American Institute of Mathematical Sciences

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

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

Sigve Hovda 1,

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.
Keywords: Tridiagonal matrices, mass-spring model, lumped element model., matrix inversion, stiffness matrix.
Mathematics Subject Classification: Primary: 15A0.
Citation: Sigve Hovda. Closed-form expression for the inverse of a class of tridiagonal matrices. Numerical Algebra, Control & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

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