A recursive computation scheme for multivariate rational interpolants
SIAM Journal on Numerical Analysis
Greatest common divisors of polynomials given by straight-line programs
Journal of the ACM (JACM)
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields
SIAM Journal on Computing
On zero-testing and interpolation of k -sparse multivariate polynomials over finite fields
Theoretical Computer Science
Algorithms for sparse rational interpolation
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
Computational Complexity of Sparse Rational Interpolation
SIAM Journal on Computing
FOXBOX: a system for manipulating symbolic objects in black box representation
ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
Modern computer algebra
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Early termination in Ben-Or/Tiwari sparse interpolation and a hybrid of Zippel's algorithm
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Early termination in sparse interpolation algorithms
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Maximal quotient rational reconstruction: an almost optimal algorithm for rational reconstruction
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Algorithms for the non-monic case of the sparse modular GCD algorithm
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Symbolic-numeric sparse interpolation of multivariate polynomials
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Fast rational function reconstruction
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
On probabilistic analysis of randomization in hybrid symbolic-numeric algorithms
Proceedings of the 2007 international workshop on Symbolic-numeric computation
On exact and approximate interpolation of sparse rational functions
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Interpolation of sparse rational functions without knowing bounds on exponents
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
A new algorithm for sparse interpolation of multivariate polynomials
Theoretical Computer Science
Symbolic-numeric sparse interpolation of multivariate polynomials
Journal of Symbolic Computation
Parallel sparse polynomial interpolation over finite fields
Proceedings of the 4th International Workshop on Parallel and Symbolic Computation
Sparse multivariate function recovery from values with noise and outlier errors
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
Hi-index | 5.23 |
Consider the black box interpolation of a @t-sparse, n-variate rational function f, where @t is the maximum number of terms in either numerator or denominator. When numerator and denominator are at most of degree d, then the number of possible terms in f is O(d^n) and explodes exponentially as the number of variables increases. The complexity of our sparse rational interpolation algorithm does not depend exponentially on n anymore. It still depends on d because we densely interpolate univariate auxiliary rational functions of the same degree. We remove the exponent n and introduce the sparsity @t in the complexity by reconstructing the auxiliary function's coefficients via sparse multivariate interpolation. The approach is new and builds on the normalization of the rational function's representation. Our method can be combined with probabilistic and deterministic components from sparse polynomial black box interpolation to suit either an exact or a finite precision computational environment. The latter is illustrated with several examples, running from exact finite field arithmetic to noisy floating point evaluations. In general, the performance of our sparse rational black box interpolation depends on the choice of the employed sparse polynomial black box interpolation. If the early termination Ben-Or/Tiwari algorithm is used, our method achieves rational interpolation in O(@td) black box evaluations and thus is sensitive to the sparsity of the multivariate f.