Abstract: This paper first reviews some results on the structure of inverses of nonsingular and irreducible tridiagonal matrices. Next, explicit inversion formulas are given for certain, not necessarily symmetric, tridiagonal matrices. The results are then applied to matrices arising from discretization of two-point boundary value problems of the Sturm-Liouville type. The results show harmonic relations between the Green functions and the discrete Green functions for the problems. Finally the results are extended to block tridiagonal matrices. This work was supported by Scientific Research Grant-in-Aid from JSPS.

Abstract: A two-way chasing algorithm to reduce a diagonal plus a symmetric semi-separable matrix to a symmetric tridiagonal one and an algorithm to reduce a diagonal plus an unsymmetric semi-separable matrix to a bidiagonal one are considered. Both algorithms are fast and stable, requiring a computational cost of N2, where N is the order of the considered matrix.

Abstract: The popular sequential Thomas algorithm for the numerical solution of tridiagonal linear algebraic equation systems is extended on a class of quasi-block-tridiagonal equation systems arising from finite-difference discretisations of boundary value and initial boundary value problems of reaction-migration-advection-diffusion type in one space dimension, occurring in electrochemistry. The extension allows for a simultaneous consideration of: (a) multiple space intervals with common boundaries; (b) additional algebraic or differential-algebraic equations coupled with mixed boundary conditions, that may express e.g. adsorption at the boundaries; (c) three-point finite-difference approximations to the gradients of the solutions of the initial/boundary value problems at the boundaries; (d) periodic or non-periodic boundary conditions at the external boundaries. The resulting equation matrix may include nonzero off-diagonal corner blocks associated with periodic boundary conditions, may be locally block-pentadiagonal at a number of isolated rows corresponding to internal spatial boundaries, and its blocks may have variable dimensions. Testing calculations are performed.

Abstract: There are many articles on symmetric tridiagonal Toeplitz and circulant systems. Such systems in areas including numerical methods for solving boundary value differential equations and in graph theory. These matrices can often be written as the product of bidiagonal matrices. In this article, nonsymmetric Toepliz systems and nonsymmetric circulant systems are examined. The coefficient matrix is split into two bidiagonal matrices and the efficient solution of the resulting systems is considered.

Abstract: This paper proposes an accurate integral-based scheme for solving the advection–diffusion equation. In the proposed scheme the advection–diffusion equation is integrated over a computational element using the quadratic polynomial interpolation function. Then elements are connected by the continuity of first derivative at boundary points of adjacent elements. The proposed scheme is unconditionally stable and results in a tridiagonal system of equations which can be solved efficiently by the Thomas algorithm. Using the method of fractional steps, the proposed scheme can be extended straightforwardly from one-dimensional to multi-dimensional problems without much difficulty and complication. To investigate the computational performances of the proposed scheme five numerical examples are considered: (i) dispersion of Gaussian concentration distribution in one-dimensional uniform flow; (ii) one-dimensional viscous Burgers equation; (iii) pure advection of Gaussian concentration distribution in two-dimensional uniform flow; (iv) pure advection of Gaussian concentration distribution in two-dimensional rigid-body rotating flow; and (v) three-dimensional diffusion in a shear flow. In comparison not only with the QUICKEST scheme, the fully time-centred implicit QUICK scheme and the fully time-centred implicit TCSD scheme for one-dimensional problem but also with the ADI-QUICK scheme, the ADI-TCSD scheme and the MOSQUITO scheme for two-dimensional problems, the proposed scheme shows convincing computational performances. Copyright © 2001 John Wiley & Sons, Ltd.

Abstract: A framework based on graph theoretic notations is described for the design and analysis of a wide range of parallel tridiagonal matrix algorithms. It comprises of three basic types of graph transformation operations: partition, selection, elimination and update. We use the framework to present a unified description of many known parallel algorithms for the solution of tridiagonal systems. We also discuss the use of this framework to design parallel algorithms.

Abstract: The physical properties of basalt, such as density, viscosity, permeability, and heat conductivity, were studied experimentally. By introducing the measured parameters into the governing equations of the finite differential method (FDM), the melting processes of basalt, in relation to the thermal boundary conditions and particle sizes, were simulated for the purpose of designing a melting furnace for basalt. The governing equations were discretized in a tri-diagonal matrix form and were solved using the tridiagonal matrix algorithm (TDMA) and the alternative direction implicit (ADI) solver. The temperature distribution, position of the two-dimensional phase-change boundary, and melting time were calculated during the numerical simulation. These provide the basis for determining the heating procedure, for controlling the furnace temperature, and for predicting basalt melting states. In the experiment, an electrical furnace was designed based on the computations. It is demonstrated that the simulation results are reasonably consistent with the observed data.

