Erich L. Kaltofen
ABSTRACT
The discipline of symbolic computation contributes to mathematical model synthesis in several ways. One is the pioneering creation of interpolation algorithms that can account for sparsity in the resulting multi-dimensional models, for example, by Zippel [12], Ben-Or and Tiwari [1], and in their recent numerical counterparts by Giesbrecht-Labahn-Lee [5] and Kaltofen-Yang-Zhi [9].
- Ben-Or, M., and Tiwari, P. A deterministic algorithm for sparse multivariate polynomial interpolation. In Proc. Twentieth Annual ACM Symp. Theory Comput. (New York, N.Y., 1988), ACM Press, pp. 301--309. Google Scholar
Digital Library
- Blahut, R. E. A universal Reed-Solomon decoder. IBM J. Res. Develop. 18, 2 (Mar. 1984), 943--959. Google ScholarDigital Library
- Boyer, B., Comer, M. T., and Kaltofen, E. L. Sparse polynomial interpolation by variable shift in the presence of noise and outliers in the evaluations. In Proc. Tenth Asian Symposium on Computer Mathematics (ASCM 2012) (Oct. 2013). Submitted to SLNCS; URL: http://www.math.ncsu.edu/~kaltofen/bibliography/13/BCK13.pdf.Google Scholar
- Comer, M. T., Kaltofen, E. L., and Pernet, C. Sparse polynomial interpolation and Berlekamp/Massey algorithms that correct outlier errors in input values. In ISSAC 2012 Proc. 37th Internat. Symp. Symbolic Algebraic Comput. (New York, N. Y., July 2012), J. van der Hoeven and M. van Hoeij, Eds., Association for Computing Machinery, pp. 138--145. URL: http://www.math.ncsu.edu/~kaltofen/bibliography/12/CKP12.pdf. Google ScholarDigital Library
- Giesbrecht, M., Labahn, G., and Lee, W. Symbolic-numeric sparse interpolation of multivariate polynomials. J. Symbolic Comput. 44 (2009), 943--959. Google ScholarDigital Library
- Kaltofen, E., and Pernet, C. Sparse polynomial interpolation codes and their decoding beyond half the minimal distance. In Nabeshima {10}. URL: http://www.math.ncsu.edu/~kaltofen/bibliography/14/KaPe14.pdf. Google ScholarDigital Library
- Kaltofen, E., and Yang, Z. Sparse multivariate function recovery from values with noise and outlier errors. In ISSAC 2013 Proc. 38th Internat. Symp. Symbolic Algebraic Comput. (New York, N. Y., 2013), M. Kauers, Ed., Association for Computing Machinery, pp. 219--226. URL: http://www.math.ncsu.edu/~kaltofen/bibliography/13/KaYa13.pdf. Google ScholarDigital Library
- Kaltofen, E., and Yang, Z. Sparse multivariate function recovery with a high error rate in evaluations. In Nabeshima {10}. URL: http://www.math.ncsu.edu/~kaltofen/bibliography/14/KaYa14.pdf. Google ScholarDigital Library
- Kaltofen, E., Yang, Z., and Zhi, L. On probabilistic analysis of randomization in hybrid symbolic-numeric algorithms. In SNC'07 Proc. 2007 Internat. Workshop on Symbolic-Numeric Comput. (New York, N. Y., 2007), J. Verschelde and S. M. Watt, Eds., ACM Press, pp. 11--17. URL: http://www.math.ncsu.edu/~kaltofen/bibliography/07/KYZ07.pdf. Google ScholarDigital Library
- Nabeshima, K., Ed. ISSAC 2014 Proc. 39th Internat. Symp. Symbolic Algebraic Comput. (New York, N. Y., 2014), Association for Computing Machinery.Google Scholar
- Welch, L. R., and Berlekamp, E. R. Error correction of algebraic block codes. US Patent 4,633,470, 1986. Filed 1983; see http://patft.uspto.gov/.Google Scholar
- Zippel, R. E. Probabilistic algorithms for sparse polynomials. PhD thesis, Massachusetts Inst. of Technology, Cambridge, USA, Sept. 1979.Google Scholar
Index Terms
- Cleaning-up data for sparse model synthesis: when symbolic-numeric computation meets error-correcting codes
Recommendations
Overdetermined Systems of Sparse Polynomial Equations
In this paper, we show that for a system of univariate polynomials given in sparse encoding we can compute a single polynomial defining the same zero set in quasilinear time in the logarithm of the degree. In particular, it is possible to decide whether ...
Sparse interpolation of multivariate rational functions
Consider the black box interpolation of a -sparse, n-variate rational function f, where 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 ...
On solving univariate sparse polynomials in logarithmic time
Special issue: Foundations of computational mathematics 2002 workshopsLet f be a degree D univariate polynomial with real coefficients and exactly m monomial terms. We show that in the special case m = 3 we can approximate within ε all the roots of f in the interval [0,R] using just O(log(D)log(D log R/ε)) arithmetic ...
Comments