Case studies of choosing a numerical differentiation method under uncertainty

Abstract

In many real-life situations (including computer-aided design and radiotelescope network design), it is necessary to estimate the derivative of a function from approximate measurement results. Usually, there exist several (approximate) models that describe measurement errors; these models may have different numbers of parameters. If we use different models, we may get estimates of different accuracy. In the design stage, we often have little information about these models, so, it is necessary to choose a model based only on the number of parameters n and on the number of measurements N. In mathematical terms, we want to estimate how having N equations Σ j c ij a j = y i with n ( n < N ) unknowns a j influences the accuracy of the result ( c ij are known coefficients, and y i are known with a standard deviation σ[ y ]). For that, we assume that the coefficients c ij are independent random variables with 0 average and standard deviation 1 (this assumption is in good accordance with real-life situations). Then, we can use computer simulations to find the standard deviation σ' of the resulting error distribution for a i . For large n , this distribution is close to Gaussian (see, e.g., [21], pp. 2.17, 6.5, 9.8, and reference therein), so, we can safely assume that the actual errors are within the 3σ' limit.