For Numerical Differentiation , Dimensionality Can Be

Finite diierence methods, like the mid-point rule, have been applied successfully to the numerical solution of ordinary and partial diierential equations. If such formulas are applied to observational data, in order to determine derivatives, the results can be disastrous. The reason for this is that measurement errors, and even rounding errors in computer approximations, are strongly ampliied in the diierentiation process, especially if small step-sizes are chosen and higher derivatives are required. A number of authors have examined the use of various forms of averaging which allows the stable computation of low order derivatives from observational data. The size of the averaging set acts like a regularization parameter and has to be chosen as a function of the grid size h. In this paper, it is initially shown how rst (and higher) order single-variate numerical diierentiation of higher dimensional observational data can be stabilized with a reduced loss of accuracy than occurs for the corresponding diierentiation of one-dimensional data. The result is then extended to the multivariate diierentiation of higher dimensional data. The nature of the trade-oo between convergence and stability is explicitly characterized, and the complexity of various implementations is examined.