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

Sigve  Hovda

doi:10.3934/naco.2016019

Article Contents

2016, Volume 6, Issue 4: 437-445. Doi: 10.3934/naco.2016019

This issue Previous Article A POD projection method for large-scale algebraic Riccati equations Next Article Solving higher index DAE optimal control problems

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

Received: June 2016

Revised: November 2016

Published: December 2016

Abstract / Introduction Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 15A09.

Citation:

Related Papers

Cited by

References

[1]	E. Asplund, Inverse of matrices {a_ij} which satisfy a_j= 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.