Differentiating matrices for arbitrarily spaced grid points

Differentiating matrices allow the numerical differentiation of functions defined at points of a discrete grid. A type of differentiating matrix based on local approximation on a sequence of sliding subgrids is considered. Previous derivations of this type of matrix have been restricted to grids with uniformly spaced points, and the resulting derivative approximations have lacked precision, especially at endpoints. The new formulation allows grids which have arbitrarily space points. It is shown that high accuracy can be achieved through use of differentiating matrices on non-uniform grids which include near-boundary points. Use of the differentiating matrix as an operator to solve eigenvalue problems involving ordinary differential equations is also considered.