2016, 6(4): 437-445. doi: 10.3934/naco.2016019

Closed-form expression for the inverse of a class of tridiagonal matrices

1. 

Department of Petroleum and Applied Geophysics, Norwegian University of Science and Technology, Trondheim, Norway

Received  June 2016 Revised  November 2016 Published  December 2016

Despite the simplicity of tridiagonal matrices, they have shown to be very resilient to closed-form solutions. We consider a class of tridiagonal stiffness matrices that stems from a variety of lumped element models in mechanical, acoustical and electrical systems. The computational efforts in such models are related to solving the generalized eigenvalue problem and finding the inverse of the stiffness matrix. To improve accuracy, it is desired to discretisize the problem as much as possible at the expense of growing matrices. This paper improves the efficiency of finding the inverse by a factor of at least three and the computational memory involved is at least halved. Moreover, the result provides an analytical expression for where the stable position is, which might be used in control systems. Surprisingly, it is the practical application itself that guides the proof.
Citation: Sigve Hovda. Closed-form expression for the inverse of a class of tridiagonal matrices. Numerical Algebra, Control and Optimization, 2016, 6 (4) : 437-445. doi: 10.3934/naco.2016019
References:
[1]

E. Asplund, Inverse of matrices {aij} which satisfy aj= 0 for j > i+p, Mathematica Scandinavia, 7 (1959), 57-60.

[2]

W. W. Barrett, A theorem on inverse of tridiagonal matrices, Linear Algebra and its Applications, 27 (1979), 211-217. doi: 10.1016/0024-3795(79)90043-0.

[3]

J. W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997. doi: 10.1137/1.9781611971446.

[4]

M. E. A. El-Mikkawy, On the inverse of a general tridiagonal matrix, Applied Mathematics and Computation, 150 (2004), 669-679. doi: 10.1016/S0096-3003(03)00298-4.

[5]

D. K. Fadeev, Properties of a matrix, inverse to a hessenberg matrix, Journal of Sovjet Mathematics, 24 (1984), 118-120.

[6]

C. D. Fonseca, On the eigenvalues of some tridiagonal matrices, Journal of Computational and Applied Mathematics, 200 (2007), 283-286. doi: 10.1016/j.cam.2005.08.047.

[7]

G. Hu and R. F. O'Connell, Analytical inversion of symmetric tridiagonal matrices, Journal of Physics A: Mathematical and General, 29 (1996), 1511-1513. doi: 10.1088/0305-4470/29/7/020.

[8]

E. Kilic, Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions, Applied Mathematics and Computation, 197 (2008), 345-357. doi: 10.1016/j.amc.2007.07.046.

[9]

R. K. Mallik, The inverse of a tridiagonal matrix, Linear Algebra and its Applications, 325 (2001), 109-139. doi: 10.1016/S0024-3795(00)00262-7.

[10]

G. Meurant, A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM Journal on Matrix Analysis and Applications, 13 (1992), 707-728. doi: 10.1137/0613045.

[11]

K. S. Narendra and A. M. Annaswarny, Stable Adaptive Systems, Prentice Hall, 1989.

[12]

K. U. Siddiqui and M. K. Singh, Mechanical System Design, New Age International, 2007.

[13]

T. L. Smith and K. S. Smith, Mechanical Vibrations : Modeling and Measurement, Springer, 2011. doi: 10.1007/978-1-4614-0460-6.

[14]

F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, Society of Industrial and Applied Mathematics, Review, 43 (2001), 235-286. doi: 10.1137/S0036144500381988.

[15]

R. Usmani, Inversion of a tridiagonal jacobi matrix, Computers & Mathematics with Applications, 27 (1994), 59-66. doi: 10.1016/0898-1221(94)90066-3.

[16]

R. Vandebril, M. V. Barel and N. Mastronardi, Matrix Computations and Semiseparable Matrices: Linear Systems, Johns Hopkins University Press, 2007.

show all references

References:
[1]

E. Asplund, Inverse of matrices {aij} which satisfy aj= 0 for j > i+p, Mathematica Scandinavia, 7 (1959), 57-60.

[2]

W. W. Barrett, A theorem on inverse of tridiagonal matrices, Linear Algebra and its Applications, 27 (1979), 211-217. doi: 10.1016/0024-3795(79)90043-0.

[3]

J. W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997. doi: 10.1137/1.9781611971446.

[4]

M. E. A. El-Mikkawy, On the inverse of a general tridiagonal matrix, Applied Mathematics and Computation, 150 (2004), 669-679. doi: 10.1016/S0096-3003(03)00298-4.

[5]

D. K. Fadeev, Properties of a matrix, inverse to a hessenberg matrix, Journal of Sovjet Mathematics, 24 (1984), 118-120.

[6]

C. D. Fonseca, On the eigenvalues of some tridiagonal matrices, Journal of Computational and Applied Mathematics, 200 (2007), 283-286. doi: 10.1016/j.cam.2005.08.047.

[7]

G. Hu and R. F. O'Connell, Analytical inversion of symmetric tridiagonal matrices, Journal of Physics A: Mathematical and General, 29 (1996), 1511-1513. doi: 10.1088/0305-4470/29/7/020.

[8]

E. Kilic, Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions, Applied Mathematics and Computation, 197 (2008), 345-357. doi: 10.1016/j.amc.2007.07.046.

[9]

R. K. Mallik, The inverse of a tridiagonal matrix, Linear Algebra and its Applications, 325 (2001), 109-139. doi: 10.1016/S0024-3795(00)00262-7.

