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Symbolic–numeric sparse interpolation of multivariate polynomials

Mark Giesbrecht, George Labahn, Wen-shin Lee
2009 Journal of symbolic computation  
We consider the problem 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.  ... 
doi:10.1016/j.jsc.2008.11.003 fatcat:hfy7ywxe2rbszp6wkqv7ruukoi

Symbolic-numeric sparse interpolation of multivariate polynomials

Mark Giesbrecht, George Labahn, Wen-shin Lee
2006 Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06  
We consider the problem 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.  ... 
doi:10.1145/1145768.1145792 dblp:conf/issac/GiesbrechtLL06 fatcat:sjord5m22zddxmkzvjbcyc322a

The "Seven Dwarfs" of Symbolic Computation [chapter]

Erich L. Kaltofen
2011 Texts & Monographs in Symbolic Computation  
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 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.  ... 
doi:10.1007/978-3-7091-0794-2_5 fatcat:y6fjj6ghrzek5ps4vig7uv5vpy

Preface

Dario Andrea Bini, Victor Y. Pan, Jan Verschelde
2008 Theoretical Computer Science  
., besides the latter areas, our special issue treats decomposition 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.  ... 
doi:10.1016/j.tcs.2008.09.001 fatcat:uwwutxg5svdkholg7mtcu2pvzy

Efficient techniques for multipolynomial resultant algorithms

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

Modular rational sparse multivariate polynomial interpolation

E. Kaltofen, Y. N. Lakshman, J.-M. Wiley
1990 Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '90  
The problem 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.  ... 
doi:10.1145/96877.96912 dblp:conf/issac/KaltofenLW90 fatcat:hhpu6eq7tjflbhhren66cvwyle

Sparse interpolation of multivariate rational functions

Annie Cuyt, Wen-shin Lee
2011 Theoretical Computer Science  
In general, the performance 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.  ... 
doi:10.1016/j.tcs.2010.11.050 fatcat:pmnhbkkj3zethawvfmovzdnohq

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.  ... 

Sparse multivariate function recovery from values with noise and outlier errors

Erich L. Kaltofen, Zhengfeng Yang
2013 Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation - ISSAC '13  
Our 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  ... 
doi:10.1145/2465506.2465524 dblp:conf/issac/KaltofenY13 fatcat:ccdgfxh7cjckvj5lneljnwblue

What Can (and Can't) we Do with Sparse Polynomials?

Daniel S. Roche
2018 Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '18  
In this tutorial we examine the state 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  ... 
doi:10.1145/3208976.3209027 dblp:conf/issac/Roche18 fatcat:fygzzsxjwrdk7kfxw4z4g7zsjq

MultiPolynomial Resultant Algorithms

Dinesh Manocha, John F. Canny
1993 Journal of symbolic computation  
The algorithm can also be used for 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.  ... 
doi:10.1006/jsco.1993.1009 fatcat:esj7dbw5kfh5njozjiktvn5arq

On exact and approximate interpolation of sparse rational functions

Erich Kaltofen, Zhengfeng Yang
2007 Proceedings of the 2007 international symposium on Symbolic and algebraic computation - ISSAC '07  
The black box algorithm for separating the 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  ... 
doi:10.1145/1277548.1277577 dblp:conf/issac/KaltofenY07 fatcat:vumng65q5zdyvof4qg3z3gu3xu

Diversification improves interpolation

Mark Giesbrecht, Daniel S. Roche
2011 Proceedings of the 36th international symposium on Symbolic and algebraic computation - ISSAC '11  
Some 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.  ... 
doi:10.1145/1993886.1993909 dblp:conf/issac/GiesbrechtR11 fatcat:7rmfxz4iwfaeto7vwpfxhc4bcq

Special issue computational algebraic complexity editorial

Erich Kaltofen, Bruno Buchberger
1990 Journal of symbolic computation  
result in a collection 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.  ... 
doi:10.1016/s0747-7171(08)80010-7 fatcat:wrri7sabmjb4tpqtqyfbwuoa6e
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