Mark Giesbrecht
ABSTRACT
We consider the problem of sparse interpolation of an approximate multivariate black-box polynomial in floating-point arithmetic. That is, both the inputs and outputs of the black-box polynomial have some error, and all numbers are represented in standard, fixed-precision, floating point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give efficient and numerically robust solutions. We note the similarity between the exact Ben-Or/Tiwari sparse interpolation algorithm and the classical Prony's method for interpolating a sum of exponential functions, and exploit the generalized eigenvalue reformulation of Prony's method. We analyze the numerical stability of our algorithms and the sensitivity of the solutions, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques in practice through numerical experiments and applications.
- B. Beckermann. The condition number of real Vandermonde, Krylov and positive definite Hankel matrices. Numeriche Mathematik, 85:553--577, 2000.]]Google Scholar
Cross Ref
- B. Beckermann, G. Golub, and G. Labahn. On the numerical condition of a generalized Hankel eigenvalue problem. submitted to Numerische Matematik, 2005.]] Google ScholarDigital Library
- M. Ben-Or and P. Tiwari. A deterministic algorithm for sparse multivariate polynomial interpolation. In Proc. Twentieth Annual ACM Symp. Theory Comput., pages 301--309, New York, N.Y., 1988. ACM Press.]] Google ScholarDigital Library
- A. Córdova, W. Gautschi, and S. Ruscheweyh. Vandermonde matrices on the circle: spectral properties and conditioning. Numerische Mathematik, 57:577--591, 1990.]]Google ScholarDigital Library
- R.M. Corless, M. Giesbrecht, I. Kotsireas, and S.M. Watt. Numerical implicitization of parametric hypersurfaces with linear algebra. In E. Roanes-Lozano, editor, Artificial Intelligence and Symbolic Computation: International Conference AISC 2000, pages 174--183, Heidelberg, Germany, 2001. Springer Verlag.]] Google ScholarDigital Library
- A. Díaz and E. Kaltofen. FoxBox a system for manipulating symbolic objects in black box representation. In O. Gloor, editor, Proc. 1998 Internat. Symp. Symbolic Algebraic Comput. (ISSAC'98), pages 30--37, New York, N. Y., 1998. ACM Press.]] Google ScholarDigital Library
- S. Gao, E. Kaltofen, J. May, Z. Yang, and L. Zhi. Approximate factorization of multivariate polynomials via differential equations. In ISSAC 2004 Proc. 2004 Internat. Symp. Symbolic Algebraic Comput., pages 167--174, 2004.]] Google ScholarDigital Library
- M. Gasca and T. Sauer. On the history of multivariate polynomial interpolation. J. Computational and Applied Mathematics, 122:23--35, 2000.]] Google ScholarDigital Library
- W. Gautschi. Norm estimates for inverses of Vandermonde matrices. Numerische Mathematik, 23:337--347, 1975.]]Google ScholarDigital Library
- W. Gautschi and G. Inglese. Lower bounds for the condition numbers of Vandermonde matrices. Numerische Mathematik, 52:241--250, 1988.]]Google ScholarCross Ref
- K. O. Geddes, S. R. Czapor, and G. Labahn. Algorithms for Computer Algebra. Kluwer Academic Publ., Boston, Massachusetts, USA, 1992.]] Google ScholarDigital Library
- G. H. Golub, P. Milanfar, and J. Varah. A stable numerical method for inverting shape from moments. SIAM J. Sci. Comput., 21(4):1222--1243, 1999.]] Google ScholarDigital Library
- G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins University Press, Baltimore, Maryland, third edition, 1996.]]Google Scholar
- D. Yu. Grigoriev, M. Karpinski, and M. F. Singer. Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. SIAM J. Comput., 19(6):1059--1063, 1990.]] Google ScholarDigital Library
- E. Kaltofen and Lakshman Yagati. Improved sparse multivariate polynomial interpolation algorithms. In P. Gianni, editor, Symbolic Algebraic Comput. Internat. Symp. ISSAC '88 Proc., volume 358 of Lect. Notes Comput. Sci., pages 467--474, Heidelberg, Germany, 1988. Springer Verlag.]] Google ScholarDigital Library
- E. Kaltofen and B. Trager. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. J. Symbolic Comput., 9(3):301--320, 1990.]] Google ScholarDigital Library
- L. Kronecker. Über einige Interpolationsformeln für ganze Funktionen mehrerer Variabeln, Lecture at the academy of sciences, December 21, 1865, volume H. Hensel (Ed.), L. Kroneckers Werke, Vol. I. Teubner, Stuttgart, 1895. reprinted by Chelsea, New York, 1968.]]Google Scholar
- R. Lorentz. Multivariate Hermite interpolation by algebaic polynomials: a survey. J. Computational and Applied Mathematics, 122:167--201, 2000.]] Google ScholarDigital Library
- Y. Mansour. Randomized approximation and interpolation of sparse polynomials. SIAM Journal on Computing, 24(2):357--368, 1995.]] Google ScholarDigital Library
- P. Milanfar, G. C. Verghese, W. C. Karl, and A. S. Wilsky. Reconstructing polygons from moments with connections to array processing. IEEE Trans. Signal Processing, 43(2):432--443, 1995.]]Google ScholarDigital Library
- Baron de Prony, Gaspard-Clair-François-Marie Riche. Essai expérimental et analytique sur les lois de la Dilatabilité des fluides élastique et sur celles de la Force expansive de la vapeur de l'eau et de la vapeur de l'alkool, à différentes températures. J. de l'École Polytechnique, 1:24--76, 1795.]]Google Scholar
- A. Sommese, J. Verschelde, and C. Wampler. Numerical factorization of multivariate complex polynomials. Theoretical Computer Science, 315(2-3):651--669, 2004.]] Google ScholarDigital Library
- J. H. Wilkinson. Rounding errors in algebraic processes. Prentice-Hall, Englewood Cliffs, N.J., 1963.]] Google ScholarDigital Library
- Z. Zilic and K. Radecka. On feasible multivariate polynomial interpolations over arbitrary fields. In S. Dooley, editor, ISSAC 99 Proc. 1999 Internat. Symp. Symbolic Algebraic Comput., pages 67--74, New York, N. Y., 1999. ACM Press.]] Google ScholarDigital Library
- R. Zippel. Probabilistic algorithms for sparse polynomials. In Proc. EUROSAM '79, volume 72 of Lect. Notes Comput. Sci., pages 216--226, Heidelberg, Germany, 1979. Springer Verlag.]] Google ScholarDigital Library
- R. Zippel. Interpolating polynomials from their values. J. Symbolic Comput., 9(3):375--403, 1990.]] Google ScholarDigital Library
Index Terms
- Symbolic-numeric sparse interpolation of multivariate polynomials
Recommendations
Symbolic-numeric sparse interpolation of multivariate polynomials
We consider the problem of sparse interpolation of an approximate multivariate black-box polynomial in floating point arithmetic. That is, both the inputs and outputs of the black-box polynomial have some error, and all numbers are represented in ...
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 ...
Sparse multivariate polynomial interpolation on the basis of Schubert polynomials
Schubert polynomials were discovered by A. Lascoux and M. Schützenberger in the study of cohomology rings of flag manifolds in 1980s. These polynomials generalize Schur polynomials and form a linear basis of multivariate polynomials. In 2003, Lenart and ...
Comments