Improved sparse multivariate polynomial interpolation algorithms
We consider the problem of interpolating sparse multivariate polynomials from their values. We discuss two algorithms for sparse interpolation, one due to Ben-Or and Tiwari (1988) and the other due to Zippel (1988). We present efficient algorithms for finding the rank of certain special Toeplitz systems arising in the Ben-Or and Tiwari algorithm and for solving transposed Vandermonde systems of equations, the use of which greatly improves the time complexities of the two interpolation algorithms.
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- Aho, A., Hopcroft, J., and Ullman, J., The Design and Analysis of Algorithms; Addison and Wesley, Reading, MA, 1974.Google Scholar
- Ben-Or, M. and Tiwari, P., “A deterministic algorithm for sparse multivariate polynomial interpolation,” 20th Annual ACM Symp. Theory Comp., pp. 301–309 (1988).Google Scholar
- Blahut, R. E., Theory and Practice of Error Control Codes; Addison-Wesley, Reading, MA, 1983.Google Scholar
- Brent, R. P., Gustavson, F. G., and Yun, D. Y. Y., “Fast solution of Toeplitz systems of equations and computation of Padé approximants,” J. Algorithms1, pp. 259–295 (1980).Google Scholar
- Brown, W. S. and Traub, J. F., “On Euclid's algorithm and the theory of subresultants,” J. ACM18, pp. 505–514 (1971).Google Scholar
- Canny, J., Kaltofen, E., and Lakshman, Yagati, “Solving systems of non-linear polynomial equations faster,” Manuscript, 1988.Google Scholar
- Cantor, D. G. and Kaltofen, E., “Fast multiplication of polynomials with coefficients from an arbitrary ring,” Manuscript, March 1987.Google Scholar
- Grigoryev, D. Yu. and Karpinski, M., “The matching problem for bipartite graphs with polynomially bounded permanents is in NC,” Proc. 28th IEEE Symp. Foundations Comp. Sci., pp. 166–172 (1987).Google Scholar
- Kaltofen, E. and Trager, B., “Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators,” Proc. 29th Annual Symp. Foundations of Comp. Sci., (1988 (to appear)).Google Scholar
- Loos, R., “Computing rational zeros of integral polynomials by p-adic expansion,” SIAM J.Comp.12, pp. 286–293 (1983).Google Scholar
- Schönhage, A. and Strassen, V., “Schnelle Multiplikation grosser Zahlen,” Computing7, pp. 281–292 (1971). (In German).Google Scholar
- Zippel, R. E., “Probabilistic algorithms for sparse polynomials,” Proc. EUROSAM '79, Springer Lec. Notes Comp. Sci.72, pp. 216–226 (1979).Google Scholar
- Zippel, R. E., “Interpolating polynomials from their values,” Manuscript, Symbolics Inc., January 1988.Google Scholar