Sparse Polynomial Hermite Interpolation

We present Hermite polynomial interpolation algorithms that for a sparse univariate polynomial f with coefficients from a field compute the polynomial from fewer points than the classical algorithms. If the interpolating polynomial f has t terms, our algorithms, which use randomization, require argument/value triples (wi,f(wi),f'(wi)) for i=0, ..., t + ↾(t+1)/2↿ - 1, where w is randomly sampled and the probability of a correct output is determined from a degree bound for f. With f' we denote the derivative of f. Our algorithms generalize to multivariate polynomials, higher derivatives and sparsity with respect to Chebyshev polynomial bases. We have algorithms that can correct errors in the points by oversampling at a limited number of good values. If an upper bound B ≥ t for the number of terms is given, our algorithms use a randomly selected w and, with high probability, t/2 + B triples, but then never return an incorrect output.