Erich Kaltofen
ABSTRACT
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 a sparse rational function. domization and early termination strategies are exploited to minimize the number of black box evaluations. In addition, rational number coefficients are recovered from modular images by rational vector recovery. The need for separate numerator and denominator size bounds is avoided via correction, and the modulus is minimized by use of lattice basis reduction, a process that can be applied to sparse rational function vector recovery itself. Finally, one can deploy sparse rational function interpolation algorithm in the hybrid symbolic-numeric setting when the black box for the function returns real and complex values with noise. We present and analyze five new algorithms for the above problems and demonstrate their effectiveness on a mark implementation.
- Becuwe, S., Cuyt, A., and Verdonk, B. Multivariate rational interpolation of scattered data. In Large-Scale Scientific Computing (Heidelberg, Germany, 2004), I. Lirkov, S. Margenov, J. Wasniewski, and Y. Plamen, Eds., vol. 2907 of Lect. Notes Comput. Sci., Springer Verlag, pp. 204--213.Google Scholar
- 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 ScholarDigital Library
- DeMillo, R.A., and Lipton, R.J. probabilistic remark on algebraic program testing. Information Process. Letters 7,4 (1978), 193--195.Google ScholarCross Ref
- Dílaz, A., and Kaltofen, E. FoxBox system for manipulating symbolic objects in black box representation. In Proc. 1998 Internat. Symp. Symbolic Algebraic Comput. (ISSAC'98) (New York, N.Y., 1998), O. Gloor, Ed., ACM Press, pp.30--37. Google ScholarDigital Library
- Dumas, J.-G. Ed. ISSAC MMVI Proc. 2006 Internat. Symp. Symbolic Algebraic Comput. (New York, N.Y., 2006), ACM Press.Google Scholar
- von zur Gathen, J., and Gerhard, J. Modern Computer Algebra. Cambridge University Press, Cambridge, New York, Melbourne, 1999. Second edition 2003. Google ScholarDigital Library
- Giesbrecht, M., Labahn, G., and Lee, W. Symbolic-numeric sparse interpolation of multivariate polynomials. In Dumas {5}, pp.116--123. Google ScholarDigital Library
- Grigoriev, D. Y., Karpinski, M., and Singer, M. F. Computational complexity of sparse rational function interpolation. SIAM J. Comput. 23 (1994), 1--11. Google ScholarDigital Library
- Kai, H. Rational function approximation and its ill-conditioned property. In Wang and Zhi {26}, pp.47--53. Preliminary version in {25}, pp. 62--64.Google Scholar
- Kaltofen, E. Greatest common divisors of polynomials given by straight-line programs. J. ACM 35,1 (1988), 231--264. Google ScholarDigital Library
- Kaltofen, E. Asymptotically fast solution of Toeplitz-like singular linear systems. In Proc. 1994 Internat. Symp. Symbolic Algebraic Comput.(ISSAC'94) (New York, N.Y., 1994), ACM Press, pp. 297--304. Journal version in {12}. Google ScholarDigital Library
- Kaltofen, E. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comput. 64,210 (1995), 777--806. Google ScholarDigital Library
- Kaltofen, E., and Lee, W. Early termination in sparse interpolation algorithms. J. Symbolic Comput. 36,3-4 (2003), 365--400. Special issue Internat. Symp. Symbolic Algebraic Comput.(ISSAC 2002). Guest editors: M. Giusti & L.M. Pardo. Google ScholarDigital Library
- Kaltofen, E., Lee, W.-s., and Lobo, A. A. Early termination in Ben-Or/Tiwari sparse interpolation and a hybrid of Zippel's algorithm. In Proc. 2000 Internat. Symp. Symbolic Algebraic Comput.(ISSAC'00)(New York, N.Y., 2000), C. Traverso, Ed., ACM Press, pp. 192--201. Google ScholarDigital Library
- Kaltofen, E., and Rolletschek, H. Computing greatest common divisors and factorizations in quadratic number fields. Math. Comput. 53,188 (1989), 697--720.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. J. Symbolic Comput. 9,3 (1990), 301--320. Google ScholarDigital Library
- Kaltofen, E., Yang, Z., and Zhi, L. Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials. In Dumas {5}, pp.169--176. Full version, 21 pages. Submitted, December 2006. Google ScholarDigital Library
- Kaltofen, E., Yang, Z., and Zhi, L. On probabilistic analysis of randomization in hybrid symbolic-numeric algorithms, 2007. Manuscript in preparation.Google Scholar
- Kaltofen, E., Yang, Z., and Zhi, L. Structured low rank approximation of a Sylvester matrix. In Wang and Zhi {26}, pp.69--83. Preliminary version in {25 }, pp. 188--201.Google Scholar
- Lagarias, J. C. The computational complexity of simultaneous diophantine approximation problems. SIAM J. Comp. 14 (1985), 196--209. Google ScholarDigital Library
- Monagan, M. Maximal quotient rational reconstruction: An almost optimal algorithm for rational reconstruction. In ISSAC 2004 Proc. 2004 Internat. Symp. Symbolic Algebraic Comput.(New York, N.Y., 2004), J. Gutierrez, Ed., ACM Press, pp.243--249. Google ScholarDigital Library
- Olesh, Z., and Storjohann, A. The vector rational function reconstruction problem, Sept. 2006. Manuscript, 14 pages.Google Scholar
- Park, H., Zhang, L., and Rosen, J. B. Low rank approximation of a Hankel matrix by structured total least norm. BIT 39,4 (1999), 757--779.Google ScholarCross Ref
- Schwartz, J. T. Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27 (1980), 701--717. Google ScholarDigital Library
- Wang, D., and Zhi, L. Eds. Internat. Workshop on Symbolic-Numeric Comput. SNC 2005 Proc.(2005). Distributed at the Workshop in Xi'an, China, July 19-21.Google Scholar
- Wang, D., and Zhi, L. Eds. Symbolic-Numeric Computation. Trends in Mathematics. Birkhäuser Verlag, Basel, Switzerland, 2007. Google ScholarDigital Library
- Wang, P. S., Guy, M. J. T., and Davenport, J. H. P-adic reconstruction of rational numbers. SIGSAM Bulletin 16,2 (May 1982), 2--3. Google ScholarDigital Library
- Zippel, R. Probabilistic algorithms for sparse polynomials. In Symbolic and Algebraic Computation (Heidelberg, Germany, 1979), vol. 72 of Lect. Notes Comput. Sci., Springer Verlag, pp.216--226. Proc. EUROSAM '79. Google ScholarDigital Library
- Zippel, R. Interpolating polynomials from their values. J. Symbolic Computation 9,3 (1990), 375--403. Google ScholarDigital Library
Index Terms
- On exact and approximate interpolation of sparse rational functions
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