Abstract
Multipoint polynomial evaluation and interpolation are fundamental for modern numerical and symbolic computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grows to quadratic for numerical solution. By combining a variant of the Multipole celebrated numerical techniques with transformations of matrix structures of [10] we achieve dramatic speedup and for a large class of inputs yield solution algorithms running in nearly linear time as well. The algorithms support similar speedup of approximation of the products of a Vandermonde matrix, its transpose, inverse, and the transpose of the inverse by a vector.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bella, T., Eidelman, Y., Gohberg, I., Olshevsky, V.: Computations with quasiseparable polynomials and matrices. TCS 409(2), 158–179 (2008)
Bini, D.A., Fiorentino, G.: Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numer. Algs. 23, 127–173 (2000)
Chandrasekaran, S., Gu, M., Sun, X., Xia, J., Zhu, J.: A superfast algorithm for Toeplitz systems of linear equations. SIMAX 29, 1247–1266 (2007)
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)
Eidelman, Y., Gohberg, I., Haimovici, I.: Separable Type Representations of Matrices and Fast Algorithms, vol. 1. Birkhäuser, Basel (2013)
Gu, M.: Stable and efficient algorithms for structured systems of linear equations. SIAM J. Matrix Anal. Appl. 19, 279–306 (1998)
Gohberg, I., Kailath, T., Olshevsky, V.: Fast Gaussian elimination with partial pivoting for matrices with displacement structure. Mathematics of Computation 64, 1557–1576 (1995)
Kirrinnis, P.: Polynomial factorization and partial fraction decomposition by Newton’s iteration. J. Complexity 14, 378–444 (1998)
Martinsson, P.G., Rokhlin, V., Tygert, M.: A fast algorithm for the inversion of Toeplitz matrices. Comput. Math. Appl. 50, 741–752 (2005)
Pan, V.Y.: On computations with dense structured matrices. Math. Computation 55(191), 179–190 (1990); Proceedings version in Proc. ISSAC 1989, pp. 34–42. ACM Press, New York (1989)
Priest, D.: Algorithms for arbitrary precision floating point arithmetic. In: Kornerup, P., Matula, D. (eds.) Proc. 10th Symp. Computer Arithmetic, pp. 132–145. IEEE Computer Society Press, Los Angeles (1991)
Pan, V.Y.: Structured Matrices and Polynomials: Unified Superfast Algorithms. Birkhäuser/Springer, Boston/New York (2001)
Pan, V.Y.: Transformations of Matrix Structures Work Again. Tech. Report TR 2013004, PhD Program in Comp. Sci., Graduate Center, CUNY (2013), http://www.cs.gc.cuny.edu/tr/techreport.php?id=449
Pan, V.Y.: Polynomial Evaluation and Interpolation and Transformations of Matrix Structures 1. Tech. Report TR 2013007, PhD Program in Comp. Sci. Graduate Center, CUNY (2013), http://www.cs.gc.cuny.edu/tr/techreport.php?id=452
Pan, V.Y., Tsigaridas, E.P.: On the Boolean complexity of the real root refinement. In: Kauers, M. (ed.) Proc. ISSAC 2013. ACM Press, New York (2013)
Stewart, G.W.: Matrix Algorithms. Basic Decompositions, vol. I. SIAM, Philadelphia (1998)
Vandebril, R., van Barel, M., Mastronardi, N.: Matrix Computations and Semiseparable Matrices: Linear Systems, vol. 1. The Johns Hopkins University Press, Baltimore (2007)
Xia, J., Xi, Y., Gu, M.: A superfast structured solver for Toeplitz linear systems via randomized sampling. SIMAX 33, 837–858 (2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this paper
Cite this paper
Pan, V.Y. (2013). Polynomial Evaluation and Interpolation and Transformations of Matrix Structures. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2013. Lecture Notes in Computer Science, vol 8136. Springer, Cham. https://doi.org/10.1007/978-3-319-02297-0_23
Download citation
DOI: https://doi.org/10.1007/978-3-319-02297-0_23
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-02296-3
Online ISBN: 978-3-319-02297-0
eBook Packages: Computer ScienceComputer Science (R0)