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Sparse Polynomial Interpolation and Berlekamp/Massey algorithms that . . .
, 2012
"... We propose algorithms performing sparse interpolation with errors, based on Prony’s–BenOr’s & Tiwari’s algorithm, using a Berlekamp/Massey algorithm with early termination. First, we present an algorithm that can recover a tsparse polynomial f from a sequence of values, where some of the val ..."
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We propose algorithms performing sparse interpolation with errors, based on Prony’s–BenOr’s & Tiwari’s algorithm, using a Berlekamp/Massey algorithm with early termination. First, we present an algorithm that can recover a tsparse polynomial f from a sequence of values, where some of the values are wrong, spoiled by either random or misleading errors. Our algorithm requires bounds T ≥ t and E ≥ e, where e is the number of evaluation errors. It interpolates f(ω i) for i = 1,...,2T(E + 1), where ω is a field element at which each nonzero term evaluates distinctly. We also investigate the problem of recovering the minimal linear generator from a sequence of field elements that are linearly generated, but where again e ≤ E elements are erroneous. We show that there exist sequences of < 2t(2e + 1) elements, such that two distinct generators of length t satisfy the linear recurrence up to e faults, at least if the field has characteristic ̸ = 2. Uniqueness can be proven (for any field characteristic) for length ≥ 2t(2e + 1) of the sequence with e errors. Finally, we present the Majority Rule Berlekamp/Massey algorithm, which can recover the unique minimal linear generator of degree t when given bounds T ≥ t and E ≥ e and the initial sequence segment of 2T(2E + 1) elements. Our algorithm also corrects the sequence segment. The Majority Rule algorithm yields a unique sparse interpolant for the first problem. The algorithms are applied to sparse interpolation algorithms with numeric noise, into which we now can bring outlier errors in the values.
Parallel Sparse Polynomial Interpolation over Finite Fields
, 2010
"... We present a probabilistic algorithm to interpolate a sparse multivariate polynomial over a finite field, represented with a black box. Our algorithm modifies the algorithm of BenOr and Tiwari from 1988 for interpolating polynomials over rings with characteristic zero to characteristic p by doing a ..."
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We present a probabilistic algorithm to interpolate a sparse multivariate polynomial over a finite field, represented with a black box. Our algorithm modifies the algorithm of BenOr and Tiwari from 1988 for interpolating polynomials over rings with characteristic zero to characteristic p by doing additional probes. To interpolate a polynomial in n variables with t nonzero terms, Zippel’s (1990) algorithm interpolates one variable at a time using O(ndt) probes to the black box where d bounds the degree of the polynomial. Our new algorithm does O(nt) probes. It interpolates each variable independently using O(t) probes which allows us to parallelize the main loop giving an advantage over Zippel’s algorithm. We have implemented both Zippel’s algorithm and the new algorithm in C and we have done a parallel implementation of our algorithm using Cilk [2]. In the paper we provide benchmarks comparing the number of probes our algorithm does with both Zippel’s algorithm and Kaltofen and Lee’s hybrid of the Zippel and BenOr/Tiwari algorithms.
Supersparse black box rational function interpolation
 Manuscript
, 2011
"... We present a method for interpolating a supersparse blackbox rational function with rational coefficients, for example, a ratio of binomials or trinomials with very high degree. We input a blackbox rational function, as well as an upper bound on the number of nonzero terms and an upper bound on the ..."
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We present a method for interpolating a supersparse blackbox rational function with rational coefficients, for example, a ratio of binomials or trinomials with very high degree. We input a blackbox rational function, as well as an upper bound on the number of nonzero terms and an upper bound on the degree. The result is found by interpolating the rational function modulo a small prime p, and then applying an effective version of Dirichlet’s Theorem on primes in an arithmetic progression progressively lift the result to larger primes. Eventually we reach a prime number that is larger than the inputted degree bound and we can recover the original function exactly. In a variant, the initial prime p is large, but the exponents of the terms are known modulo larger and larger factors of p − 1. The algorithm, as presented, is conjectured to be polylogarithmic in the degree, but exponential in the number of terms. Therefore, it is very effective for rational functions with a small number of nonzero terms, such as the ratio of binomials, but it quickly becomes ineffective for a high number of terms. The algorithm is oblivious to whether the numerator and denominator have a common factor. The algorithm will recover the sparse form of the rational function, rather than the reduced form, which could be dense. We have experimentally tested the algorithm in the case of under 10 terms in numerator and denominator combined and observed its conjectured high efficiency.
The “Seven Dwarfs ” of Symbolic Computation*
, 2010
"... We present the Seven Dwarfs of Symbolic Computation, which are sequential and parallel algorithmic methods that today carry a great majority of all exact and hybrid symbolic compute cycles. SymDwf 1. Exact linear algebra, integer lattices SymDwf 2. Exact polynomial and differential algebra, Gröbner ..."
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We present the Seven Dwarfs of Symbolic Computation, which are sequential and parallel algorithmic methods that today carry a great majority of all exact and hybrid symbolic compute cycles. SymDwf 1. Exact linear algebra, integer lattices SymDwf 2. Exact polynomial and differential algebra, Gröbner bases SymDwf 3. Inverse symbolic problems, e.g., interpolation and parameterization SymDwf 4. Tarski’s algebraic theory of real geometry SymDwf 5. Hybrid symbolicnumeric computation SymDwf 6. Computation of closed form solutions SymDwf 7. Rewrite rule systems and computational group theory We will elaborate on each dwarf and compare with Colella’s seven and the Berkeley team’s thirteen dwarfs of scientific computing.
Symbolic Polynomials with Sparse Exponents
"... Earlier work has presented algorithms to factor and compute GCDs of symbolic Laurent polynomials, that is multivariate polynomials whose exponents are integervalued polynomials. These earlier algorithms had the problem of high computational complexity in the number of exponent variables and their d ..."
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Earlier work has presented algorithms to factor and compute GCDs of symbolic Laurent polynomials, that is multivariate polynomials whose exponents are integervalued polynomials. These earlier algorithms had the problem of high computational complexity in the number of exponent variables and their degree. The present paper solves this problem, presenting a method that preserves the structure of sparse exponent polynomials. 1
Interpolation of ShiftedLacunary Polynomials
, 2010
"... Given a “black box” function to evaluate an unknown rational polynomial f ∈ Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t ∈ Z>0, the shift α ∈ Q, the exponents 0 ≤ e1 ..."
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Given a “black box” function to evaluate an unknown rational polynomial f ∈ Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t ∈ Z>0, the shift α ∈ Q, the exponents 0 ≤ e1 <e2 < ·· · <et, and the coefficients c1,...,ct ∈ Q \{0} such that f(x) =c1(x − α) e1 + c2(x − α) e2 + ···+ ct(x − α) et. The computed sparsity t is absolutely minimal over any shifted power basis. The novelty of our algorithm is that the complexity is polynomial in the (sparse) representation size, which may be logarithmic in the degree of f. Our method combines previous celebrated results on sparse interpolation and computing sparsest shifts, and provides a way to handle polynomials with extremely high degree which are, in some sense, sparse in information.