Sparse Polynomial Hermite Interpolation

Abstract

We present Hermite polynomial interpolation algorithms that for a sparse univariate polynomial with coefficients from a field compute the polynomial from fewer points than the classical algorithms. If the interpolating polynomial has terms, our algorithms, require argument/value triples ( , ( ), ′ ( )) for = 0, . . . , + ⌈( +1)/2⌉ −1, where is randomly sampled and the probability of a correct output is determined from a degree bound for . With ′ we denote the derivative of . 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 ≥ for the number of terms is given, our algorithms use a randomly selected and, with high probability, ⌈ /2⌉ + triples, but then never return an incorrect output.The algorithms are based on Prony's sparse interpolation algorithm. While Prony's algorithm and its variants use fewer values, namely, 2 + 1 and + values ( ), respectively, they need more arguments . The situation mirrors that in algebraic error correcting codes, where the Reed-Solomon code requires fewer values than the multiplicity code, which is based on Hermite interpolation, but the Reed-Solomon code requires more distinct arguments. Our sparse Hermite interpolation algorithms can interpolate polynomials over finite fields and over the complex numbers, and from floating point data. Our Prony-based approach does not encounter the Birkhoff phenomenon of Hermite interpolation, when a gap in the derivative values causes multiple interpolants. We can interpolate from + 1 values of and 2 − 1 values of ′ . CCS CONCEPTS• Mathematics of computing → Interpolation.

Hermite Interpolation With Dickson Polynomials and Bernstein Basis Polynomials

“…The algorithms in the present manuscript perform sparse Hermite interpolation with Dickson polynomials (both the first and the second kind) and Bernstein basis polynomials. The algorithms use transformations from Dickson polynomials to Laurent polynomials, a transformation from Bernstein basis polynomials to Laurent polynomials, and the algorithm given in [2] as a middle step.…”

“…In [13], it is introduced that We can make use of Equation ( 6) and can design an algorithm that solves Problem 1.1.iii. Algorithm 2.3.1 first uses Equation (6) to convert Problem 1.1.iii to another problem that the Algorithm in [2] can solve, then uses Algorithm [2], and then recovers the coefficient-degree tuples (𝑐 𝑗 , 𝑒 𝑗 ) such that 𝑓(𝑥) = ∑ 𝑐 𝑗 𝐵 𝑒 𝑗 ,𝑛 (𝑥) 𝑡 𝑗=0…”

“…Let black boxes for 𝑓(𝑥) and 𝑓 ′ (𝑥) be given. [2] introduces a procedure to rebuild the unknown polynomial 𝑓(𝑥) from the data sets {(𝑘 𝑠 , 𝑓(𝑘 𝑠 ))} 𝑠=0 𝑚 and {(𝑘 𝑠 , 𝑓′(𝑘 𝑠 ))} 𝑠=0 𝑚 where 𝑚 = 𝑡 + ⌈ 𝑡+1 2 ⌉ − 1. Here the tuples ( * , 𝑓( * )) and ( * , 𝑓′( * )) can be computed with given black boxes.…”

“…Here the tuples ( * , 𝑓( * )) and ( * , 𝑓′( * )) can be computed with given black boxes. The algorithm presented in [2], which is based on Prony's sparse polynomial interpolation algorithm (a.k.a. Ben-or & Tiwari's Algorithm) [3,4], performs sparse Hermite interpolation using those 2𝑚 + 2, where 𝑡 ≪ deg(𝑓), data points above.…”

“…Ben-or & Tiwari's Algorithm) [3,4], performs sparse Hermite interpolation using those 2𝑚 + 2, where 𝑡 ≪ deg(𝑓), data points above. The method in [2] uses "less data points" than the previously known Hermite interpolation algorithms use to reconstruct the unknown polynomial 𝑓(𝑥).…”