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. Finite dierence methods, such as the mid-point rule, have been applied successfully to the numerical solution of ordinary and partial dierential 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 amplied in the dierentiation 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 dierentiation of higher dimensional observational data can be stabilized with a reduced loss of accuracy than occurs for the corresponding differentiation of one-dimensional data. The result is then extended to the multivariate dierentiation of higher dimensional data. The nature of the trade-o between convergence and stability is explicitly characterized, and the complexity of various implementations is examined.

For numerical differentiation, dimensionality can be a blessing!