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Symbolic–numeric sparse interpolation of multivariate polynomials

2009
*
Journal of symbolic computation
*

We consider the problem

doi:10.1016/j.jsc.2008.11.003
fatcat:hfy7ywxe2rbszp6wkqv7ruukoi
*of**sparse**interpolation**of*an approximate*multivariate*black-box*polynomial*in floating-point arithmetic. ... By*interpolating*the black box evaluated at random primitive roots*of*unity, we give efficient and*numerically*robust solutions. ...*Numerical*methods for*sparse**interpolation*In this section we present two methods for black-box*interpolation**of**sparse**multivariate**polynomials*in floating-point arithmetic. ...##
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Symbolic-numeric sparse interpolation of multivariate polynomials

2006
*
Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06
*

We consider the problem

doi:10.1145/1145768.1145792
dblp:conf/issac/GiesbrechtLL06
fatcat:sjord5m22zddxmkzvjbcyc322a
*of**sparse**interpolation**of*an approximate*multivariate*black-box*polynomial*in floating-point arithmetic. ... By*interpolating*the black box evaluated at random primitive roots*of*unity, we give efficient and*numerically*robust solutions. ...*Numerical*methods for*sparse**interpolation*In this section we present two methods for black-box*interpolation**of**sparse**multivariate**polynomials*in floating-point arithmetic. ...##
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The "Seven Dwarfs" of Symbolic Computation
[chapter]

2011
*
Texts & Monographs in Symbolic Computation
*

Exact

doi:10.1007/978-3-7091-0794-2_5
fatcat:y6fjj6ghrzek5ps4vig7uv5vpy
*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*symbolic*-*numeric*computation SymDwf 6. Computation*of*closed form solutions SymDwf 7. ... Zippel's [1990] and Ben-Or and Tiwari's [1988]*sparse**multivariate**polynomial*algorithms are a fundamental contribution from*symbolic*computation to the task*of*function/model recovery. ...##
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Preface

2008
*
Theoretical Computer Science
*

., besides the latter areas, our special issue treats decomposition

doi:10.1016/j.tcs.2008.09.001
fatcat:uwwutxg5svdkholg7mtcu2pvzy
*of*a*multivariate**polynomial*into the sum*of*squares,*sparse**multivariate**polynomial**interpolation*, ill conditioned linear systems*of*... set*of*univariate or*multivariate**polynomials*. ... Cuyt and Lee present a deterministic*numerical*algorithm for*sparse**multivariate**polynomial**interpolation*, which requires no information on the number*of*terms or the partial degree in each variable. ...##
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Efficient techniques for multipolynomial resultant algorithms

1991
*
Proceedings of the 1991 international symposium on Symbolic and algebraic computation - ISSAC '91
*

F.(1987) figuration
space for curved objects for robot motion
planning.
6 "A deter-
ministic algorithm
for

doi:10.1145/120694.120706
dblp:conf/issac/ManochaC91
fatcat:moothyqmyrbmbpuuyqow5azm5y
*sparse**multivariate**polynomial**interpolation*", 20th Annual ACM Symp. ... upon the application). 3.1*Multivariate**Interpolation*oft he resulting matrix is a*polynomial*. ...##
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Modular rational sparse multivariate polynomial interpolation

1990
*
Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '90
*

The problem

doi:10.1145/96877.96912
dblp:conf/issac/KaltofenLW90
fatcat:hhpu6eq7tjflbhhren66cvwyle
*of**interpolating**multivariate**polynomials*whose coefficient domain is the rational numbers is considered. ... The effect*of*intermediate number growth on a speeded Ben-Or and Tiwari algorithm is studied. Then the newly developed modular algorithm is presented. ... Here we report how these techniques can be applied to the problem*of**interpolating**sparse**multivariate**polynomials*. ...##
###
Sparse interpolation of multivariate rational functions

2011
*
Theoretical Computer Science
*

In general, the performance

doi:10.1016/j.tcs.2010.11.050
fatcat:pmnhbkkj3zethawvfmovzdnohq
*of*our*sparse*rational black box*interpolation*depends on the choice*of*the employed*sparse**polynomial*black box*interpolation*. ... Consider the black box*interpolation**of*a τ -*sparse*, n-variate rational function f , where τ is the maximum number*of*terms in either*numerator*or denominator. ... Acknowledgements We thank Erich Kaltofen and Zhengfeng Yang for valuable remarks and providing their*sparse*rational*interpolation*codes and benchmarks. ...##
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Page 4451 of Mathematical Reviews Vol. , Issue 97G
[page]

1997
*
Mathematical Reviews
*

(English summary)
Parallel

*symbolic*computation. J.*Symbolic*Comput. 21 (1996), no. 4-6, 377-396. Summary: “A new algorithm for*sparse**multivariate**polynomial**interpolation*is presented. ...*multivariate**polynomial**interpolation*and its parallel implementation. ...##
###
Page 376 of Mathematical Reviews Vol. , Issue 91A
[page]

