enhance the efficiency of linear system solving, thereby making it a valuable addition to the toolkit of numerical methodologies in scientific and engineering domains.

A THIRD REFINEMENT OF JACOBI METHOD FOR SOLUTIONS TO SYSTEM OF LINEAR EQUATIONS

Solving linear systems of equations stands as one of the fundamental challenges in linear algebra, given their prevalence across various fields. The demand for an efficient and rapid method capable of addressing diverse linear systems remains evident. In scenarios involving large and sparse systems, iterative techniques come into play to deliver solutions. This research paper contributes by introducing a refinement to the existing Jacobi method, referred to as the "Third Refinement of Jacobi Method." This novel iterative approach exhibits its validity when applied to coefficient matrices exhibiting characteristics such as symmetry, positive definiteness, strict diagonal dominance, and -matrix properties. Importantly, the proposed method significantly reduces the spectral radius, thereby curtailing the number of iterations and substantially enhancing the rate of convergence. Numerical experiments were conducted to assess its performance against the original Jacobi method, the second refinement of Jacobi, and the Gauss-Seidel method. The outcomes underscore the "Third Refinement of Jacobi" method's potential to enhance the efficiency of linear system solving, thereby making it a valuable addition to the toolkit of numerical methodologies in scientific and engineering domains.

Решение интегральных уравнений Фредгольма методом замены интеграла квадратурой с двенадцатым порядком погрешности в матричном виде

An algorithm for the numerical solution of the Fredholm equation of the second kind with a continuous kernel by the method of integral replacement and the matrix solution of SLAE with a quadrature formula of the twelfth order of error with a number of integration intervals divisible by ten is proposed. The new formula, compared to Simpson's formula, gives 15 significant digits for the nodal values of the solution function, even with a small number of intervals of 10.20 on a segment in a finite number of elementary operations. The resulting algorithm has double precision and minimal computation time. While Simpson's formula,together with the matrix method for solving SLAE, gives only 6 significant digits with twenty integration intervals. Moreover, double precision is not available for Simpson's formula (15 zeros in the infinite norm of the solution residual), since the FORTRAN language allows maximum matrix arrays of 200×200. Estimates are obtained for the upper bound of the admissible parameter |λ| for the matrix of the Fredholm equation with strict diagonal dominance or with a small norm of the integral kernel.

Β-matrices and β-tensors

In this manuscript we introduce the class of ?-matrices, which gives a new sufficient condition for the positivity of the determinant. However, we show that nonnegative ?-matrices are not necessarily P-matrices. For column stochastic matrices, the property of being a ?-matrix is weaker than strict diagonal dominance. We extend ?-matrices to tensors and call them ?-tensors. Although they are not in general P-tensors, we prove that nonnegative ?-tensors of odd order are P-tensors

Construction of an Effective Preconditioner for the Even-odd Splitting of Cubic Spline Wavelets

In this study, the method for decomposing splines of degree m and smoothness C^m-1 into a series of wavelets with zero moments is investigated. The system of linear algebraic equations connecting the coefficients of the spline expansion on the initial scale with the spline coefficients and wavelet coefficients on the embedded scale is obtained. The originality consists in the application of some preconditioner that reduces the system to a simpler band system of equations. Examples of applying the method to the cases of first-degree spline wavelets with two first zero moments and cubic spline wavelets with six first zero moments are presented. For the cubic case after splitting the system into even and odd rows, the resulting matrix acquires a seven-diagonals form with strict diagonal dominance, which makes it possible to apply an effective sweep method to its solution

On the seven-diagonals splitting for the cubic spline wavelets with six vanishing moments on an interval

This study uses a zeroing property of the first six moments for constructing a splitting algorithm for the cubic spline wavelets. First, we construct a system of cubic basic spline-wavelets, realizing orthogonal conditions to all polynomials up to any degree. Then, using the homogeneous Dirichlet boundary conditions, we adapt spaces to the orthogonality to all polynomials up to the fifth degree on the closed interval. The originality of the study consists of obtaining implicit finite relations connecting the coefficients of the spline decomposition at the initial scale with the spline coefficients and wavelet coefficients at the nested scale by a tape system of linear algebraic equations with a non-degenerate matrix. After excluding the even rows of the system, the resulting transformation matrix has seven diagonals, instead of five as in the previous case with four zero moments. A modification of the system is performed, which ensures a strict diagonal dominance, and, consequently, the stability of the calculations. The comparative results of numerical experiments on approximating and calculating the derivatives of a discrete function are presented.

