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Variant of the thomas algorithm for opposite-bordered tridiagonal systems of equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2010年第6期26卷 752-759页

作者： Martin, Alexandre Boyd, Iain D. Univ Michigan Dept Aerosp Engn Ann Arbor MI 48109 USA

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 O(n) operations, instead of O(n(3)). 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 (C) 2008 John Wiley & Sons, Ltd.

关键词： linear solver thomas algorithm elementary column operations

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A generalized symbolic thomas algorithm for the solution of opposite-bordered tridiagonal linear systems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2015年 290卷 423-432页

作者： Jia, Jiteng Sogabe, Tomohiro Li, Sumei Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shaanxi Peoples R China Aichi Prefectural Univ Grad Sch Informat Sci & Technol Nagakute Aichi 4801198 Japan

In the current paper, we present a generalized symbolic thomas algorithm, that never suffers from breakdown, for solving the opposite-bordered tridiagonal (OBT) linear systems. The algorithm uses a fill-in matrix factorization and can solve an OBT linear system in 0(n) operations. Meanwhile, an efficient method of evaluating the determinant of an opposite-bordered tridiagonal matrix is derived. The computational costs of the proposed algorithms are also discussed. Moreover, three numerical examples are provided in order to demonstrate the performance and effectiveness of our algorithms and their competitiveness with some already existing algorithms. All of the experiments are performed on a computer with the aid of programs written in Matlab. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Opposite-bordered tridiagonal matrices Matrix factorization Linear solver Determinant Computational cost thomas algorithm

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Extension of the thomas algorithm to a class of algebraic linear equation systems involving quasi-block-tridiagonal matrices with isolated block-pentadiagonal rows, assuming variable block dimensions

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COMPUTING 2001年第4期67卷 269-285页

作者： Bieniasz, LK Polish Acad Sci Inst Phys Chem Molten Salts Lab PL-30318 Krakow Poland

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.

关键词： quasi-block-tridiagonal matrices thomas algorithm boundary value problems initial boundary value problems finite-difference methods

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An efficient implementation of the thomas-algorithm for block penta-diagonal systems on vector computers

An efficient implementation of the Thomas-algorithm for bloc...

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7th International Conference on Computational Science (ICCS 2007)

作者： Benkert, Katharina Fischer, Rudolf Univ Stuttgart High Performance Comp Ctr HLRS D-70569 Stuttgart Germany NEC Corp Ltd High Performance Comp Europe GmbH D-40549 Dusseldorf Germany

ISBN: (纸本)9783540725831

In simulations of supernovae, linear systems of equations with a block penta-diagonal matrix possessing small, dense matrix blocks occur. For an efficient solution, a compact multiplication scheme based on a restructured version of the thomas algorithm and specifically adapted routines for LU factorization as well as forward and backward substitution are presented. On a NEC SX-8 vector system, runtime could be decreased between 35 % and 54 % for block sizes varying from 20 to 85 compared to the original code with BLAS and LAPACK routines.

关键词： thomas algorithm vector architecture

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Computational analysis of a normalized time-fractional Fokker-Planck equation

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PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS 2025年 665卷

作者： Wang, Jian Chen, Keyong Kim, Junseok Nanjing Univ Informat Sci & Technol Sch Math & Stat Nanjing 210044 Peoples R China Korea Univ Dept Math Seoul 02841 South Korea

We propose a normalized time-fractional Fokker-Planck equation (TFFPE). A finite ence method is used to develop a computational method for solving the equation, system's dynamics are investigated through computational simulations. The proposed demonstrates accuracy and efficiency in approximating analytical solutions. Numerical validate the method's effectiveness and highlight the impact of various fractional the dynamics of the normalized time-fractional Fokker-Planck equation. The numerical emphasize the significant impact of different fractional orders on the temporal evolution the system. Specifically, the computational results demonstrate how varying the order influences the diffusion process, with lower orders exhibiting stronger memory and slower diffusion, while higher orders lead to faster propagation and a transition classical diffusion behavior. This work contributes to the understanding of fractional and provides a robust tool for simulating time-fractional systems.

关键词： Normalized time-fractional Fokker-Planck equation thomas algorithm Probability distribution

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An asymptotic approximation of the solution for nearly tridiagonal quasi-Toeplitz linear systems

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MATHEMATICS AND COMPUTERS IN SIMULATION 2025年 234卷 359-367页

作者： Kim, Philsu Park, Sangbeom Kim, Seonghak Bak, Soyoon Kyungpook Natl Univ Dept Math Daegu 41566 South Korea

We introduce an asymptotic approximate algorithm for solving nearly tridiagonal quasi-Toeplitz linear systems. When addressing low-rank perturbations of a tridiagonal Toeplitz matrix system based on the Sherman-Morrison-Woodbury formula (or Woodbury identity), conventional methods require solving at least two simpler systems. The proposed algorithm overcomes this limitation by providing an explicit asymptotic formula for one of these systems. This asymptotic approximation enables a rapid resolution of the original system with minimal additional computation. To validate the accuracy and efficiency of the proposed algorithm, we conduct numerical experiments on two cases, comparing the results with those of existing methods. The results demonstrate that the proposed algorithm significantly reduces computation time while maintaining accuracy compared to the existing methods.