1991
*
Mathematical Reviews
*

Summary: “We consider the problem

*of**interpolating**sparse*mul- tivariate*polynomials*from their values. ... Miihlbach (D-HANN-AM) 91a:65014 65D05 Kaltofen, Erich (1-RSP-C); Yagati, Lakshman (1-RSP-C) Improved*sparse**multivariate**polynomial**interpolation*algorithms. ...##
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Sparse multivariate function recovery from values with noise and outlier errors

2013
*
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation - ISSAC '13
*

Our

doi:10.1145/2465506.2465524
dblp:conf/issac/KaltofenY13
fatcat:ccdgfxh7cjckvj5lneljnwblue
*multivariate**polynomial*and rational function*interpolation*algorithm combines Zippel's*symbolic**sparse**polynomial**interpolation*technique [Ph.D. ... Our*multivariate*algorithm can build a*sparse*model from a number*of*evaluations that is linear in the sparsity*of*the model. ... Note that in [2, Section 3] we have shown stability for a*numerical*version*of*Blahut's errorcorrecting*polynomial**interpolation*algorithm for E = 1. 2. an exact*interpolation*algorithm for*sparse**multivariate*...##
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What Can (and Can't) we Do with Sparse Polynomials?

2018
*
Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '18
*

In this tutorial we examine the state

doi:10.1145/3208976.3209027
dblp:conf/issac/Roche18
fatcat:fygzzsxjwrdk7kfxw4z4g7zsjq
*of*the art for*sparse**polynomial*algorithms in three areas: arithmetic,*interpolation*, and factorization. ... Simply put, a*sparse**polynomial*is one whose zero coefficients are not explicitly stored. ... An even more difficult problem is*sparse*rational function*interpolation*, which is the same as*sparse**polynomial**interpolation*except that the unknown f is a fraction*of*two*sparse**multivariate**polynomials*...##
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MultiPolynomial Resultant Algorithms

1993
*
Journal of symbolic computation
*

The algorithm can also be used for

doi:10.1006/jsco.1993.1009
fatcat:esj7dbw5kfh5njozjiktvn5arq
*interpolating**polynomials*from their values and expanding*symbolic*determinants. ... Recently, the techniques*of*Gr} obner bases and*polynomial*continuationhave received much attention as algorithmic methods for these*symbolic*and*numeric*applications. ... This problem arises in*symbolic*and*numeric*techniques used to manipulate sets*of**polynomial*equations. ...##
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On exact and approximate interpolation of sparse rational functions

2007
*
Proceedings of the 2007 international symposium on Symbolic and algebraic computation - ISSAC '07
*

The black box algorithm for separating the

doi:10.1145/1277548.1277577
dblp:conf/issac/KaltofenY07
fatcat:vumng65q5zdyvof4qg3z3gu3xu
*numerator*from the denominator*of*a*multivariate*rational function can be combined with*sparse**multivariate**polynomial**interpolation*algorithms to*interpolate*... Finally, one can deploy the*sparse*rational function*interpolation*algorithm in the hybrid*symbolic*-*numeric*setting when the black box for the rational function returns real and complex values with noise ... Acknowledgement: We thank Wen-shin Lee for providing her*numeric**sparse**interpolation*code to us, Arne Storjohann for sending us [22] and discussions on rational vector recovery, and Lihong Zhi for discussions ...##
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Diversification improves interpolation

2011
*
Proceedings of the 36th international symposium on Symbolic and algebraic computation - ISSAC '11
*

Some

doi:10.1145/1993886.1993909
dblp:conf/issac/GiesbrechtR11
fatcat:7rmfxz4iwfaeto7vwpfxhc4bcq
*numerical*ingredients We show that the*sparse**interpolation*problem is well-posed for evaluations at low-order roots*of*unity: Theorem Suppose f , g ∈ C[x] , p is a randomly-chosen "good prime", ∈ ... Cost*of*our algorithm: O(n 2 t 2 log 2 d).Zippel's method. Go variable-by-variable; at each*of*n steps perform univariate*interpolation*t times on degree-d*polynomials*. ...##
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Special issue computational algebraic complexity editorial

1990
*
Journal of symbolic computation
*

result in a collection

doi:10.1016/s0747-7171(08)80010-7
fatcat:wrri7sabmjb4tpqtqyfbwuoa6e
*of*surprisingly effective algorithms for*sparse**multivariate**polynomial**interpolation*, which are presented here. ... The article by Zippel deals with the problem*of**interpolating*a*sparse**multivariate**polynomial*over fields*of*characteristic zero from its values. ...
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