Estimates for the inverse of tridiagonal matrices arising in boundary-value problems☆

By considering tridiagonal matrices as three-term recurrence relations with Dirichlet boundary conditions, one can formulate their inverses in terms of Green's functions. This analysis is applied to three-point difference schemes for 1-D problems, and five-point difference schemes for 2-D problems. We derive either an explicit inverse of the Jacobian or a sharp estimate for both uniform and nonuniform grids.