A general solution for a special tri-diagonal linear system
Peerayuth Charnsethikulfengprc@ku.ac.thDepartment of Industrial EngineeringKasetsart UniversityBangkok, 10900, ThailandView full profile → , Chusana Nuntanartdew_159@hotmail.comDepartment of Industrial EngineeringKasetsart UniversityBangkok, 10900, ThailandView full profile → , *Aphisak WitthayapraphakornCorresponding authoraphisak.wi@up.ac.thDepartment of Industrial EngineeringFaculty of Engineering University of PhayaoPhayao, 56000, ThailandView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 10 Oct 2023
- Published Online:
- 07 May 2024
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-1860
- Pages:
- 601–617
Abstract
Keywords
Subject Classifications
References
[1] Strang, G., Linear Algebra and its Applications, 4th edition, Brook Coles (2005).
[2] Zienkiewicz, O.C., Taylor, R. L. and Zhu, J.Z., The Finite Element Method: Its Basis and Fundamentals, 6th Edition, Butterworth-Heinemann (2005).
[3] http://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm.
[4] Dubeau, F., “Linear algebra and the sums of powers of integers”, Electronic Journal Linear Algebra, Vol. 17, pp. 577-596 (2008).
[5] Knuth, D., “Johann Faulhaber and sums of powers”, Mathematics of Computation, Vol. 61, no. 203, pp.277-294 (1993).
[6] Parag V. Patil et al., “Algorithm: Three Dimensional Finite Volume Numerical Grid Technique”, Global Journal of Pure and Applied Mathematics, Vol. 13, Number 9, pp. 5655-5671 (2017).
[7] T. Wang, Y. Wang, and Z. Cheng, “Stability and Hopf Bifurcation Analysis of a General Tri-diagonal BAM Neural Network with Delays”, Neural Processing Letters, vol. 53, pp. 4571–4592, Aug. (2021).
[8] Patil, Kulkarni, “A numerical study on MHD double diffusive nonlinear mixed convective nanofluid flow around a vertical wedge with diffusion of liquid hydrogen”, Journal of the Egyptian Mathematical Society, Vol. 29, Article number: 24 (2021).
[9] M.E.A. El-Mikkawy, “A new computational algorithm for solving periodic tri-diagonal linear systems”, Applied Mathematics and Computation, Vol. 161, Issue 2, pp. 691-696 (2005).
[10] D. Yambangwai, W. Cholamjiak, T. Thianwan, H. Dutta, “On a new weight tri-diagonal iterative method and its applications”, Soft Computing, vol 25, pp.725–740 (2020).
[11] Rihuan Ke, Michael K. Ng, Hai-Wei Sun, “A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations”, Journal of Computational Physics, Vol. 303, pp. 203-211 (2015).
[12] D. J. Warne, N. A. Kelson, R. F. Hayward, “Solving Tri-Diagonal Linear Systems using Field Programmable Gate Arrays”, 4th International Conference on Computational Methods (ICCM 2012) (2012).
[13] Di Zhao, Jinhang Yu, “Efficiently solving tri-diagonal system by chunked cyclic reduction and single-GPU shared memory”, The Journal of Supercomputing, Vol. 71, pp. 369–390 (2014).
[14] J. S. V. R. Krishna Prasad, Parag V. Patil, “Algorithm for Solving Tri-diagonal Finite Volume Discretized Linear Systems”, Applications and Applied Mathematics: An International Journal, Vol. 10, Issue 2, pp. 995-1006 (2015).
[15] R.K. Mohanty, “An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients”, Applied Mathematics and Computation, Vol. 165, Issue 1, pp. 229-236 (2005).
[16] Yao Huang, Bing-Jia Xiao, Zheng-Ping Luo, “Fast parallel Grad–Shafranov solver for real-time equilibrium reconstruction in EAST tokamak using graphic processing unit”, Chinese Physics B, Vol. 26, Number 8 (2017).
[17] Meichen Guo, Adair Lang, Michael Cantoni, “Structured Moving Horizon Estimation for Linear System Chains”, 18th European Control Conference (ECC) (2019).
[18] R.K. Mohanty, “A class of non-uniform mesh three point arithmetic average discretization for y’’ = f(x,y,y’) and the estimates of y’ “, Applied Mathematics and Computation, Vol. 183, Issue 1, pp. 477-485 (2006).
[19] Serguei Patchkovskii, H.G. Muller, “Simple, accurate, and efficient implementation of 1-electron atomic time-dependent Schrödinger equation in spherical coordinates”, Computer Physics Communications, Vol. 199, pp. 153-169 (2016).
[20] J. F. Barbero G., J. Margalef-Bentabol, and E. J. S. Villaseñor, “A two-sided Faulhaber-like formula involving Bernoulli polynomials”, Comptes Rendus Mathématique, Vol. 358, issue 1, pp. 41-44 (2020).
[21] M. Gnewuch, H. Pasing, and C. Weiß, “A Generalized Faulhaber Inequality, Improved Bracketing Covers, and Applications to Discrepancy”, Mathematics of Computation, Vol. 90, pp. 2873-2898 (2021).
[22] M. Ono and S. Yamamoto, “On the refined Kaneko–Zagier conjecture for general integer indices”, Mathematische Zeitschrift, vol. 305, no. 1, pp. 1-13, Aug. (2023).
[23] K. J. McGown and H. R. Parks, “The generalization of Faulhaber’s formula to sums of non-integral powers”, Journal of Mathematical Analysis and Applications, Vol. 330, Issue 1, pp. 571-575 (2007).
[24] W. Y. C. Chen, A. M. Fu, and I. F. Zhang, “Faulhaber’s theorem on power sums”, Discrete Mathematics, Vol. 309, Issue 10, pp. 2974-2981 (2009).
[25] B. C. Kellner and J. Sondow, “Power-Sum Denominators”, The American Mathematical Monthly, Vol. 124, No. 8, pp. 695-709 (2017).
[26] D. B. Fairlie and A. P. Veselov, “Faulhaber and Bernoulli Polynomials and Solitons”, Physica D: Nonlinear Phenomena, Vol. 152–153, pp. 47-50 (2001).
[27] N. Kilar and Y. Simsek, “Formulas Involving Sums of Powers, Special Numbers and Polynomials Arising from p-Adic Integrals, Trigonometric and Generating Functions”, Publications De L’institut Mathématique, Nouvelle série, tome 108(122), pp. 103–120 (2020).
[28] H. R. Parks, “Sums of non-integral powers”, Journal of Mathematical Analysis and Applications, Vol. 297, Issue 1, pp. 343-349 (2004).
[29] V. J. W. Guo, M. Rubey, and J. Zeng, “Combinatorial interpretations of the q-Faulhaber and q-Salié coefficients”, Journal of Combinatorial Theory, Series A, Vol. 113, Issue 7, pp. 1501-1515 (2006).
[30] M. J. Wang, S. Goel, and G. Mao, “A New Algorithm to Generate A Formula for the Sum of Integer Powers”, Proceedings of the 2014 ACM Southeast Regional Conference, Article No.: 50, pp. 1–4 (2014).
