A Low-cost and Numerically Stable Algorithm to Solve Tridiagonal Systems via Quasiseparable Matrices

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(n3). 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.