Abstract: This paper seeks to develop an efficient B-spline scheme for solving Fisher's equation, which is a nonlinear reaction-diffusion equation describing the relation between the diffusion and nonlinear multiplication of a species. To find the solution, domain is partitioned into a uniform mesh and then cubic B-spline function is applied to Fisher's equation. The method yields stable and accurate solutions. The results obtained are acceptable and in good agreement with some earlier studies. An important advantage is that the method is capable of greatly reducing the size of computational work.

Abstract: To solve tridiagonal systems of linear equations, the Thomas Algorithm is a much more efficient method than, for instance, Gaussian elimination. The algorithm uses a series of elementary row operations and can solve a system of n equations in (n) operations, instead of (n3) . Many variations of the Thomas Algorithm have been developed over the years to solve very specific near-tridiagonal matrix. However, none of these methods address the situation of a system of linear equations that could easily be solved if elementary operations on columns are applied, instead of elementary operations on rows. The present paper proposes an efficient method that allows the use of elementary column operations to solve linear systems of equations using vector multiplication techniques, such as the one proposed by Thomas. Copyright © 2008 John Wiley & Sons, Ltd.

Abstract: Using a suitable Laguerre basis set that ensures a tridiagonal matrix representation of the reference Hamiltonian, we were able to evaluate exactly the matrix elements of the Yukawa potential in this representation This enabled us to compute the bound state spectrum as the eigenvalues of the associated analytical matrix representing the full Hamiltonian We also used the complex scaling method to evaluate the resonance energies and compared our results with those obtained using the Gauss quadrature approach and the corresponding results from the literature

Abstract: A parallel algorithm for solving a series of matrix equations with a constant tridiagonal matrix and different right-hand sides is proposed and studied. The process of solving the problem is represented in two steps. The first preliminary step is calculating some rows of the inverse matrix of system of linear algebraic equations. The second step consists in calculating solutions for all right-hand sides. For reducing the communication interactions, based on the formulated and proved the main Gaussian Parallel Elimination Theorem for tridiagonal system of equations, we propose an original algorithm for calculating share components of the solution vector. Theoretical estimates validating the efficiency of the approach for both the common- and distributed-memory supercomputers are obtained. Direct and iterative methods of solving a 2D Poisson equation, which include procedures of tridiagonal matrix inversion, are realized using the MPI paradigm. Results of computational experiments on a multicomputer demonstrate a high efficiency and scalability of the parallel Dichotomy Algorithm.

Abstract: We make a relativistic extension of the one-dimensional J-matrix method of scattering. The relativistic potential matrix is a combination of vector, scalar, and pseudo-scalar components. These are non-singular short-range potential functions (not necessarily analytic) such that they are well represented by their matrix elements in a finite subset of a square integrable basis set that supports a tridiagonal symmetric matrix representation for the free Dirac operator. Transmission and reflection coefficients are calculated for different potential coupling modes. This is the first of a two-paper sequence where we develop the theory in this part then follow it with applications in the second.

Abstract: In this paper a fitted fourth-order finite difference scheme is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at both end (left and right) points. We have introduced a fitting factor in Numerov fourth-order tridiagonal finite difference scheme and is obtained from the theory of singular perturbations. Thomas algorithm is used to solve the tridiagonal system. Several numerical examples are solved and compared with exact solution. It is observed that the present method approximates the exact solution very well.

Abstract: It has been shown that a nonsingular symmetric tridiagonal linear system of the form Tx = b can be solved in a backward-stable manner using diagonal pivoting meth- ods, where the LBL T decomposition of T is computed, i.e., T = LBL T , where L is unit lower triangular and B is block diagonal with 1 1 and 2 2 blocks. In this paper, we generalize these methods for solving unsymmetric tridiagonal matrices. We present three algorithms that compute the LBM T factorization, where L and M are unit lower triangular and B is block diago- nal with 1 1 and 2 2 blocks. These algorithms are normwise backward stable and reduce to the LBL T factorization when applied to symmetric matrices. We demonstrate the robustness of the algorithms for the unsymmetric case using a wide range of well-conditioned and ill-conditioned linear systems. Numerical results suggest that these algorithms are comparable to Gaussian elimination with partial pivoting (GEPP). However, unlike GEPP, these algorithms do not require row interchanges, and thus, may be used in applications where row interchanges are not possible. In addition, substantial computational savings can be achieved by carefully managing the nonzero elements of the factors L, B, and M.

Abstract: The Schrodinger equation with the Eckart potential is studied in this paper by working in a complete square integrable basis that carries a tridiagonal matrix representation of the wave operator. The discrete spectrum of the bound states is obtained by diagonalization of the recursion relation and the wavefunctions are expressed in terms of the Jocobi polynomial.

Abstract: The paper presents a type of tridiagonal preconditioners for solving linear system Ax=b with nonsingular M-matrix A, and obtains some important convergent theorems about preconditioned Jacobi and Gauss-Seidel type iterative methods. The main results theoretically prove that the tridiagonal preconditioners cannot only accelerate the convergence of iterations, but also generalize some known results.

