Tridiagonal Systems Of Linear Equations Research Articles

Algorithms for Solving Linear Systems of Equations of Tridiagonal Type via Transformations

Numeric algorithms for solving the linear systems of tridiagonal type have already existed. The well-known Thomas algorithm is an example of such algorithms. The current paper is mainly devoted to constructing symbolic algorithms for solving tridiagonal linear systems of equations via transformations. The new symbolic algorithms remove the cases where the numeric algorithms fail. The computational cost of these algorithms is given. MAPLE procedures based on these algorithms are presented. Some illustrative examples are given.

Solving a kind of boundary-value problem for ordinary differential equations using Fermi—The next generation CUDA computing architecture

The aim of this paper is to show that a special kind of boundary value problem for solving second-order ordinary differential equations can be efficiently solved on modern heterogeneous computer architectures based on CPU and GPU Fermi processors. Such a problem reduces to the problem of solving a large tridiagonal system of linear equations with an almost Toeplitz structure. The considered algorithm is based on the recently developed divide and conquer method for solving linear recurrence systems with constant coefficients.

Numerical Solution of the Boltzmann Based Lubrication Equation for the Air-Bearing Interface Between a Skewed Slider and a Disk With Discrete Data Tracks

Discrete track recording (DTR) is a method for increasing the recording density of a data storage disk by use of a pattern arrangement of discrete tracks. The DTR track structure consists of a pattern of very narrow concentric raised areas and recessed areas underneath a magnetic recording layer. In order to design the air-bearing slider platform that houses the magnetic transducer for DTR application at very low fly heights, the influence of the disk surface topography as a surface roughness effect must be taken into account. This paper is focused on the numerical solution of the roughness averaged lubrication equation reported recently in the work of White (2010, “A Gas Lubrication Equation for High Knudsen Number Flows and Striated Rough Surfaces,” ASME J. Tribol., 132, p. 021701) and is specialized for the influence of discrete disk data tracks on the recording head slider-disk air-bearing interface subject to a nonzero skew angle formed between the slider longitudinal axis and the direction of disk motion. The generalized lubrication equation for a smooth surface bearing and appropriate for high Knudsen number analysis is quite nonlinear. And including the averaging process required for treatment of a nonsmooth disk surface, as well as the rotational transformation required to allow for a nonzero skew angle, increases further the nonlinearity and general complexity of the lubrication equation. Emphasis is placed on development of a numerical algorithm that is fast, accurate, and robust for air-bearing analysis of complex slider surfaces. The numerical solution procedure developed utilizes a time integration of the lubrication equation for both steady-state and dynamic analyses. The factored-implicit scheme, a form of the more general alternating-direction-implicit numerical approach, was chosen to deal with the two-dimensional and highly nonlinear aspects of the problem. Factoring produces tightly banded coefficient matrices and results in an algorithm that is second-order accurate in time while requiring only the solution of tridiagonal systems of linear equations in advancing the computation from one time level to the next. Numerical solutions are presented that demonstrate the performance of the computational scheme and illustrate the influence of some discrete track parameters on skewed air-bearing performance as compared with a flat surface data storage disk.

High-performance modeling acoustic and elastic waves using the parallel Dichotomy Algorithm

A high-performance parallel algorithm is proposed for modeling the propagation of acoustic and elastic waves in inhomogeneous media. An initial boundary-value problem is replaced by a series of boundary-value problems for a constant elliptic operator and different right-hand sides via the integral Laguerre transform. It is proposed to solve difference equations by the conjugate gradient method for acoustic equations and by the GMRES( k) method for modeling elastic waves. A preconditioning operator was the Laplace operator that is inverted using the variable separation method. The novelty of the proposed algorithm is using the Dichotomy Algorithm [26], which was designed for solving a series of tridiagonal systems of linear equations, in the context of the preconditioning operator inversion. Via considering analytical solutions, it is shown that modeling wave processes for long instants of time requires high-resolution meshes. The proposed parallel fine-mesh algorithm enabled to solve real application seismic problems in acceptable time and with high accuracy. By solving model problems, it is demonstrated that the considered parallel algorithm possesses high performance and efficiency over a wide range of the number of processors (from 2 to 8192).

Parallel Dichotomy Algorithm for solving tridiagonal system of linear equations with multiple right-hand sides

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.

