Numerical Differentiation by Finite-Dimensional Regularition

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Abstract
Tikhonov regularization of order one is applied to the differentiation operation. By minimizing the regularization functional over a finite-dimensional space there results a procedure for numerical differentiation. This finite-dimensional regularization results in a sparse, symmetric, positive definite matrix problem when cubic splines are chosen as the finite-dimensional space. The effects of error in the data on the values of the regularized derivative can be estimated in terms of the norm of the regularization operator. Numerical experiments are presented which illustrate the stability and obtainable accuracy of the method.

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