Abstract: In this paper, we derive the general expression of the r th power (r ∈ N) for one type of tridiagonal matrix.

Abstract: Non-symmetric and symmetric twisted block factorizations of block tridiagonal matrices are discussed. In contrast to non-blocked factorizations of this type, localized pivoting strategies can be integrated which improves numerical stability without causing any extra fill-in. Moreover, the application of such factorizations for approximating an eigenvector of a block tridiagonal matrix, given an approximation of the corresponding eigenvalue, is outlined. A heuristic strategy for determining a suitable starting vector for the underlying inverse iteration process is proposed.

Abstract: Binary LFSRs with tridiagonal matrices are interesting for their application to the design of very fast stream ciphers. Recently, explicit existence conditions for tridiagonal matrices with given characteristic polynomial have been reported. Here, these conditions are reviewed with the aim of giving an easily accessible and unified view of methods and proofs.

Abstract: The cyclic reduction method is a direct method for solving tridiagonal linear systems. At the first step of this method, a tridiagonal coefficient matrix is transformed into a pentadiagonal form. In this article, we prove that the condition number for eigenvalues of some classes of coefficient matrices always decreases after the first step of the cyclic reduction method.

Abstract: A parallel algorithm for computing the finite difference solution to the elliptic equations with non-separable variables is presented. The resultant matrix is symmetric positive definite, thus the preconditioning conjugate gradient or the chebyshev method can be applied. A differential analog to the Laplace operator is used as preconditioner. For inversion of the Laplace operator we implement a parallel version of the separation variable method, which includes the sequential FFT algorithm and the parallel solver for tridiagonal matrix equations (dichotomy algorithm). On an example of solving acoustic equations by the integral Laguerre transformation method, we show that the algorithm proposed is highly efficient for a large number of processors.

Abstract: A Newton-Krylov type algorithm is designed and implemented for a pseudo compressible Navier-Stokes solver in an incompressible Cavity flow. Both GMRES and BICGSTAB (Krylov-subspace) techniques are employed for the solving the linear solver resulting from the residual linearization. ILU-0 and ILU-1 and Thomas algorithm are used for preconditioning. The results show promising convergence acceleration especially for the GMRES/ILU-1 case compared to the classic Approximate Factorization method.Copyright © 2010 by ASME

Abstract: In this paper,a new recursive algorithm is proposed for computing the inverse of a periodic tridiagonal matrix.The new algorithm make the most of the special structure of the periodic tridiagonal matrix,by using the recursive computational method,a high-order periodic tridiagonal is transformed into a low-order periodic tridiagonal for computing the inverse.Then the simplified calculation method is obtained,it can cut down the amount of calculation and storing capacity,and it has some advangtages evidently in accuracy of the calculation.From the numerical experiments it has been known that the methods is effective.

Abstract: In this note we examine the algebraic variety of complex tridiagonal matrices , such that , where is a fixed real diagonal matrix. If then is , the set of tridiagonal normal matrices. For , we identify the structure of the matrices in and analyze the suitability for eigenvalue estimation using normal matrices for elements of . We also compute the Frobenius norm of elements of , describe the algebraic subvariety consisting of elements of with minimal Frobenius norm, and calculate the distance from a given complex tridiagonal matrix to .

Abstract: For linear systems with coefficient matrices of classical structure, WZ factorizations for matrices are basic mathematical theories to design a class of parallel algorithms. Based on WZ factorization, a parallel algorithm is provided for symmetric tridiagonal linear systems. The method estimates the computation task carefully so that it assigns the system skillfully to get even load balance. In addition, the algorithm makes full use of the overlap between computation and communication to reduce waiting time in each processor. Both the subsystem assigned in each processor and the reduced subsystem have the same computing logic, as a result, a two-level method forms. By theory analysis and experiment results, it can be concluded that our method is effective in load balance and efficiency.

Abstract: The computation of the eigenvalues and orthogonal eigenvectors of an N × N real symmetric tridiagonal matrix is a well known problem in numerical analysis. The problem frequently arises in the determination of eigenvalues and eigenvectors of dense and banded symmetric matrices and in connection with various families of orthogonal polynomials and special functions satisfying three term recurrence relations. Numerous algorithms exist for the solution of this problem, which typically require O(N2) operations for the determination of eigenvalues and O(N3) operations for the determination of orthogonal eigenvectors. In this thesis we propose a new class of fast algorithms for the computation of the eigenvalues of a symmetric tridiagonal matrix in O( N ln N) operations. Such an algorithm may be combined with any one of the existing methods for the determination of eigenvectors of a symmetric tridiagonal matrix with known eigenvalues. The underlying technique is a divide-and-conquer approach which determines eigenvalues of a larger tridiagonal matrix from those of constituent matrices by the use of relations of their characteristic polynomials. The evaluation of characteristic polynomials is accelerated by the use of a technique known as the Fast Multipole Method. We provide a detailed presentation of a prototype for this class of algorithms and discuss several generalizations. An implementation of a prototype for this class of algorithms has been developed in FORTRAN, which serves to provide a comparison with existing techniques in terms of running time and accuracy. We present numerical results which demonstrate the effectiveness of the method.