Abstract: Several relative eigenvalue condition numbers that exploit tridiagonal form are derived. Some of them use triangular factorizations instead of the matrix entries and so they shed light on when eigenvalues are less sensitive to perturbations of factored forms than to perturbations of the matrix entries. A novel empirical condition number is used to show when perturbations are so large that the eigenvalue response is not linear. Some interesting examples are examined in detail.

Abstract: This paper provides an inverse formula freed of determinant expressions for a general tridiagonal matrix. This is viewed as an alternative version of the Usmani formula, which easily tends to blow up computationally. We discuss a number of different viewpoints regarding the proposed and Usmani’s formulas, such as the proof method and the meaning of included terms, although our formula itself may be obtained by a simple transformation of Usmani’s. A study of the limit elements based on the inverse formula and a numerical experiment for comparison with the other inverse methods are provided. In addition, we briefly discuss the inverse formula in the case of zero minors, which is illustrated by a numerical example. Mathematics subject classification (2010): 15A09, 15A06.

Abstract: SUMMARY A high-order alternating direction implicit (ADI) method for solving the unsteady convection-dominated diffusion equation is developed. The fourth-order Pade scheme is used for the discretization of the convection terms, while the second-order Pade scheme is used for the diffusion terms. The Crank–Nicolson scheme and ADI factorization are applied for time integration. After ADI factorization, the two-dimensional problem becomes a sequence of one-dimensional problems. The solution procedure consists of multiple use of a one-dimensional tridiagonal matrix algorithm that produces a computationally cost-effective solver. Von Neumann stability analysis is performed to show that the method is unconditionally stable. An unsteady two-dimensional problem concerning convection-dominated propagation of a Gaussian pulse is studied to test its numerical accuracy and compare it to other high-order ADI methods. The results show that the overall numerical accuracy can reach third or fourth order for the convection-dominated diffusion equation depending on the magnitude of diffusivity, while the computational cost is much lower than other high-order numerical methods. Copyright © 2011 John Wiley & Sons, Ltd.

Abstract: In this paper we present an application of second order homogeneous linear dif- ference equations with constant coefficients to evaluate the determinant of tridiagonal ma- trices. Comparing the obtained results with a certain alternative approach (1) some formulae for the finite sum are derived.

Abstract: In this paper, a novel linear equation solution method is proposed based on a row elimination back-substitution method (REBSM) The elimination and back-substitution procedures are carried out on individual row levels The advantage of the proposed method is that it is much faster and requires less storage than the Gaussian elimination algorithm and, therefore, is capable of solving larger systems of equations The method is particularly efficient for solving band diagonal sparse systems with symmetric or nonsymmetric coefficient matrices, and can be easily applied to popular numerical methods, such as the finite element method and the boundary element method Detailed Fortran codes and examples are given to demonstrate the robustness and efficiency of the proposed method

Abstract: In this paper, we presented an asymptotic fitted approach to solve singularly perturbed delay differential equations of second order with left and right boundary. In this approach, the singularly perturbed delay differential equations is modified by approximating the term containing negative shift using Taylor series expansion. After approximating the coefficient of the second derivative of the new equation, we introduced a fitting parameter and determined its value using the theory of singular Perturbation; O’Malley [1]. The three term recurrence relation obtained is solved using Thomas algorithm. The applicability of the method is tested by considering five linear problems (two problems on left layer and one problem on right layer) and two nonlinear problems.

Abstract: A new algorithm for inversing block periodic tridiagonal matrices is obtained,using the LU and UL decomposition of block tridiagonal matrix and the special structure of its inversion,the Sheman-Morrison-Woodbury formula is used during this process.Based on this algorithm,new algorithms for inversing periodic tridiagonal matrix and symmetric periodic tridiagonal matrix are also obtained.The computing complexity and the computing time of these algorithms are lower than the traditional algorithms.

Abstract: Based on the study of the tridiagonal matrix and the block tridiagonal matrix calculation,a new algorithm of diagonal decoupling is proposed for a typical class of multi-variable coupled tridiagonal industrial systems.The key of the algorithm is to construct two compensation matrices,the former series compensation matrix L(s)and after series compensation matrix R(s).Thus the multi-variable coupled system is transferred to a diagonal system and decoupling purposes are achieved.Considering zero components may exist on sub-diagonal,two construction algorithms of L(s)and R(s)are discussed.The simulation not only confirms the validity of the method,but obtains the L(s)and R(s)have the feature of many same components,which will greatly reduce the workload in industry.

Abstract: The matrix-form LSQR method is presented in this paper for solving the least squares problem of the matrix equation AXB = C with tridiagonal matrix constraint. Based on a matrix-form bidiagonalization procedure, the least squares problem associated with the tridiagonal constrained matrix equation AXB = C reduces to a unconstrained least squares problem of linear system, which can be solved by using the classical LSQR algorithm. Furthermore, the preconditioned matrix-form LSQR method is adopted for solving the corresponding least squares problem.

Abstract: A well-known property of an $M$-matrix is that its inverse is elementwise nonnegative, which we write as $M^{-1} \geq 0$. In a previous paper [Linear Algebra Appl., 434 (2011), pp. 131--143], we gave sufficient bounds on single element perturbations so that monotonicity persists for a perturbed tridiagonal $M$-matrix. Here we extend these results, presenting the actual maximum upper bounds on single element perturbations, as well as sufficient and necessary conditions for the maximum allowable higher rank perturbations. Perturbed Toeplitz tridiagonal $M$-matrices are considered as a special case. We compare our results to existing normwise bounds due to Bouchon and an iterative algorithm provided by Buffoni. We demonstrate the utility of these results by considering an application: ensuring a nonnegative solution of a discrete analogue of an integro-differential population model.

