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About 522 papers for “Tridiagonal matrix algorithm”

DETERMINATION OF TRIDIAGONAL MATRIX INVERSE USING THELEWIS ALGORITHM

David, Aji Saputra · Digilib Repository Unila (Lampung University) · 2025

A matrix is a rectangular array of numbers consisting of entries, with its size (order) determined by the number of rows and columns. There are various types of matrices, one of which is the tridiagonal matrix. A tridiagonal matrix is a square matrix that only has non-zero elements non-zero elements on the main diagonal, superdiagonal, and subdiagonals. One of the main challenges in using tridiagonal matrices is determining their inverse. This study aims to determine the inverse of a tridiagonal matrix using the Lewis Algorithm, a recursive-based method that utilises patterns of relationships between elements to generate the inverse, particularly for large-sized matrices and sparsely populated matrices. This research was conducted analytically and reinforced with implementation using the Python programming language. The results of the research show that the Lewis Algorithm is able to determine the inverse of a tridiagonal matrix systematically with validation through multiplication of the initial matrix and the inverse result, which produces an identity matrix. Keywords: Matrix, Tridiagonal Matrix, Matrix Inverse, Lewis Algorithm. Matriks adalah susunan bilangan berbentuk segi empat siku-siku yang terdiri dari entri-entri, dengan ukuran (ordo) ditentukan oleh banyaknya baris dan kolom. Terdapat berbagai jenis matriks, salah satunya adalah matriks tridiagonal. Matriks tridiagonal merupakan jenis matriks bujursangkar yang hanya memiliki elemen tidak nol pada diagonal utama, superdiagonal, dan subdiagonal. Salah satu tantangan utama dalam penggunaan matriks tridiagonal adalah menentukan inversnya. Penelitian ini bertujuan untuk menentukan invers matriks tridiagonal menggunakan Algoritma Lewis, yaitu metode berbasis rekursif yang memanfaatkan pola hubungan antar elemen untuk menghasilkan invers, khususnya untuk matriks berdimensi besar dan jarang terisi. Penelitian ini dilakukan secara analitik dan diperkuat dengan implementasi menggunakan bahasa pemrograman Python. Hasil penelitian menunjukkan bahwa Algoritma Lewis mampu menentukan invers matriks tridiagonal secara sistematis dengan validasi melalui perkalian kembali matriks awal dan hasil invers yang menghasilkan matriks identitas. Kata-kata kunci: Matriks, Matriks Tridiagonal, Invers Matriks, Algoritma Lewis.

Cited by 0

Performance Optimization of Tridiagonal Matrix Algorithm [TDMA] on Multicore Architectures

Anishchandran Chathalingath, M. Arun · International Journal of Grid and High Performance Computing · 2019

Fast and efficient tridiagonal solvers are highly appreciated in scientific and engineering domain, but challenging optimization task for computer engineers. The state-of-the-art developments in multi-core computing paves the way to meet this challenge to an extent. The technical advances in multi-core computing provide opportunities to exploit lower levels of parallelism and concurrency for inherently sequential algorithms. In this article, the authors present an optimal performance pipelined parallel variant of the conventional Tridiagonal Matrix Algorithm (TDMA), aka the Thomas algorithm, on a multi-core CPU platform. The implementation, analysis and performance comparison of the proposed pipelined parallel TDMA and the conventional version are performed on an Intel SIMD multi-core architecture. The results are compared in terms of elapsed time, speedup, cache miss rate. For a system of ‘n' linear equations where n = 2^36 in presented pipelined parallel TDMA achieves speedup of 1.294X with a parallel efficiency of 43% initially and inclines towards linear speed up as the system grows.

Cited by 1

THE APPLICATION OF TRIDIAGONAL MATRIX ALGORITHM IN CUBIC SPLIN INTERPOLATION

Elvathna Syafwan, Efend Efend, Narwen Narwen et al. · 2020

Tridiagonal matrix algorithm is known to be more efficient in the computational process than the Gaussian elimination method in solving linear system problems involving tridiagonal matrices. In this paper, the tridiagonal matrix algorithm is applied to the cubic spline interpolation problem with natural boundary conditions. In this case, the tridiagonal matrix algorithm plays a role in finding the second derivative of each cubic spline sub-function so that it is more efficient in obtaining the coefficients of the third order polynomials that form the cubic spline function

Cited by 0

Parallel tridiagonal matrix inversion with a hybrid multigrid–Thomas algorithm method

Parker, Joseph · UKAEA Open Data · 2020

Tridiagonal matrix inversion is an important operation with many applications. It arises frequently in solving discretized one-dimensional elliptic partial differential equations, and forms the basis for many algorithms for block tridiagonal matrix inversion for discretized PDEs in higher-dimensions. In such systems, this operation is often the scaling bottleneck in parallel computation. In this paper, we derive a hybrid multigrid–Thomas algorithm designed to efficiently invert tridiagonal matrix equations in a highly-scalable fashion in the context of time evolving partial differential equation systems. We decompose the domain between processors, using multigrid to solve on a grid consisting of the boundary points of each processor’s local domain. We then reconstruct the solution on each processor using a direct solve with the Thomas algorithm. This algorithm has the same theoretical optimal scaling as cyclic reduction and recursive doubling. We use our algorithm to solve Poisson’s equation as part of the spatial discretization of a time-evolving PDE system. Our algorithm is faster than cyclic reduction per inversion and retains good scaling efficiency to twice as many cores.