[10]

G. Meurant, A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM Journal on Matrix Analysis and Applications, 13 (1992), 707-728. doi: 10.1137/0613045.

[11]

K. S. Narendra and A. M. Annaswarny, Stable Adaptive Systems, Prentice Hall, 1989.

[12]

K. U. Siddiqui and M. K. Singh, Mechanical System Design, New Age International, 2007.

[13]

T. L. Smith and K. S. Smith, Mechanical Vibrations : Modeling and Measurement, Springer, 2011. doi: 10.1007/978-1-4614-0460-6.

[14]

F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, Society of Industrial and Applied Mathematics, Review, 43 (2001), 235-286. doi: 10.1137/S0036144500381988.

[15]

R. Usmani, Inversion of a tridiagonal jacobi matrix, Computers & Mathematics with Applications, 27 (1994), 59-66. doi: 10.1016/0898-1221(94)90066-3.

[16]

R. Vandebril, M. V. Barel and N. Mastronardi, Matrix Computations and Semiseparable Matrices: Linear Systems, Johns Hopkins University Press, 2007.

[1]

Eric Bedford, Kyounghee Kim. Degree growth of matrix inversion: Birational maps of symmetric, cyclic matrices. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 977-1013. doi: 10.3934/dcds.2008.21.977

[2]

Gero Friesecke, Karsten Matthies. Geometric solitary waves in a 2D mass-spring lattice. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 105-114. doi: 10.3934/dcdsb.2003.3.105

[3]

Travis G. Draper, Fernando Guevara Vasquez, Justin Cheuk-Lum Tse, Toren E. Wallengren, Kenneth Zheng. Matrix valued inverse problems on graphs with application to mass-spring-damper systems. Networks and Heterogeneous Media, 2020, 15 (1) : 1-28. doi: 10.3934/nhm.2020001

[4]

Rafael del Rio, Mikhail Kudryavtsev, Luis O. Silva. Inverse problems for Jacobi operators III: Mass-spring perturbations of semi-infinite systems. Inverse Problems and Imaging, 2012, 6 (4) : 599-621. doi: 10.3934/ipi.2012.6.599

[5]

Seung-Yeal Ha, Hansol Park. Emergent behaviors of the generalized Lohe matrix model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4227-4261. doi: 10.3934/dcdsb.2020286

[6]

Anthony Tongen, María Zubillaga, Jorge E. Rabinovich. A two-sex matrix population model to represent harem structure. Mathematical Biosciences & Engineering, 2016, 13 (5) : 1077-1092. doi: 10.3934/mbe.2016031

[7]

Marco Scianna, Luigi Preziosi, Katarina Wolf. A Cellular Potts model simulating cell migration on and in matrix environments. Mathematical Biosciences & Engineering, 2013, 10 (1) : 235-261. doi: 10.3934/mbe.2013.10.235

[8]

Vassilios A. Tsachouridis, Georgios Giantamidis, Stylianos Basagiannis, Kostas Kouramas. Formal analysis of the Schulz matrix inversion algorithm: A paradigm towards computer aided verification of general matrix flow solvers. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 177-206. doi: 10.3934/naco.2019047

[9]

Scott R. Pope, Laura M. Ellwein, Cheryl L. Zapata, Vera Novak, C. T. Kelley, Mette S. Olufsen. Estimation and identification of parameters in a lumped cerebrovascular model. Mathematical Biosciences & Engineering, 2009, 6 (1) : 93-115. doi: 10.3934/mbe.2009.6.93

[10]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247

[11]

Robert Stephen Cantrell, Brian Coomes, Yifan Sha. A tridiagonal patch model of bacteria inhabiting a Nanofabricated landscape. Mathematical Biosciences & Engineering, 2017, 14 (4) : 953-973. doi: 10.3934/mbe.2017050

[12]

Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479

[13]

Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic and Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707

[14]

Panagiotes A. Voltairas, Antonios Charalambopoulos, Dimitrios I. Fotiadis, Lambros K. Michalis. A quasi-lumped model for the peripheral distortion of the arterial pulse. Mathematical Biosciences & Engineering, 2012, 9 (1) : 175-198. doi: 10.3934/mbe.2012.9.175

[15]

Nalin Fonseka, Ratnasingham Shivaji, Jerome Goddard, Ⅱ, Quinn A. Morris, Byungjae Son. On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3401-3415. doi: 10.3934/dcdss.2020245

[16]

Gianluca D'Antonio, Paul Macklin, Luigi Preziosi. An agent-based model for elasto-plastic mechanical interactions between cells, basement membrane and extracellular matrix. Mathematical Biosciences & Engineering, 2013, 10 (1) : 75-101. doi: 10.3934/mbe.2013.10.75

[17]

Zhuchun Li, Xiaoping Xue, Seung-Yeal Ha. A revisit to the consensus for linearized Vicsek model under joint rooted leadership via a special matrix. Networks and Heterogeneous Media, 2014, 9 (2) : 335-351. doi: 10.3934/nhm.2014.9.335

[18]

Jacek Banasiak, Amartya Goswami. Singularly perturbed population models with reducible migration matrix 1. Sova-Kurtz theorem and the convergence to the aggregated model. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 617-635. doi: 10.3934/dcds.2015.35.617

[19]

Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5509-5517. doi: 10.3934/dcdsb.2020356

[20]

Marie Turčičová, Jan Mandel, Kryštof Eben. Score matching filters for Gaussian Markov random fields with a linear model of the precision matrix. Foundations of Data Science, 2021, 3 (4) : 793-824. doi: 10.3934/fods.2021030

 Impact Factor: 

Metrics

  • PDF downloads (985)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]