Sparse Polynomial Interpolation Based on Derivative

Huang, Qiao-Long

doi:10.48550/arxiv.2002.03708

Cited by 2 publications

(7 citation statements)

References 18 publications

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“…Finding candidate exponents Like in the recent line of work of Gao and Huang [27,31,28,29], our overall approach is to generate candidate terms of the unknown sparse polynomials . is is achieved by interpolating mod − 1 for tiny primes , where ∈ ( log ) is so small that even performing ̃ ( ) operations is allowable within the targeted complexity.…”

Section: Main Ideasmentioning

confidence: 99%

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Sparse Polynomial Interpolation and Division in Soft-linear Time

Giorgi,

Grenet,

Cray

et al. 2022

Preprint

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Given a way to evaluate an unknown polynomial with integer coefficients, we present new algorithms to recover its nonzero coefficients and corresponding exponents. As an application, we adapt this interpolation algorithm to the problem of computing the exact quotient of two given polynomials. ese methods are efficient in terms of the bit-length of the sparse representation, that is, the number of nonzero terms, the size of coefficients, the number of variables, and the logarithm of the degree. At the core of our results is a new Monte Carlo randomized algorithm to recover an integer polynomial ( ) given a way to evaluate ( ) mod for any chosen integers and . is algorithm has nearly-optimal bit complexity, meaning that the total bit-length of the probes, as well as the computational running time, is so ly linear (ignoring logarithmic factors) in the bit-length of the resulting sparse polynomial. To our knowledge, this is the first sparse interpolation algorithm with so -linear bit complexity in the total output size. For integer polynomials, the best previously known results have at least a cubic dependency on the bit-length of the exponents. * In this work, we do not consider the case of unbalanced bit lengths, where the differing sizes of each coefficient and exponent are considered in the complexity.

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Section: Main Ideasmentioning

confidence: 99%

“…Hence, there is no restriction on the evaluation domain, but the evaluation cost has to be taken into account. Subsequent works have refined the complexity bounds of this algorithm when the ring of coefficients is a finite field, the ring of integers or rational numbers [4,5,31,28].…”

Section: Introductionmentioning

confidence: 99%

Sparse Polynomial Interpolation and Division in Soft-linear Time

Giorgi,

Grenet,

Cray

et al. 2022

Preprint

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“…In particular, we compute the term degrees even when evaluating at powers of a root of unity whose order is below the degree of , provided that the term values remain distinct. The idea was used before for fast sparse interpolation algorithms of polynomial products [3] and of polynomials given by straight line programs [10,11] where the asymptotic complexity was optimized, and the number of samples could be increased by a constant factor.…”

Section: The Number Of Arguments At Which the Algorithm Computesmentioning

confidence: 99%

“…where , is defined in (11) below for = and = are the Laurent terms in . The last 2 matrix factors in (10) are nonsingular because = ≠ = for 1 ≤ < ≤ and the last factor is a transposed Vandermonde matrix.…”

Section: Probabilistic Analysismentioning

confidence: 99%

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Sparse Polynomial Hermite Interpolation

Kaltofen

2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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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.

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Sparse Polynomial Interpolation Based on Derivative

Cited by 2 publications

References 18 publications

Sparse Polynomial Interpolation and Division in Soft-linear Time

Sparse Polynomial Interpolation and Division in Soft-linear Time

Sparse Polynomial Hermite Interpolation

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