Cited by 0Open access

A Parallel Algorithm for Computing the Eigenvalues of a Symmetric Tridiagonal Matrix

Paul N. Swarztrauber · Mathematics of Computation · 1993

A parallel algorithm, called polysection, is presented for computing the eigenvalues of a symmetric tridiagonal matrix.The method is based on a quadratic recurrence in which the characteristic polynomial is constructed on a binary tree from polynomials whose degree doubles at each level.Intervals that contain exactly one zero are determined by the zeros of polynomials at the previous level which ensures that different processors compute different zeros.The signs of the polynomials at the interval endpoints are determined a priori and used to guarantee that all zeros are found.The use of finite-precision arithmetic may result in multiple zeros; however, in this case, the intervals coalesce and their number determines exactly the multiplicity of the zero.For an NxN matrix the eigenvalues can be determined in 0(log2 A^ time with N2 processors and O(N) time with N processors.The method is compared with a parallel variant of bisection that requires 0(N2) time on a single processor, O(N) time with A^ processors, and 0(log N) time with A^2 processors.

Cited by 3Open access

Convergence of LR algorithm for a one-point spectrum tridiagonal matrix. Numerische Mathematik

Carla Ferreira, Beresford Parlett · 2009

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. Key words. unsymmetric tridiagonal matrices, multiple eigenvalues, LR algorithm AMS subject classifications. 65F15 1. Introduction. This paper presents a rigorous proof that the LR algorithm, without shifts, applied to an unreduced tridiagonal matrix with a one-point spectrum converges to an upper bidi-agonal matrix. The rate of convergence is very slow, like 1/k after k steps, but what is remarkable is that the algorithm actually converges. We hasten to say that this result is not exactly new. In the middle of the 1960’s J.H. Wilkinson sketched out the underlying reason for this surprising result,

Cited by 0

AH-TDMA: An Adaptive Heterogeneous Tridiagonal Matrix Algorithm on the New Sunway Supercomputer

Min Tian, Qi Liu, Yanlong Li et al. · 2023

The solution of tridiagonal linear systems is used in in various fields and plays a crucial role in numerical simulations. However, there is few efficient solver for tridiagonal linear systems on the new Sunway supercomputer. Based on a three-dimensional heat conduction problem, we propose an adaptive heterogeneous tridiagonal matrix algorithm (AH-TDMA). Our major innovations include: (1) To address computational hotspots within AH-TDMA, a multi-level parallel approach involving MPI+Athread has been adopted. (2) An adaptive data partitioning scheme has been set up to achieve load balance. (3) Employing direct memory access and establishing shared space between registers and main memory, instead of employing discrete memory access, is done to enhance memory access efficiency. (4) The optimization of the loop structure has been made in adjusting the sequencing of the dual-layered loops to reduce communication overhead. The experimental results show that, with one core group, the AH-TDMA achieves a speedup of 99.6 times for hotspot and total time speedup up to 59.4 times compared to the Parallel and Scalable Library for Tridiagonal Matrix Algorithm (PaScal TDMA). The AH-TDMA is scalable up to 2048 core groups, with a parallel efficiency of 69.2%.

Cited by 0

Tridiagonal Matrix Algorithm for Real-Time Simulation of a Two-Dimensional PEM Fuel Cell Model

Daming Zhou, Fei Gao, Elena Breaz et al. · IEEE Transactions on Industrial Electronics · 2017

This paper presents a novel two-dimensional real-time modeling approach for a proton exchange membrane fuel cell (PEMFC) based on a tridiagonal matrix algorithm (Thomas algorithm). The Thomas algorithm consists of a forward elimination and a backward substitution, its arithmetic complexity of computations being much lower than the Gaussian elimination. In order to use this advanced numerical solver, the differential equations of reactant gas convection and diffusion phenomena in serpentine channels are transformed into a tridiagonal equations system. In addition, a three-level bisection algorithm has been developed to solve spatial physical quantities distribution for electrochemical domain. The real-time computing methods developed in this paper are then implemented in C language for a fast execution time in a real-time processor. The proposed real-time model is experimentally validated using a 1.2 kW Ballard NEXA fuel cell system, and its practical feasibilities in advanced real-time control for PEMFC systems have been experimentally demonstrated in an RT-LAB real-time simulator.

