Tridiagonal matrix algorithm

Abstract: Abstract This paper presents an approach to efficiently solve a system of linear equations characterized by n × n non-singular tridiagonal matrices utilizing quasiseparable structures. By employing sparse factorization of the quasiseparable matrices, we obtain a low-cost, i.e., O(n) , in contrast to the brute-force computations associated with solving tridiagonal systems with complexity O(n 3 ) . Furthermore, the proposed algorithm provides an alternative method for solving systems of equations having tridiagonal Toeplitz coefficient matrices achieving O(n) complexity algorithm.To ensure the stability and accuracy of the algorithm, we present backward and forward error results in solving the tridiagonal system of equations. Finally, the paper presents signal flow graphs to demonstrate the proposed algorithm's reliability and simplicity and realize it as an architecture for very large-scale integrated circuits. To sum up, the paper offers efficient, exact, and numerically stable algorithms in solving systems of linear equations having non-singular tridiagonal and tridiagonal Toeplitz matrices, providing a compelling alternative to brute-force calculation with a significantly reduced computational cost and digital signal processing architecture of a physical system.

Abstract: In this article, the determinant of tridiagonal Toeplitz matrices is determined recursively and explicitly. The method used is descriptive exploratory the journal written by Fitri Aryani. The inverse of tridiagonal Toeplitz matrices is calculated using the adjoint method, but the determinant and adjoint of the matrices are based on the recursive calculation of the determinant. With this approach, the formulas for the determinant and inverse of tridiagonal Toeplitz matrices can be formulated clearly and efficiently. This study demonstrates the effectiveness of the method used in simplifying computations and provides an algorithm for the formulation.

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.