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.

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

Upper and lower bounds for inverse elements of finite and infinite tridiagonal matrices

Finite difference schemes have been widely studied because of their fundamental role in numerical analysis. However, most finite difference formulas in the literature are not suitable for discrete time-varying problems because of intrinsic limitations and their relatively low precision. In this paper, a high-precision 1-step-ahead finite difference formula is developed. This 5-instant finite difference (5-IFD) formula is used to approximate and discretize first-order derivatives, and it helps us to compute discrete time-varying generalized matrix inverses. Furthermore, as special cases of generalized matrix inverses, time-varying matrix inversion, and scalar reciprocals are generally deemed as independent problems and studied separately, which are solved unitedly in this paper. The precision of the 5-IFD formula and the convergence behavior of the corresponding discrete-time models are derived theoretically and shown in numerical experiments. Conventional useful formulas, such as the Euler forward finite difference (EFFD) formula and the 4-instant finite difference (4-IFD) formula are also used for comparisons and to show the superiority of the 5-IFD formula.

A 5-instant finite difference formula to find discrete time-varying generalized matrix inverses, matrix inverses, and scalar reciprocals

In this paper, for a class of perturbed Toeplitz periodic tridiagonal (PTPT) matrices, some properties, including the determinant, the inverse matrix, the eigenvalues and the eigenvectors, are studied in detail. Specifically, the determinant of the PTPT matrix can be explicitly expressed using the well-known Fibonacci numbers; the inverse of the PTPT matrix can also be explicitly expressed using the Lucas number and only four elements in the PTPT matrix. Eigenvalues and eigenvectors can be obtained under certain conditions. In addition, some algorithms are presented based on these theoretical results. Comparison of our new algorithms and some recent works is given. Numerical results confirm our new theoretical results and show that the new algorithms not only can obtain accurate results but also have much better computing efficiency than some existing algorithms studied recently.

Properties of a class of perturbed Toeplitz periodic tridiagonal matrices

In the current paper, we present a novel symbolic algorithm for solving periodic tridiagonal linear systems without imposing any restrictive conditions. The computational cost of the algorithm is less than or almost equal to those of three well-known algorithms given by Chawla and Khazal (Int. J. Comput. Math. 79(4):473–484, 2002) and by El-Mikkawy (Appl. Math. Comput. 161:691–696, 2005), respectively. In addition, the solution of periodic anti-tridiagonal linear systems is also discussed. Two numerical experiments are provided in order to illustrate the performance and effectiveness of our algorithm. All of the experiments were performed on a computer with aid of programs written in MATLAB.

A symbolic algorithm for periodic tridiagonal systems of equations

We introduce a new quadrant interlocking factorization (Q.I.F.) for use with the partition method for the solution of tridiagonal linear systems. A given linear system of size N is partitioned into r blocks each of size n (N = rn n even). The new factorization WZ of the coefficient matrix in each block has the properties that the vector is invariant under the transformation W and the solution process with coefficient matrix Z proceeds from the first and the last unknowns towards the middle ones. These properties of the new factorization help us uncouple the partitioned systems for parallel execution once a core system of order 2r is solved. The scalar count for the resulting algorithm is O(17N) which is the count for cyclic reduction and the partition method of Wang [9]. We also note that for the parallel algorithm based on modified Q.I.F. given in Evans et al. [6] the number of processors required for the algorithm is in terms of semibandwidth and the size of the system; on the other hand, the number of ...

A new quadrant interlocking factorization for parallel solution of tridiagonal linear systems

Motivated by the recursive partitioning algorithm of Evans [2], we present a new algorithm for inverting tridiagonal matrices. Our derivation of the algorithm is different but elementary. The present algorithm has potential for its vector and parallel implementation.

A recursive partitioning algorithm for inverting tridiagonal matrices

We present a parallel algorithm for the solution of tridiagonal linear systems based on the method of partitioning and backward elimination. A given N × N system is partitioned into r blocks each of size n × n (N = rn n even). Within each block the system is rewritten by collecting pairs of equations from the top and bottom (written backwards); this makes it suitable to solve the subsystem by a UL-factorization of its coefficient matrix. Once a core system of size 2r × 2r has been solved, solutions of the r subsystems can be computed in parallel. Interestingly, the serial arithmetical operations count for the present algorithm is 0(1612;N); this makes it faster than the quadrant interlocking factorization method of Chawla and Passi [1], the partitioning method of Wang [7] and cyclic reduction (see Hockney and Jesshope [6, p. 479]), each of which has a serial count of 0(17N).

A fast parallel algorithm for the solution of tridiagonal linear systems

Evans [2, 3] introduced the method of recursive point partitioning algorithm for the solution of sparse banded matrix systems and investigated the “one-line at a time” strategy for the solution of tridiagonal linear systems. Recursive block partitioning schemes resulting from variation in the size of the block structure using “two-lines at a time” have been investigated for both the tridiagonal and the quindiagonal matrix systems in Okolie [6]. The case of partitioning strategy for an nth order system has been considered by Evans and Okolie [4] resulting in a recursive decoupling algorithm for tridiagonal linear systems. Following the recursive point partitioning algorithm of Evans [2, 3], Chawla et al [1] developed a recursive partitioning algorithm for inverting tridiagonal matrices. In the present paper we present a method for inverting tridiagonal matrices by adopting the strategy resulting in a recursive doubling algorithm; the present algorithm has a highly parallel structure.

A recursive doubling algorithm for inverting tridiagonal matrices

A recursive decoupling method was introduced by Evans [2] to form the basis of a parallel tridiagonal linear system solver. Recently, Evans [3] has described another version of the recursive decoupling strategy making it into an efficient parallel algorithm by using a series of rank one updating stages performed in parallel to combine the solutions of the 2 ×2 subsystems. However, in his version of the decoupling method the diagonal elements of the coefficient matrix of the system are modified. In the present paper we present an alternative implementation of the recursive decoupling method which is in the same vein with the essential difference that our implementation obviates the need to modify the diagonal elements of the coefficient matrix. The total arithmetical operations count for the present version of recursive decoupling method is , while that for the parallel variant of cyclic reduction, termed PARACR, is (Hockney and Jesshope [4]).

On recursive decoupling method for solving tridiagonal linear systems

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

K. Passi

Author Tools

Chat about Author