关键词： Tridiagonal Toeplitz matrix thomas algorithm LU decomposition Sherman-Morrison-Woodbury formula

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Fast and accurate recursive algorithms for solving cyclic tridiagonal linear systems

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JOURNAL OF MATHEMATICAL CHEMISTRY 2025年第5期63卷 1211-1228页

作者： Li, Su-Mei Fan, Xin Wang, Yi-Fan Xian Univ Sci & Technol Sch Sci Xian 710054 Shaanxi Peoples R China Xidian Univ Sch Math & Stat Xian 710071 Shaanxi Peoples R China

Cyclic tridiagonal matrices, a specific subclass of quasi-tridiagonal matrices, frequently arise in theoretical and computational chemistry. This paper addresses the solution of cyclic tridiagonal linear systems with coefficient matrices that are subdiagonally dominant, superdiagonally dominant and weakly diagonally dominant. For the subdiagonally dominant case, we perform an elementary transformation to convert the matrix into a block 2-by-2 form, then solve the system using block LU factorization. For the superdiagonally dominant and weakly diagonally dominant cases, we extend this approach using block LU factorization and matrix similarity transformations. Our proposed algorithms outperform existing methods in terms of floating-point operations, memory storage, and data transmission. Numerical experiments, implemented in MATLAB, demonstrate the accuracy and efficiency of the proposed algorithms.

关键词： Cyclic tridiagonal matrices Linear systems Block LU factorization Matrix transformation thomas algorithm

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Accelerated thomas Solver for (Quasi-)Block-Tridiagonal Linear Algebraic Equation Systems, Using SSE/AVX Instruction Sets for Vectorizing Dense Block Operations

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INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS 2016年第6期13卷

作者： Barnas, Dawid Bieniasz, Leslaw K. Cracow Univ Technol Fac Math Phys & Comp Sci Ul Warszawska 24 PL-31155 Krakow Poland

Streaming SIMD Extensions (SSE) and Advanced Vector Extensions (AVX) are additional processor instruction sets available in contemporary personal computers, designed for vectorized floating point calculations. Unfortunately, in order to utilize the advantages of these instructions, one cannot rely on automatic options of high level language compilers. Instead, handwritten assembly language or intrinsic function call insertions are necessary. By using this idea an accelerated C++ code is devised, for solving (quasi-) block-tridiagonal linear algebraic equation systems by means of an extended thomas algorithm. Speedups reaching 3.5 and 3 (relative to C++ without using SSE/AVX) are demonstrated for single and double precision calculations, respectively.

关键词： Quasi-block-tridiagonal thomas algorithm Streaming SIMD Extensions Advanced Vector Extensions computational electrochemistry

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Efficient graphic processing unit implementation of the chemical-potential multiphase lattice Boltzmann method

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INTERNATIONAL JOURNAL OF HIGH PERFORMANCE COMPUTING APPLICATIONS 2021年第1期35卷 78-96页

作者： Ye, Yutong Zhu, Hongyin Zhang, Chaoying Wen, Binghai Guangxi Normal Univ Guangxi Key Lab Multisource Informat Min & Secur Guilin 541004 Peoples R China East China Normal Univ Shanghai Key Lab Trustworthy Comp Shanghai Peoples R China

The chemical-potential multiphase lattice Boltzmann method (CP-LBM) has the advantages of satisfying the thermodynamic consistency and Galilean invariance, and it realizes a very large density ratio and easily expresses the surface wettability. Compared with the traditional central difference scheme, the CP-LBM uses the thomas algorithm to calculate the differences in the multiphase simulations, which significantly improves the calculation accuracy but increases the calculation complexity. In this study, we designed and implemented a parallel algorithm for the chemical-potential model on a graphic processing unit (GPU). Several strategies were used to optimize the GPU algorithm, such as coalesced access, instruction throughput, thread organization, memory access, and loop unrolling. Compared with dual-Xeon 5117 CPU server, our methods achieved 95 times speedup on an NVIDIA RTX 2080Ti GPU and 106 times speedup on an NVIDIA Tesla P100 GPU. When the algorithm was extended to the environment with dual NVIDIA Tesla P100 GPUs, 189 times speedup was achieved and the workload of each GPU reached 96%.

关键词： Multi-strategy optimization lattice Boltzmann method graphic processing unit chemical potential thomas algorithm

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Restricted gene expression programming: a new approach for parameter identification inverse problems of partial differential equation

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SOFT COMPUTING 2017年第10期21卷 2651-2663页

作者： Chen, Yan Li, Kangshun Chen, Zhangxing Wang, Jinfeng South China Agr Univ Coll Math & Informat Guangzhou 510642 Guangdong Peoples R China Univ Calgary Dept Chem & Petr Engn Calgary AB Canada

Traditionally, solving the parameter identification inverse problems of partial differential equation encountered many difficulties and insufficiency. In this article, we proposed a restricted gene expression programming (GEP) for parameter identification inverse problems of partial differential equation based on the self-adaption, self-organization, and self-learning characters of GEP. The algorithm simulates parametric function itself of partial differential equation directly through the observed values by taking effect to inverse results caused by noise of the measured value into full consideration. Modeling is unnecessary to add regularization in modeling process aiming at special problems again. The experiment results also showed that the algorithm has good noise-immunity. In case there is no noise or noise is very low, the identified parametric function is almost the same as the original accurate value;when noise is very high, accurate result can still be obtained, which successfully realizes automation of parameter modeling process for partial differential equation.

关键词： Gene expression programming Partial differential equation Inverse problems thomas algorithm

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