Abstract: The characterization of inverses of symmetric tridiagonal and block tridiagonal matrices and the development of algorithms for finding the inverse of any general non-singular tridiagonal matrix are subjects that have been studied by many authors. The results of these research usually depend on the existence of the LU factorization of a non-sigular matrix A, such that A = LU. Besides, the conditions that ensure the nonsingularity of A and its LU factorization are not promptly obtained. Then, we are going to present in this work two extremely simple sufficient conditions for existence of the LU factorization of a Toeplitz symmetric tridiagonal matrix A. We take into consideration the roots of the modified Chebyshev polynomial, and we also present an analysis based on the parameters of the Crout’s method.

Abstract: The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we improve the theoretical results presented in a previous paper “A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices” and provide a fast algorithm to find the upper and lower bounds of the interval eigenvalues of a class of symmetric tridiagonal interval matrices.

Abstract: A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent factorizations are established, leading to semi-closed-form formulas for the inverse sum of two single-pair matrices. An application to derive the symbolic inverse of a particular Gram matrix is presented.

Abstract: A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent factorizations are established, leading to semi-closed-form formulas for the inverse sum of two single-pair matrices. An application to derive the symbolic inverse of a particular Gram matrix is presented.

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 chapter, we derive a general expression for the entries of the mth (m 2 N) power for two certain types of tridiagonal matrices of arbitrary order. We also present a method for computing the positive integer powers of the one type of the anti-tridiagonal matrices corresponding to the matrix. Additionlly, we provide Maple 18 procedures in order to verify our calculations.

Abstract: Doubly bordered k-tridiagonal interval linear systems play a crucial role in various mathematical and engineering applications where uncertainty is inherent in the system’s parameters. In this paper, we propose a novel symbolic algorithm for solving such systems efficiently. Our approach combines symbolic computation techniques with interval arithmetic to provide rigorous solutions in the form of tight interval enclosures. By exploiting the tridiagonal structure and employing a divide-and-conquer strategy, our algorithm achieves significantly reduced computational complexity compared to existing numerical methods. We also present theoretical analysis and provide numerical experiments to demonstrate the effectiveness and accuracy of our algorithm. The proposed symbolic algorithm offers a valuable tool for handling doubly bordered k-tridiagonal interval linear systems and opens up possibilities for addressing uncertainty in real-world problems with improved efficiency and reliability.