An incomplete tridiagonalization-based determinant evaluation for a generalized periodic tridiagonal matrix

In this paper, we first propose a novel incomplete tridiagonalization approach for evaluating the determinant of the generalized periodic tridiagonal matrix. By the incomplete tridiagonalization approach and Laplace’s theorem we transform the determinant of the generalized periodic tridiagonal matrix into the determinants of tridiagonal matrix and periodic tridiagonal matrix with lower-order. Then, we present a breakdown-free recursive algorithm for computing the determinant of the generalized periodic tridiagonal matrix based on the incomplete tridiagonalization approach. Even though the algorithm is not a symbolic algorithm, it never suffers from breakdown. Furthermore, we give an explicit formula for the determinant of the generalized periodic tridiagonal matrix with Toeplitz structure. Some numerical results with simulations in MATLAB implementation are provided to demonstrate the accuracy and effectiveness of the proposed algorithm, and its competitiveness with MATLAB built-in function.