  This study investigates tridiagonal near-Toeplitz matrices in which the
Toeplitz part is strictly diagonally dominant. The focus is on determining the
exact inverse of these matrices and establishing upper bounds for the infinite
norms of the inverse matrices. For cases with $b > 2$ and $b < -2$, we derive
the compact form of the entries of the exact inverse. These results remain
valid even when the matrices' corners are not diagonally dominant, specifically
when $|\widetilde{b}| < 1$.
  Furthermore, we calculate the traces and row sums of the inverse matrices.
Afterwards, we present upper bound theorems for the infinite norms of the
inverse matrices. To demonstrate the effectiveness of the bounds and their
application, we provide numerical results for solving Fisher's problem.
  Our findings reveal that the converging rates of fixed-point iterations
closely align with the expected rates, and there is minimal disparity between
the upper bounds and infinite norm of the inverse matrix. Specifically, this
observation holds true when $b > 2$ with $\widetilde{b} \leq 1$ and $b < -2$
with $\widetilde{b} \geq -1$. For other cases, there is potential for further
improvement in the obtained upper bounds.
  This study contributes to the field of numerical analysis of fixed-point
iterations by improving the convergence rate of iterations and reducing the
computing time of the inverse matrices.