Cited by 25Open access

THE EIGENVALUES OF SYMMETRIC TRIDIAGONAL MATRIX METHOD WITH ACCELERATION MBY THE QR SHIFT

Wadia Faid Hassan Al-shameri · Journal of the faculty of Education · 2023

This research investigates the application of the QR - method for computing all the eigenvalues of the real symmetric tridiagonal matrix. The Householder method will be used for reduction of the real symmetric matrix to symmetric tridiagonal form, and then the so called QR - method with acceleration shift applies a sequence of orthogonal transformations to the symmetric tridiagonal matrix which converges to a similar matrix that is tridiagonal. This tridiagonal matrix possesses an eigenvalues similar to the eigenvalues of the symmetric tridiagonal matrix. Particular attention is paid to the shift technique that accelerates the rate of convergence. Computer algorithms for implementing the Householder's method and QR – method are presented. Computer Matlab programs for performing the Householder algorithm and the QR algorithm (with acceleration shift) are listed in the Appendix.

Cited by 0Open access

A New Algorithm for Computing the Inverse of Tridiagonal and Periodic Tridiagonal Matrices

Zhao Li-qun · 2010

A periodic tridiagonal matrix is transformed into a tridiagonal matrix for computing the inverse,by making the most of the structural characteristics and using the appropriate partitioned matrices. At the same time a new algorithm is proposed by crout factorization method for computing the inverse of a tridiagonal matrix,and it is applied to obtain the inverse of a periodic tridiagonal matrix. The algorithm can cut down the amount of calculation and storing capacity. From the numerical experiments it is known that the methods is effective.

Cited by 0

A Stable, Rational QR Algorithm for the Computation of the Eigenvalues of an Hermitian, Tridiagonal Matrix

C. Reinsch · Mathematics of Computation · 1971

The most efficient program for finding all the eigenvalues of a symmetric matrix is a combination of the Householder tridiagonalization and the QR algorithm.The latter, if carried out in a natural way, requires An additions, 10« multiplications, In divisions, and n square roots per iteration (n the order of the matrix).In 1963, Ortega and Kaiser showed that the process can be carried out using no square roots (and saving In multiplications).However, their algorithm is unstable and several modifications were suggested to increase its accuracy.We, too, want to give such a modification together with some examples demonstrating the achieved accuracy.1. Introduction.In 1961 Francis [4] proposed the QR transformation, an offspring of Rutishauser's LR transformation [8], for the computation of the eigenvalues of a general matrix.He considered his method to be inefficient for Hermitian matrices but, fortunately, it soon turned out that, contrary to his original opinion, the method is especially efficient for this class of matrices, provided the given matrix is first reduced by Householder's method to real tridiagonal form and provided that shifts are used to accelerate the rate of convergence.(A description of this technique can be found in [10], for tested ALGOL programs see [6], [3], [2], the properties of the now generally adopted shift are described in [11].)Ortega and Kaiser [7] pointed out that by avoiding square roots the efficiency of this algorithm can be further increased (though if all eigenvalues are to be computed, it is already superior to all other known methods).The algorithm which they proposed, however, was unstable and several modifications were suggested (e.g., [9], [5] and others, not published).We, too, want to give such a modification here, together with some examples demonstrating the achieved accuracy.2. The Algorithm.Let A be the shifted matrix with diagonal entries au ■ ■ ■ , a" and subdiagonal entries bu ■ •• , &»_!.As is well known, a QR step consists of the orthogonal-triangular decomposition, A = QR, and the recombination in reversed order, Ä = RQ.For tridiagonal matrices, the decomposition is usually done by the application of n -1 plane rotations P^, • • • , Pj_! from the left to A to produce the upper triangular matrix P with diagonal rx, ••• , r", first superdiagonal qu • • • , qn-u

Cited by 1

Files and plotting scripts for "Parallel tridiagonal matrix inversion with a hybrid multigrid--Thomas algorithm method"

Joseph Thomas Parker, P. Hill, David Dickinson et al. · Zenodo (CERN European Organization for Nuclear Research) · 2020

This archive contains the files required to reproduce the results and figures presented in <em>Parallel tridiagonal matrix inversion with a hybrid multigrid--Thomas algorithm method</em>, J. T. Parker, P. A. Hill, D. Dickinson and B. D. Dudson. Also available at the repository: https://gitlab.com/JosephThomasParker/files-and-plotting-scripts-for-parallel-tridiagonal-matrix-inversion-with-a-hybrid-multigrid-thomas-algorithm-method/ Questions to joseph.parker@ukaea.uk. This version is before submission to journal.

Cited by 1Open access