  This paper analyzes the convergence of fixed-point iterations of the form u =
f(u) and the properties of the inverse of the related pentadiagonal matrices,
associated with the fourth-order nonlinear beam equation. This nonlinear
problem is discretized using the finite difference method with the clamped-free
and clamped-clamped boundary conditions in the one dimension. Explicit formulas
for the inverse of the matrices and norms of the inverse are derived. In
iterative process, the direct computation of inverse matrix allows to achieve
an efficiency. Numerical results were provided.
