Communication Dans Un Congrès Année : 2020

Hermite Rational Function Interpolation with Error Correction

Résumé

We generalize Hermite interpolation with error correction, which is the methodology for multiplicity algebraic error correction codes, to Hermite interpolation of a rational function over a field K from function and function derivative values. We present an interpolation algorithm that can locate and correct ≤ E errors at distinct arguments ξ ∈ K where at least one of the values or values of a derivative is incorrect. The upper bound E for the number of such ξ is input. Our algorithm sufficiently oversamples the rational function to guarantee a unique interpolant. We sample (f /g) (j) (ξ i ) for 0 ≤ j ≤ ℓ i , 1 ≤ i ≤ n, ξ i distinct, where (f /g) (j) is the j-th derivative of the rational function f /g, f, g ∈ K[x], GCD(f, g) = 1, g = 0, and where N = n i=1 (ℓ i + 1) ≥ D f + D g + 1 + 2E + 2 E k=1 ℓ k ; D f is an upper bound for deg(f ) and D g an upper bound for deg(g), which are input to our algorithm. The arguments ξ i can be poles, which is truly or falsely indicated by a function value ∞ with the corresponding ℓ i = 0. Our results remain valid for fields K of characteristic ≥ 1 + max i ℓ i . Our algorithm has the same asymptotic arithmetic complexity as that for classical Hermite interpolation, namely N (log N ) O(1) . For polynomials, that is, g = 1, and a uniform derivative profile ℓ 1 = • • • = ℓ n , our algorithm specializes to the univariate multiplicity code decoder that is based on the 1986 Welch-Berlekamp algorithm.

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Dates et versions

hal-04925033 , version 1 (25-02-2025)

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Erich Kaltofen, Clément Pernet, Zhi-Hong Yang. Hermite Rational Function Interpolation with Error Correction. Computer Algebra in Scientific Computing, Sep 2020, Linz, Austria. pp.335-357, ⟨10.1007/978-3-030-60026-6_19⟩. ⟨hal-04925033⟩
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