Minimal Numerical Differentiation Formulas
Provides improved numerical differentiation tools for applications with irregular data points, though incremental in nature.
The paper develops numerical differentiation formulas for irregular centers that minimize absolute seminorms of weight vectors, achieving high accuracy in experiments with weighted ℓ1 and least squares methods.
We investigate numerical differentiation formulas on irregular centers in two or more variables that are exact for polynomials of a given order and minimize an absolute seminorm of the weight vector. Error bounds are given in terms of a growth function that carries the information about the geometry of the centers. Specific forms of weighted $\ell_1$ and weighted least squares minimization are proposed that produce numerical differentiation formulas with particularly good performance in numerical experiments.