Abstract: An alternating direction implicit (ADI), scheme for the wide-angle finite-difference beam propagation method (FD-BPM) based on the wide angled Pade multistep method is presented. The scheme incorporates an iterative technique for correction of the operator splitting error. The resulting equations are efficiently solved by the Thomas algorithm for tri-diagonal band matrices. The dispersion characteristics, accuracy and stability of the scheme is verified analytically and numerically for the cases of a plane wave propagating at fixed angle to the assumed propagation direction of the algorithm and the three dimensional angled propagation of a Gaussian beam. The computational requirements of the method are assessed against a standard wide-angle Pade multistep method that uses direct and iterative sparse matrix solvers.

Abstract: The Householder reduction of a member of the anti-symmetric Gaussian unitary ensemble gives an anti-symmetric tridiagonal matrix with all independent elements. The random variables permit the introduction of a positive parameter $\beta$, and the eigenvalue probability density function of the corresponding random matrices can be computed explicitly, as can the distribution of $\{q_i\}$, the first components of the eigenvectors. Three proofs are given. One involves an inductive construction based on bordering of a family of random matrices which are shown to have the same distributions as the anti-symmetric tridiagonal matrices. This proof uses the Dixon-Anderson integral from Selberg integral theory. A second proof involves the explicit computation of the Jacobian for the change of variables between real anti-symmetric tridiagonal matrices, its eigenvalues and $\{q_i\}$. The third proof maps matrices from the anti-symmetric Gaussian $\beta$-ensemble to those realizing particular examples of the Laguerre $\beta$-ensemble. In addition to these proofs, we note some simple properties of the shooting eigenvector and associated Prufer phases of the random matrices.

Abstract: In this paper, we consider the relationships between the second order linear recurrences and the permanents and determinants of tridiagonal matrices. 1. Introduction The well-known Fibonacci, Lucas and Pell numbers can be generalized as follows: Let A and B be nonzero, relatively prime integers such that D = A 4B 6= 0: De…ne sequences fung and fvng by, for all n 2 (see [10]), un = Aun 1 Bun 2 (1.1) vn = Avn 1 Bvn 2 (1.2) where u0 = 0; u1 = 1 and v0 = 2; v1 = A: If A = 1 and B = 1; then un = Fn (the nth Fibonacci number) and vn = Ln (the nth Lucas number). If A = 2 and B = 1; then un = Pn ( the nth Pell number). An alternative is to let the roots of the equation t At+B = 0 be, for n 0 un = n n and vn = n + : (1.3) Also it is well-known that + = A and = B: The permanent of an n-square matrix A = (aij) is de…ned by

Abstract: An algorithm for the inverse of a general tridiagonal matrix is presented. For a tridiagonal matrix having the Doolittle factorization, an inversion algorithm is established. The algorithm is then generalized to deal with a general tridiagonal matrix without any restriction. Comparison with other methods is provided, indicating low computational complexity of the proposed algorithm, and its applicability to general tridiagonal matrices.

Abstract: An approach is presented to solve bordered tridiagonal linear equations (BTLES) By the approach, a BTLE is first converted into three or more tridiagonal linear equations (TLES) that are independent each other, then the solution of the BTLE can be obtained via the solutions of the TLES Since the TLES are independent each other, their solution can be obtained via parallel computing under heterogeneous environments The approach costs at most O(n2) of time complexity in sequential mode Detail mathematical deduction is presented to reveal the approach and a framework is introduced to implement the approach The approach, which can also be available for grid computing, greatly increases the flexibility and agility of computations as well as the computational efficiency

Abstract: It is known that the worst case near-far resistance of optimum multiuser detectors for asynchronous Gaussian multiple-access channels can be expressed in terms of a class of block-tridiagonal matrices, and the minimal eigenvalues of such a class of block-tridiagonal matrices serve as a good measure of the worst case near-far resistance. In this paper, we focus on the two-user scenario where each block-tridiagonal matrix under consideration is a tridiagonal matrix. We derive closed-form expressions for the minimal eigenvalues of such a class of tridiagonal matrices in terms of the largest real solution of a trigonometric equation in [0,pi]. We also obtain lower bounds and upper bounds on the minimal eigenvalues which improve on previously known results in the literature.

