These days, obtaining an optimal bound for smallest and/or largest eigenvalues or singular values of a given matrix is one of the important and notable issues. Although great achievements for the estimation of eigenvalues or singular values have been discovered, bounds obtained so far are not satisfactory yet. Recently, Buchholzer et al. [H. Buchholzer, C. Kanzow, Bounds for the extremal eigenvalues of a class of symmetric tridiagonal matrices with applications, Linear Algebra and its Applications 436 (2012) 1837–1849] have been extracted some sharp bounds for the extremal eigenvalues of a class of symmetric tridiagonal matrices with Toeplits structure which destroyed by perturbing two elements on each off-diagonal. They could also obtain a lower bound for the smallest singular value of (symmetric or asymmetric) Toeplitz tridiagonal positive definite matrices and apply these bounds in solving an advection-diffusion partial differential equation and shown that such bounds are very useful and applicable. In fact, they could apply their results to the discretization of a partial differential equation where matrices arise that can be decomposed as a Kronecker product of tridiagonal matrices of the mentioned structure. It must be pointed out that the key idea of their work is based on the behavior of obtained fixed points of a recursive equation. The main theoretical results contained in their work are depending on the relative (absolute) sizes of the matrix entries. Results show that their bounds are more appropriated rather than previous bounds. At the beginning of this thesis, after preparing some preliminaries, we follow and elaborate outstanding work of Buchholzer et al. To obtain our results, we take a closer look at the class of such matrices and exploit heavily the particular structure. The main theoretical results contained in our work are based on extending the work of Buchholzer et al. In fact, we extend and generalize their work to a class of symmetric tridiagonal matrices with Toeplits structure which destroyed by perturbing four elements on each off-diagonal and obtain some sharp and useful bounds. We have been extracted some sharp bounds for the extremal eigenvalues of these symmetric tridiagonal matrices with Toeplits structure. We could also obtain a lower bound for the smallest singular value of (symmetric or asymmetric) Toeplitz tridiagonal positive definite matrices. These matrices appear in some practical problems. For example, they appear in solving some ordinary or partial differential equations, and obtaining a bound for eigenvalues or singular values of these matrices is very important issues in investigating the stability of the numerical methods. There exist many results for more general matrices like Gershgorin’s, Ostrowski’s or Brauer’s Theorem that estimate the area to which the eigenvalues belong to, however, the bounds one obtains from these results for the particular class of matrices considered here are by far too weak. On the other hand, there are many bounds obtained in the literature for the smallest singular value of some particular matrices, but we did not deal with them because most of them are based on the determinant and the process of calculation of determinant is very expensive. Some numerical results indicate that our bounds are extremely good.