Abstract: Discretizing two-point boundary value problems on an interval by finite difference method, we obtain a certain type of tridiagonal coefficient matrices. In this paper we give an explicit inversion formula for such tridiagonal matrices using Yamamoto-Ikebe’s inversion formula.

Abstract: Qualitative properties of matrix splitting methods for linear systems with tridiagonal and block tridiagonal Stieltjes-Toeplitz matrices are studied. Two particular splittings, the so-called symmetric tridiagonal splittings and the bidiagonal splittings, are considered, and conditions for qualitative properties like nonnegativity and shape preservation are shown for them. Special attention is paid to their close relation to the well-known splitting techniques like regular and weak regular splitting methods. Extensions to block tridiagonal matrices are given, and their relation to algebraic representations of domain decomposition methods is discussed. The paper is concluded with illustrative numerical experiments.

Abstract: In this paper, for inverse eigenproblems with given four eigenvalues and eigenvector are considered and are given some necessary and sufficient conditions for the uniqueness of the solution.

Abstract: Symmetric Toeplitz tridiagonal systems arise in many scientific applications. An efficient algorithm, the Simple Parallel Prefix (SPP) algorithm, was previously proposed for solving symmetric Toeplitz tridiagonal systems on SIMD and vector computers. Based on the SPP algorithm, a scalable parallel algorithm is proposed for solving periodic symmetric Toeplitz tridiagonal systems in this study. The newly proposed algorithm has the same parallel computation count as that of the SPP algorithm for non-periodic systems, and it requires only shift communication.

Abstract: 本論文では最近の高性能な算術演算パイプラインを備えたスカラプロセッサに適した,3 重対角行 列の並列ソルバーのアルゴリズムを提案する.これは特に,2重の算術パイプラインを備えたプロセッ サで効果が大きい.このアルゴリズムは行列を並べ替えることで,両端(行列を格納した配列では初 めと終わり)から真ん中に向かう順序を採用する.これにより演算量を増加することなく 2ウェイの 並列性が生まれる.この順序付けは再帰的に適用することができ,並列性は 2のべきで増やしてゆけ るが,演算量は増加する.インタフェースは現行の 3重対角ソルバーを変更することなく実装できる. さらに単一の算術演算パイプラインしか持たない機種に対しても,並列性を通分に利用することで除 算を減らせるので効果がある.4 ウェイの並列性は SMP 並列計算機向きである.また,この並列 3 重対角ソルバーのアルゴリズムの延長として,対称 3重対角行列のスツルム列と行列式を並列計算す るアルゴリズムを述べる.

Abstract: We propose a new parallel solver for nonsymmetric tridiagonal matrices, which is an improvement over the dissection method. The conventional dissection method is difficult to apply to a general nonsymmetric tridiagonal matrix, because the independence of decomposition operations in each subdomain is lost when pivoting is introduced. In our algorithm, due to the reordering of the nodes adjacent to the boundary nodes, the independence of decomposition operations in each subdomain is guaranteed even when partial pivoting is introduced. Thus, the LU decomposition of the whole matrix with partial pivoting can be done in parallel. We evaluated our algorithm on 1 node of the SR8000/F1 (a shared-memory parallel computer with 8 processors) and obtained speedup of 5.5 times compared with the conventional sequential tridiagonal solver with pivoting, when computing the LU decomposition of a nonsymmetric tridiagonal matrix of order 8000.

Abstract: The tridiagonal Toeplitz linear systems occur repeatedly in the solution of the implicit finite difference equations derived from linear first order hyperbolic equations, i.e. the Transport equation, under a variety of boundary conditions. Interest in fast direct methods for solving these kind of linear systems has long been a hot spot of research. A parallel algorithm for certain tridiagonal Toeplitz linear systems on distributed memory multicomputers is presented. Derivation of the algorithm is introduced. The algorithm is based on the factorization of the coefficient matrix and the principle of ‘divide and conquer’ in designing parallel algorithms. Authors make full use of the special structure of the coefficient matrix. By using the customary nesting procedure, Horner's formula, authors avoid the necessary of quantities α i, (-α) i(i=2,3,…,m ) and (α m) i, (-α m) i(i=2,3,…,p-1 ). This reduces the algorithm's redundancy computation caused by parallelization. The complexity of the algorithm is analyzed using Log P model. Its communication mechanism is very simple. The communication complexity is only related to p , the number of processors, and not related to n , the size of the matrix. The algorithm's parallel efficiency is high. The analysis of complexity and numerical experiments show that the algorithm's speedup satisfy: S p(n)→p(n→+∞) . This is the best a parallel algorithm can reach. The algorithm has been implemented on parallel computers. The results of numerical experiments about the algorithm on a distributed memory multicomputer are presented in this paper.