Abstract: Evaluation of the eigenvectors of symmetric tridiagonal matrices is one of the most basic tasks in numerical linear algebra. It is a widely known fact that, in the case of well separated eigenvalues, the eigenvectors can be evaluated with high relative accuracy. Nevertheless, in general, each coordinate of the eigenvector is evaluated with only high $absolute$ accuracy. In particular, those coordinates whose magnitude is below the machine precision are not expected to be evaluated with any accuracy whatsoever. It turns out that, under certain conditions, frequently ecountered in applications, small (e.g. $10^{-50}$) coordinates of eigenvectors of symmetric tridiagonal matrices can be evaluated with high $relative$ accuracy. In this paper, we investigate such conditions, carry out the analysis, and describe the resulting numerical schemes. While our schemes can be viewed as a modification of already existing (and well known) numerical algorithms, the related error analysis appears to be new. Our results are illustrated via several numerical examples.

Abstract: : A one-dimensional fluid model of a surface-type dielectric barrier discharge is created using He as the background gas. This simple model, which only considers ionizing collisions and recombination in the electropositive gas, creates an important framework for future studies into the origin of experimentally observed flow control effects of the DBD. The two methods employed in this study include the semi-implicit sequential algorithm and the fully implicit simultaneous algorithm. The first involves consecutive solutions to Poisson's, the electron continuity, ion continuity and electron energy equations. This method combines a successive over relaxation algorithm as a Poisson solver with the Thomas algorithm tridiagonal routine to solve each of the continuity equations. The second algorithm solves an Ax=b system of linearized equations simultaneously and implicitly. The coefficient matrix for the simultaneous method is constructed using a Crank-Nicholson scheme for additional stability combined with the Newton-Raphson approach to address the non-linearity and to solve the system of equations. Various boundary conditions, flux representations and voltage schemes are modeled. Test cases include modeling a transient sheath, ambipolar decay and a radio-frequency discharge. Results are compared to validated computational solutions and/or analytic results when obtainable. Finally, the semi-implicit method is used to model a DBD streamer.

Abstract: In parallel solving weak diagonal dominant tridiagonal systems,the approximate error of the Parallel Diagonal Dominant(PDD) algorithm cannot be ignored.An iterated PDD algorithm was presented.In the algorithm,the solution of the correction value was calculated by iterative method,and the computational accuracy was obviously improved.Through error analysis on the algorithm,an estimation formula of iteration number was derived for a given error tolerance.And the numerical experiment shows the validity.Based on the complexity analysis of the iterative and non-iterative PDD algorithm,the increase of iterative algorithm computational complexity is very small,but the communication complexity increases exponentially with the iteration number.

Abstract: For a general time-dependent linear competitive-cooperative tridiagonal system of differential equations, we obtain canonical Floquet invariant bundles which are exponentially separated in the framework of skew-product flows. Such Floquet bundles naturally reduce to the standard Floquet space when the system is assumed to be time-periodic. The obtained Floquet theory is applied to study the dynamics on the hyperbolic omega-limit sets for the nonlinear competitive-cooperative tridiagonal systems in time-recurrent structures including almost periodicity and almost automorphy.

Abstract: In this paper we propose a method for the numerical solution of singularly perturbed two-point boundary value problems (BVPs) with the boundary layer at left or right point. A fitting factor is introduced in a tridiagonal finite difference scheme and is obtained from the theory of singular perturbations. Thomas algorithm is used to solve the system. Error estimates are derived and numerical examples are solved to illustrate the present method.

Abstract: Fitted fourth order central difference scheme is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end point. A fitting factor is introduced in a tri-diagonal finite difference scheme and is obtained from the theory of singular perturbations. Thomas Algorithm is used to solve the system and its stability is investigated. To demonstrate the applicability of the method, we have solved linear and nonlinear problems. From the results, it is observed that the present method approximates the exact solution very well.

Abstract: In this study, we obtain the Generalized k-Fibonacci and k-Lucas numbers by using determinants of tridiagonal matrices. Therefore it has been established a new generalization for the tridiagonal matrices that represent well known numbers such as Fibonacci, Lucas, Pell and Pell-Lucas

Abstract: Examples are shown for analytical solution of engineering structures. The basic procedures for statically indeterminate systems are the force method and the deformation method. The calculation leads to a linear system of equations. The coefficients of the systems are in many cases tridiagonal or one-pair matrices. The solving of this kind of problems is based on a theorem, which states that the inverse of each tridiagonal matrix is a one-pair matrix and vice versa. This enables us to find an analytical solution in numerous particular problems. Examples are taken from theory of reinforced concrete and from different girder and cable-stayed structures.

Abstract: A block representation of the BLU factorization for block tridiagonal matrices is presented. Some properties on the factors obtained in the course of the factorization are studied. Simpler expressions for errors incurred at the process of the factorization for block tridiagonal matrices are considered.

Abstract: This paper provides a new proof of a theorem of Chandler-Wilde, Chonchaiya, and Lindner that the spectra of a certain class of infinite, random, tridiagonal matrices contain the unit disc almost surely. It also obtains an analogous result for a more general class of random matrices whose spectra contain a hole around the origin. The presence of the hole forces substantial changes to the analysis.

Abstract: We present an algorithm for computing any block of the inverse of a block tridiagonal, nearly block Toeplitz matrix (defined as a block tridiagonal matrix with a small number of deviations from the purely block Toeplitz structure). By exploiting both the block tridiagonal and the nearly block Toeplitz structures, this method scales independently of the total number of blocks in the matrix and linearly with the number of deviations. Numerical studies demonstrate this scaling and the advantages of our method over alternatives.