A generalization of irreducibility and diagonal dominance with applications to horizontal and vertical linear complementarity problems

In this paper, we generalize and analyze the concepts of diagonal dominance and irreducibility in the framework of column and row representative matrices of a set. Our analysis includes the definition of particular sets of M- and H-matrices. We also analyze the form that the matrices of the introduced irreducible sets must have and the implications of the obtained results on the solution of vertical and horizontal linear complementarity problems. In this context, we prove that the projected Jacobi and the projected Gauss-Seidel methods for horizontal linear complementarity problems converge when the matrices of the problem satisfy one of the introduced generalizations of strict diagonal dominance or of irreducible diagonal dominance. This extends current convergence results.

An Algorithm with the Even-odd Splitting of the Wavelet Transform of Non-Hermitian Splines of the Seventh Degree, II

In this study, the zeroing property of the first two moments is used to construct an algorithm for splitting spline wavelets of the seventh degree. The presentation is based on the system of basic spline wavelets of the seventh degree, constructed in the previous article, which implements the conditions of orthogonality to all polynomials of any degree. Then, using homogeneous Dirichlet boundary conditions, the system is adapted to orthogonality to all polynomials up to the first degree on a finite interval. Implicit finite relationships are obtained between the spline coefficients in the original scale, on the one hand, and the spline coefficients and wavelet coefficients in the nested scale, on the other hand. After eliminating the even rows of the system, the transformation matrix has seven diagonals instead of five, as in the previous case studied. The resulting system has been modified to ensure strict diagonal dominance and, hence, computational stability, in contrast to the fivediagonal case.

Shifted Cubic Spline Wavelets with Two Vanishing Moments on the Interval and a Splitting Algorithm

This paper deals with the use of the first two vanishing moments for constructing cubic spline-wavelets orthogonal to polynomials of the first degree. A decrease in the supports of these wavelets is shown in comparison with the classical semiorthogonal wavelets. For splines with homogeneous Dirichlet boundary conditions of the second order, an algorithm of the shifted wavelet transform is obtained in the form of a solution of a tridiagonal system of linear equations with a strict diagonal dominance

Multistability and instability of competitive neural networks with non-monotonic piecewise linear activation functions

This paper addresses the issue of multistability for competitive neural networks. First, a general class of continuous non-monotonic piecewise linear activation functions is introduced. Then, based on the fixed point theorem, the contraction mapping theorem and the eigenvalue properties of strict diagonal dominance matrix, it is shown that under some conditions, such n-neuron competitive neural networks have exactly 5n equilibrium points, among which 3n equilibrium points are locally exponentially stable and the others are unstable. Moreover, it is revealed that the neural networks with non-monotonic piecewise linear activation functions introduced in this paper can have greater storage capacity than the ones with Mexican-hat-type activation function and nondecreasing saturated activation function. In addition, unlike most existing multistability results of neural networks with nondecreasing activation functions, the location of those obtained 3n locally stable equilibrium points in this paper is more flexible. Finally, a numerical example is provided to illustrate and validate the theoretical findings via comprehensive computer simulations.

Complete Stability of Neural Networks With Nonmonotonic Piecewise Linear Activation Functions

This brief studies the complete stability of neural networks with nonmonotonic piecewise linear activation functions. By applying the fixed-point theorem and the eigenvalue properties of the strict diagonal dominance matrix, some conditions are derived, which guarantee that such $n$ -neuron neural networks are completely stable. More precisely, the following two important results are obtained: 1) The corresponding neural networks have exactly $5^n$ equilibrium points, among which $3^n$ equilibrium points are locally exponentially stable and the others are unstable; 2) as long as the initial states are not equal to the equilibrium points of the neural networks, the corresponding solution trajectories will converge toward one of the $3^n$ locally stable equilibrium points. A numerical example is provided to illustrate the theoretical findings via computer simulations.

Multistability and Instability of Neural Networks With Discontinuous Nonmonotonic Piecewise Linear Activation Functions.

In this paper, we discuss the coexistence and dynamical behaviors of multiple equilibrium points for recurrent neural networks with a class of discontinuous nonmonotonic piecewise linear activation functions. It is proved that under some conditions, such n -neuron neural networks can have at least 5(n) equilibrium points, 3(n) of which are locally stable and the others are unstable, based on the contraction mapping theorem and the theory of strict diagonal dominance matrix. The investigation shows that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibrium points and more locally stable equilibrium points than the ones with continuous Mexican-hat-type activation function or discontinuous two-level activation functions. An illustrative example with computer simulations is presented to verify the theoretical analysis.

