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

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.