Abstract: We prove convergence for the basic LR algorithm on a real unreduced tridiagonal matrix with a one-point spectrum—the Jordan form is one big Jordan block. First we develop properties of eigenvector matrices. We also show how to deal with the singular case.

Abstract: It is showed that if A is I-block diagonally dominant (II-block diagonally dominant), then the reduced matrix S preserves the same property. We also give a sufficient condition for the reduced matrix S also to be a block H-matrix when A is a block H-matrix, and some properties on the comparison matrices @m"I(A^(^k^)), @m"I"I(A^(^k^)), @m"I(L), and @m"I(U) are obtained. Finally, error analysis of block LU factorization for block tridiagonal matrix is presented.

Abstract: An algorithm for the inverse of a general tridiagonal matrix is presented.First,for the tridi- agonal matrix having Doolittle factorization,an algorithm for the inverse was established.Then the al- gorithm was generalized to a general tridiagonal marx without any restrictive condition.Some com- parison with other methods for the inverse was discussed in the end.It is shown that the arithmetic operations of the algorithm are low and it is applicable to a general tridiagonal matrix.

Abstract: This paper is concerned with the block monotone iterative schemes of numerical solutions of nonlinear parabolic systems with initial and boundary condition in two dimensional space. By using the finite difference method, the system is discretized into algebraic systems of equations, which can be represented as block matrices. Two iterative schemes, called the block Jacobi scheme and the block Gauss-Seidel scheme, are introduced to solve the system block by block. The Thomas algorithm is used to solve tridiagonal matrices system efficiently. For each scheme, two convergent sequences starting from the initial upper and lower solutions are constructed. Under a sufficient condition the monotonicity of the sequences, the existence and the uniqueness of solution are proven. To demonstrate how these method work, the numerical results of several examples with different types of nonlinear functions and different types of boundary conditions are also presented.

Abstract: A stability analysis is carried out for certain classes of switched linear systems with tridiagonal structure, under arbitrary switching signal. This analysis is made using diagonal common quadratic Lyapunov functions. Namely, necessary and sufficient conditions for the existence of such Lyapunov functions are proposed for second order switched systems and for third order switched systems with Toeplitz tridiagonal structure.

Abstract: Based on the idea of chasing method,a new algorithm is developed to solving the circular and quasi-circular tridiagonal systems in this paper.The computational costs of multiplication and division are 8N and 3N,respectively.Compared with the traditional method,the new chasing method saves the computational cost.The numerical experiments indicate that,the exact solutions can be obtained in several seconds by using this method.

Abstract: This paper researches the inverse eigenvalue problem for tridiagonal matrices with linear relation. And presents the problem that using the tridiagonal matrix and the matrix which is got rid of the last row and the last column to make the original matrix. The expression of the solution of the problem is given, and some numerical example is provided.

Abstract: Bspline and Bezier methods are two powerful methods for approximation of data in all branches of engineering problems. With some improvements and modifications they can be applied for interpolation of data. Although literature offers different approaches for the formulation of Bspline, there is a set of independent functions that defines the Bspline equation. The numbers of the coefficients of Bspline equation are equal to the numbers of pair data or control points. The advantage of Bspline method is that, the most of set functions diminishes at some control points. Therefore a simple tridiagonal linear system is obtained. The tridiagonal linear system can be easily solved by Thomas algorithm. Bezier curves can be obtained by drawing the governing parametric equations. The parametric equations are Bernstein polynomials. Bezier curve does not pass through all the data points but passes through end points. It is mainly applied for approximation approaches. By considering some complementary points between original points, the Bezier can be forced to pass through all control points, hence it can be applied for interpolation purposes. The drawn curves by this method are smoother and have less sinuosity forms. For the comparison of the interpolation properties, several problems are solved by these two methods. The results explain that both methods are powerful and robust for interpolation of highly non-uniform set of data. The Bezier model is superior to Bspline with regarding to accuracy, smoothness and less calculation operations.