A Novel Parallel Algorithm Based on the Gram-Schmidt Method for Tridiagonal Linear Systems of Equations

This paper introduces a new parallel algorithm based on the Gram-Schmidt orthogonalization method. This parallel algorithm can find almost exact solutions of tridiagonal linear systems of equations in an efficient way. The system of equations is partitioned proportional to number of processors, and each partition is solved by a processor with a minimum request from the other partitions' data. The considerable reduction in data communication between processors causes interesting speedup. The relationships between partitions approximately disappear if some columns are switched. Hence, the speed of computation increases, and the computational cost decreases. Consequently, obtained results show that the suggested algorithm is considerably scalable. In addition, this method of partitioning can significantly decrease the computational cost on a single processor and make it possible to solve greater systems of equations. To evaluate the performance of the parallel algorithm, speedup and efficiency are presented. The results reveal that the proposed algorithm is practical and efficient.

Variant of the Thomas Algorithm for opposite‐bordered tridiagonal systems of equations

AbstractTo 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 𝒪(n) operations, instead of 𝒪(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.

On the preconditioning of the block tridiagonal linear system of equations

Two algorithms for computing the inverse factors of general tridiagonal and pentadiagonal matrices are obtained. Then, these algorithms are used for computing a block ILU preconditioner for the block tridiagonal linear system of equations. Some numerical results are given to show the robustness and efficiency of the preconditioner. The performance of the proposed preconditioner is compared with a recently proposed method.

An explicit and conservative remapping strategy for semi‐Lagrangian advection

AbstractA conservative semi‐Lagrangian advection scheme has recently been proposed by Cotter et al. in their article “The remapped particle‐mesh semi‐Lagrangian advection scheme” published in The Quarterly Journal of the Royal Meteorological Society (2007, volume 133, pages 251–260). It is based on a remapped particle‐mesh implementation using bicubic B‐spline as basis functions. It has been shown that the method is of comparable accuracy as standard bicubic semi‐Lagrangian advection schemes. A potential drawback of the, so called, remapped particle‐mesh semi‐Lagrangian (RPM SL) advection scheme is that it requires the solution of tridiagonal linear systems of equations. In this note, we demonstrate that the solution of linear equations can be avoided without sacrificing the conservation and accuracy properties of the original RPM SL method. Copyright © 2007 Royal Meteorological Society

A communication-less parallel algorithm for tridiagonal Toeplitz systems

Diagonally dominant tridiagonal Toeplitz systems of linear equations arise in many application areas and have been well studied in the past. Modern interest in numerical linear algebra is often focusing on solving classic problems in parallel. In McNally [Fast parallel algorithms for tri-diagonal symmetric Toeplitz systems, MCS Thesis, University of New Brunswick, Saint John, 1999], an m processor Split & Correct algorithm was presented for approximating the solution to a symmetric tridiagonal Toeplitz linear system of equations. Nemani [Perturbation methods for circulant-banded systems and their parallel implementation, Ph.D. Thesis, University of New Brunswick, Saint John, 2001] and McNally (2003) adapted the works of Rojo [A new method for solving symmetric circulant tri-diagonal system of linear equations, Comput. Math. Appl. 20 (1990) 61–67], Yan and Chung [A fast algorithm for solving special tri-diagonal systems, Computing 52 (1994) 203–211] and McNally et al. [A split-correct parallel algorithm for solving tri-diagonal symmetric Toeplitz systems, Internat. J. Comput. Math. 75 (2000) 303–313] to the non-symmetric case. In this paper we present relevant background from these methods and then introduce an m processor scalable communication-less approximation algorithm for solving a diagonally dominant tridiagonal Toeplitz system of linear equations.

A parallel and scalable algorithm for ADI method with pre-propagation and message vectorization

It is hard to implement the ADI method in an efficient way on distributed-memory parallel computers. We propose “ P-scheme” which parallelizes a tridiagonal linear system of equations for the ADI method, but its effectiveness is limited to the cases where the problem size is large enough mainly because of the communication cost of the propagation phase of the scheme. In order to overcome this difficulty, we propose an improved version of the P-scheme with “ message vectorization” which aggregates several communication messages into one and alleviates the communication cost. Also we evaluate the effectiveness of message vectorization for the ADI method and show that the improved version of the P-scheme works well even for smaller problems and linear and super-linear speedups can be achieved for 8194 × 8194 and 16,386 × 16,386 problems, respectively.