Dynamical Behaviors of Multiple Equilibria in Competitive Neural Networks With Discontinuous Nonmonotonic Piecewise Linear Activation Functions.

This paper addresses the problem of coexistence and dynamical behaviors of multiple equilibria for competitive neural networks. First, a general class of discontinuous nonmonotonic piecewise linear activation functions is introduced for competitive neural networks. Then based on the fixed point theorem and theory of strict diagonal dominance matrix, it is shown that under some conditions, such n -neuron competitive neural networks can have 5(n) equilibria, among which 3(n) equilibria are locally stable and the others are unstable. More importantly, it is revealed that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibria and locally stable equilibria than the ones with other activation functions, such as the continuous Mexican-hat-type activation function and discontinuous two-level activation function. Furthermore, the 3(n) locally stable equilibria given in this paper are located in not only saturated regions, but also unsaturated regions, which is different from the existing results on multistability of neural networks with multiple level activation functions. A simulation example is provided to illustrate and validate the theoretical findings.

Spectrum localizations for matrix operators on [formula omitted] spaces

We investigate localization of spectra of infinite matrices that can act as linear operators from the Banach space lp,1≤p≤∞, to itself. To that end, we propose a method that is based on generalization of a strict diagonal dominance of infinite matrices that is adapted to lp and lq norms. Results are illustrated with several examples, and some interesting applications are discussed.

Strict diagonal dominance in asymptotic stability of general equilibrium

Stability of general equilibrium is usually depicted by a dynamic process of price adjustment which makes the flow of prices eventually come to rest at certain prices, so that the supply and demand of every commodity tend to equal each other. Here we construct a dynamical system of a competitive economy and find that the strict diagonal dominance of the Jacobian matrix of the excess demand function at its equilibrium guarantees the asymptotic stability.MSC:37N40, 91B55, 93D20.

Cubic multiwavelets orthogonal to polynomials and a splitting algorithm

In this paper, an implicit method of decomposition of cubic Hermite splines using a new type of multiwavelets with supercompact supports is investigated. A splitting algorithm of wavelet-transforms of solving two three-diagonal systems of linear equations with strict diagonal dominance in parallel is justified. Results of some numerical experiments are presented.

Strict double diagonal dominance in Euclidean Jordan algebras

In the first part of the paper, we deal with Euclidean Jordan algebraic generalizations of some results of Brualdi on inclusion regions for the eigenvalues of complex matrices using directed graphs. As a consequence, the theorems of Brauer–Ostrowski and Brauer on the location of eigenvalues are extended to the setting of Euclidean Jordan algebras. In the second part, motivated by the work of Li and Tsatsomeros on the class of doubly diagonally dominant matrices with complex entries and its subclasses, we present some inter-relations between the H-property, generalized strict diagonal dominance, invertibility, and strict double diagonal dominance in Euclidean Jordan algebras. In addition, we show that in a Euclidean Jordan algebra, the Schur complements of a strictly doubly diagonally dominant element inherit this property.

Correction to “Strict Diagonal Dominance and Optimal Bounds for the Skeel Condition Number”

This note is to present a correction to the proof of inequality (0.1) of Theorem 0.1 in the paper “Strict diagonal dominance and optimal bounds for the Skeel condition number” [SIAM J. Numer. Anal., 45 (2007), pp. 1107–1108].

Strict diagonal dominance and a Geršgorin type theorem in Euclidean Jordan algebras

For complex square matrices, the Levy–Desplanques theorem asserts that a strictly diagonally dominant matrix is invertible. The well-known Geršgorin theorem on the location of eigenvalues is equivalent to this. In this article, we extend the Levy–Desplanques theorem to an object in a Euclidean Jordan algebra when its Peirce decomposition with respect to a Jordan frame is given. As a consequence, we prove a Geršgorin type theorem for the spectral eigenvalues of an object in a Euclidean Jordan algebra.

Refining Gerschgorin disks through new criteria for nonsingularity

AbstractNew criteria for nonsingular matrices are provided. These criteria generalize strict diagonal dominance and they are also applied to derive new inclusion sets for the eigenvalues of a matrix. Results on exclusion sets are also presented. Copyright © 2007 John Wiley & Sons, Ltd.