Abstract: Groups of unknowns in linear systems of algebraic equations are eliminated by solving smaller systems whose coefficients are just coefficients in the original systems, with no matrix inversions and multiplications being involved. This could yield solutions with improved accuracy with respect to other solution methods. The amount of computation involved is only slightly higher than that in the Gaussian elimination.

Abstract: An exponential compact higher order scheme is developed for stationary convection‐diffusion type of differential equations which inludes incompressible flow equations in stream function vorticity formulation. The scheme is, ingeneral, fourth order accurate however for one‐dimensional constant convection and diffusion coefficients, the scheme is O(h6) and produces a tri‐diagonal system of equations that can be solved efficiently using Thomas algorithm. For two‐dimensional problems, the scheme produces an O(h4+k4) accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit (ADI) procedure. The efficiency of the developed scheme is measured using wave number analysis. The analysis shows that the developed scheme has a much better spectral resolution than any of the existing higher order schemes.

Abstract: Implicit finite-difference schemes of approximate factorization and predictor-corrector schemes based on a special splitting of operators are proposed for the numerical solution of the Navier-Stokes equations governing a viscous compressible heat-conducting gas. The schemes are based on scalar tridiagonal Gaussian elimination and are unconditionally stable. The accuracy and efficiency of the algorithms are confirmed by computing two-dimensional flows of complex geometry.

Abstract: The range of application of the method of comparison-matrices for the calculation of brackets to eigenvalues of the Schrodinger equation is extended to infinite bandmatrices, H. It is shown by construction that the elements αi and βi+1 of an infinite tridiagonal matrix may be calculated up to some finite maximum index, i, in a finite calculation. The tridiagonal matrix is a unitary transform of H and consequently has the same eigenvalues.

Abstract: Recently the study of intensive laser and atom interaction is one of the most interesting topics in laser and atom interdisciplinary physics. Study of interaction of hydrogen atom and intense laser using a matrix method is reported in this paper. This method includes Givens reduction for real symmetric tridiagonal matrix form of Hamiltonian matrix, Thomas algorithm for eigenvectors, and bisection algorithm with Sturm theorem for eigenvalues. Some preliminary results and progress for this study are presented.

Abstract: In this paper the transmittance through a quantum wire connected with two electron reservoirs is calculated and non-trivial transformation between the evolution operator method and the Green's function technique is reported. To show this equivalence an analytical nonlinear formula which concerns symmetrical tridiagonal matrices is proofed. This formula connects the cofactor and three determinants of tridiagonal matrices.

Abstract: In this paper we review the parallel solution of sparse linear systems, usually deriving by the discretization of ODE-IVPs or ODE-BVPs. The approach is based on the concept of parallel factorization of a (block) tridiagonal matrix. This allows to obtain efficient parallel extensions of many known matrix factorizations, and to derive, as a by-product, a unifying approach to the parallel solution of ODEs.

Abstract: In this paper we discuss general tridiagonal matrix models which are natural extensions of the ones given in Dumitriu and Edelman (J. Math. Phys. 43(11): 5830–5847, 2002; J. Math. Phys. 47(11):5830–5847, 2006). We prove here the convergence of the distribution of the eigenvalues and compute the limiting distributions in some particular cases. We also discuss the limit of fluctuations, which, in a general context, turn out to be Gaussian. For the case of several random matrices, we prove the convergence of the joint moments and the convergence of the fluctuations to a Gaussian family. The methods involved are based on an elementary result on sequences of real numbers and a judicious counting of levels of paths.

Abstract: In this paper, the concept of generalized spectral function is introduced for finite- order tridiagonal symmetric matrices (Jacobi matrices) with complex entries. The structure of the generalized spectral function is described in terms of spectral data consisting of the eigenvalues and normalizing numbers of the matrix. The inverse problems from generalized spectral function as well as from spectral data are investigated. In this way, a procedure for construction of complex tridiagonal matrices having real eigenvalues is obtained.