An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients

We propose a three level implicit unconditionally stable difference scheme of O( k 2 + h 2) for the difference solution of second order linear hyperbolic equation u tt + 2 α( x, t) u t + β 2( x, t) u = A( x, t) u xx + f( x, t), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where A( x, t) > 0, α( x, t) > β( x, t) ⩾ 0. The proposed formula is applicable to the problems having singularity at x = 0. The resulting tri-diagonal linear system of equations is solved by using Gauss-elimination method. Numerical examples are provided to illustrate the unconditionally stable character of the proposed method.

A direct method for solving circulant tridiagonal block systems of linear equations

This paper presents a modification of Rojo’s algorithm [Comput. Math. Appl. 20 (1990) 61] to solve block circulant tridiagonal systems of linear equations which are Toeplitz and Hermitian. This new approach gives us a general direct algorithm for solving the problem. We will show how to choose a block matrix as a parameter to describe the method. We employ the factorization of block Toeplitz tridiagonal matrices as the product of two block Toeplitz subdiagonal and superdiagonal matrices. The algorithm is based on obtaining the solution of the nonlinear matrix equation A = Γ + B* Γ −1 B. Finally, some numerical results will be given.

A BSP recursive divide and conquer algorithm to solve a tridiagonal linear system

In this paper we discuss a recursive divide and conquer method to solve a tridiagonal system of linear equations. We propose two divide and conquer algorithms using different communication schemes. The first one uses a fan-in scheme to perform communication among processors, while the second one follows a rather different model, in which all the processors communicate all data to the main one. A theoretical study of the computational cost of both algorithms is developed computing theoretical times; firstly in an IBM SP2 computer with a high performance switch and Ethernet connection, and secondly in a CRAY T3D computer. We present experimental results for the IBM SP2 computer, for 2, 4, and 8 processors, comparing these results with the theoretical predicted times.

Reliable Solution of Tridiagonal Systems of Linear Equations

In this paper we present new formulas for characterizing the sensitivity of tridiagonal systems that are independent of the condition number of the underlying matrix. We also introduce efficient algorithms for solving tridiagonal systems of linear equations which are stable and reliable (namely, stable in the backward sense and little sensitive to perturbations in the coefficients).

Vector and Parallel Computing Comparisons for Block Pre-Conditioned Conjugate Gradient

This work deals with the Block Preconditioning for the Conjugate Gradient Method on block tridiagonal systems of linear equations. The results obtained by using Transputer networks, IBM-SP2 and Cray-J90 confrrm the high perfonnance of this method with parallel and vector processing.

Stable solution of tridiagonal systems

In this paper we present three different pivoting strategies for solving general tridiagonal systems of linear equations. The first strategy resembles the classical method of Gaussian elimination with no pivoting and is stable provided a simple and easily checkable condition is met. In the second strategy, the growth of the elements is monitored so as to ensure backward stability in most cases. Finally, the third strategy also uses the right‐hand side vector to make pivoting decisions and is proved to be unconditionally backward stable.

An Improved Mapping of Cyclic Elimination onto Hypercubes Using Data Replication

In this paper, we propose a new mapping of the Cyclic Elimination (CE) algorithm for the solution of block tridiagonal linear system of equations onto hypercube multiprocessors. Unlike the previous mapping schemes, in our mapping of the CE algorithm all communications are restricted to physically adjacent processors, using the concept of data replication. The effectiveness of our mapping is demonstrated by comparing it with the existing mapping of the Cyclic Reduction algorithm onto hypercubes using both analytical and simulation methods.

The block alternating group explicit method (blage) to solve 2D-elliptic differential equations

In this paper, we present the Block Alternating Group Explicit Method (BLAGE) to solve the block tridiagonal linear system of equations derived from the discretisation of 2D-elliptic boundary value problems. The convergence of the method is proved and the case for optimum acceleration parameters is also studied. The BLAGE method is used to solve two partial differential equations for which it is required to solve block tridiagonal linear system involving the solution of simpler tridiagonal system of equations. The method is accurate and flexible.

Improved finite-difference beam-propagation method based on the generalized Douglas scheme and its application to semivectorial analysis

The generalized Douglas scheme for variable coefficients is applied to the propagating beam analysis. Once the alternating direction implicit method is used, the truncation error is reduced in the transverse directions compared with the conventional Crank-Nicholson scheme, maintaining a tridiagonal system of linear equations. Substantial improvement in the accuracy is achieved even in the TM mode propagation. As an example of the semivectorial analysis, the propagating field and the attenuation constant of a bent embedded waveguide with a trench section are calculated and discussed.