Fast Approximate Computations with Cauchy Matrices, Polynomials and Rational Functions

  • Victor Y. Pan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8476)


The papers [18], [9], [29], and [28] combine the techniques of the Fast Multipole Method of [15], [8] with the transformations of matrix structures, traced back to [19]. The resulting numerically stable algorithms approximate the solutions of Toeplitz, Hankel, Toeplitz-like, and Hankel-like linear systems of equations in nearly linear arithmetic time, versus the classical cubic time and the quadratic time of the previous advanced algorithms. We extend this progress to decrease the arithmetic time of the known numerical algorithms from quadratic to nearly linear for computations with matrices that have structure of Cauchy or Vandermonde type and for the evaluation and interpolation of polynomials and rational functions. We detail and analyze the new algorithms, and in [21] we extend them further.


Cauchy matrices Fast Multipole Method HSS matrices Vandermonde matrices Polynomial evaluation Rational evaluation Interpolation 


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  1. 1.
    Bracewell, R.: The Fourier Transform and Its Applications, 3rd edn. McGraw-Hill, New York (1999)Google Scholar
  2. 2.
    Börm, S.: Efficient Numerical Methods for Non-local Operators: \(\mathcal H^2\)-Matrix Compression, Algorithms and Analysis. European Math. Society (2010)Google Scholar
  3. 3.
    Bella, T., Eidelman, Y., Gohberg, I., Olshevsky, V.: Computations with Quasiseparable Polynomials and Matrices. Theoretical Computer Science 409(2), 158–179 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bini, D.A., Fiorentino, G.: Design, Analysis, and Implementation of a Multiprecision Polynomial Rootfinder. Numer. Algs. 23, 127–173 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bini, D., Pan, V.Y.: Polynomial and Matrix Computations, Volume 1: Fundamental Algorithms. Birkhäuser, Boston (1994)CrossRefGoogle Scholar
  6. 6.
    Barba, L.A., Yokota, R.: How Will the Fast Multipole Method Fare in Exascale Era? SIAM News 46(6), 1–3 (2013)Google Scholar
  7. 7.
    Chandrasekaran, S., Dewilde, P., Gu, M., Lyons, W., Pals, T.: A Fast Solver for HSS Representations via Sparse Matrices. SIAM J. Matrix Anal. Appl. 29(1), 67–81 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Carrier, J., Greengard, L., Rokhlin, V.: A Fast Adaptive Algorithm for Particle Simulation. SIAM J. Scientific Computing 9, 669–686 (1998)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Chandrasekaran, S., Gu, M., Sun, X., Xia, J., Zhu, J.: A Superfast Algorithm for Toeplitz Systems of Linear Equations. SIAM J. Matrix Anal. Appl. 29, 1247–1266 (2007)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Dutt, A., Gu, M., Rokhlin, V.: Fast Algorithms for Polynomial Interpolation, Integration, and Differentiation. SIAM Journal on Numerical Analysis 33(5), 1689–1711 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Dewilde, P., van der Veen, A.: Time-Varying Systems and Computations. Kluwer Academic Publishers, Dordrecht (1998)CrossRefzbMATHGoogle Scholar
  12. 12.
    Eidelman, Y., Gohberg, I.: A Modification of the Dewilde–van der Veen Method for Inversion of Finite Structured Matrices. Linear Algebra and Its Applications 343, 419–450 (2002)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Eidelman, Y., Gohberg, I., Haimovici, I.: Separable Type Representations of Matrices and Fast Algorithms. Birkhäuser (2013)Google Scholar
  14. 14.
    Gohberg, I., Kailath, T., Olshevsky, V.: Fast Gaussian Elimination with Partial Pivoting for Matrices with Displacement Structure. Mathematics of Computation 64, 1557–1576 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Greengard, L., Rokhlin, V.: A Fast Algorithm for Particle Simulation. Journal of Computational Physics 73, 325–348 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Gentelman, W., Sande, G.: Fast Fourier Transform for Fun and Profit. Full Joint Comput. Conference 29, 563–578 (1966)Google Scholar
  17. 17.
    Lipton, R.J., Rose, D., Tarjan, R.E.: Generalized Nested Dissection. SIAM J. on Numerical Analysis 16(2), 346–358 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Martinsson, P.G., Rokhlin, V., Tygert, M.: A Fast Algorithm for the Inversion of Toeplitz Matrices. Comput. Math. Appl. 50, 741–752 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Pan, V.Y.: On Computations with Dense Structured Matrices, Math. of Computation, 55(191), 179–190 (1990); Also in Proc. Intern. Symposium on Symbolic and Algebraic Computation (ISSAC 1989), 34–42. ACM Press, New York (1989)Google Scholar
  20. 20.
    Pan, V.Y.: Structured Matrices and Polynomials: Unified Superfast Algorithms. Birkhäuser/Springer, Boston/New York (2001)CrossRefGoogle Scholar
  21. 21.
    Pan, V.Y.: Transformations of Matrix Structures Work Again II, In: arxiv:1311.3729[math.NA]Google Scholar
  22. 22.
    Pan, V.Y.: Fast Approximation Algorithms for Computations with Cauchy Matrices and Extensions, in arxiv and Tech. Report TR 201400x, PhD Program in Comp. Sci., Graduate Center, CUNY (2014)Google Scholar
  23. 23.
    Pan, V.Y., Reif, J.: Fast and Efficient Parallel Solution of Sparse Linear Systems. SIAM J. on Computing 22(6), 1227–1250 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Rokhlin, V.: Rapid Solution of Integral Equations of Classical Potential Theory. Journal of Computational Physics 60, 187–207 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Vandebril, R., Van Barel, M., Mastronardi, N.: Matrix Computations and Semiseparable Matrices: Linear Systems, vol. 1. The Johns Hopkins University Press, Baltimore (2007)Google Scholar
  26. 26.
    Xia, J.: On the Complexity of Some Hierarchical Structured Matrix Algorithms. SIAM J. Matrix Anal. Appl. 33, 388–410 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Xia, J.: Randomized Sparse Direct Solvers. SIAM J. Matrix Anal. Appl. 34, 197–227 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Xia, J., Xi, Y., Cauley, S., Balakrishnan, V.: Superfast and Stable Structured Solvers for Toeplitz Least Squares via Randomized Sampling. SIAM J. Matrix Anal. and Applications 35, 44–72 (2014)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Xia, J., Xi, Y., Gu, M.: A Superfast Structured Solver for Toeplitz Linear Systems via Randomized Sampling. SIAM J. Matrix Anal. Appl. 33, 837–858 (2012)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Victor Y. Pan
    • 1
  1. 1.Department of Mathematics and Computer ScienceLehman College and the Graduate Center of the City University of New YorkBronxUSA

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