Abstract
Our new numerical algorithms approximate real and complex roots of a univariate polynomial lying near a selected point of the complex plane, all its real roots, and all its roots lying in a fixed half-plane or in a fixed rectangular region. The algorithms seek the roots of a polynomial as the eigenvalues of the associated companion matrix. Our analysis and experiments show their efficiency. We employ some advanced machinery available for matrix eigen-solving, exploit the structure of the companion matrix, and apply randomized matrix algorithms, repeated squaring, matrix sign iteration and subdivision of the complex plane. Some of our techniques can be of independent interest.
This is a preview of subscription content, log in via an institution to check access.
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
Bini, D.A., Gemignani, L., Pan, V.Y.: Inverse power and Durand/Kerner iteration for univariate polynomial root-finding. Computers and Math (with Applics.) 47(2/3), 447–459 (2004)
Bini, D., Pan, V.Y.: Graeffe’s, Chebyshev, and Cardinal’s processes for splitting a polynomial into factors. J. Complexity 12, 492–511 (1996)
Cardinal, J.P.: On two iterative methods for approximating the roots of a polynomial. Lectures in Applied Math 32, 165–188 (1996)
Emiris, I.Z., Mourrain, B., Tsigaridas, E.P.: Real Algebraic Numbers: Complexity Analysis and Experimentation. In: Hertling, P., Hoffmann, C.M., Luther, W., Revol, N. (eds.) Real Number Algorithms. LNCS, vol. 5045, pp. 57–82. Springer, Heidelberg (2008)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
Higham, N.J.: Functions of Matrices: Theory and Computations. SIAM (2008)
Pan, C.–T.: On the existence and computation of Rrank-revealing LU factorization. Linear Algebra and Its Applications 316, 199–222 (2000)
Pan, V.Y.: Structured Matrices and Polynomials: Unified Superfast Algorithms. Birkhäuser, Boston, and Springer, NY (2001)
Pan, V.Y.: Amended DSeSC power method for polynomial root-finding. Computers and Math (with Applics.) 49(9-10), 1515–1524 (2005)
Pan, V.Y., Qian, G., Murphy, B., Rosholt, R.E., Tang, Y.: Real root-finding. In: Vershelde, J., Stephen Watt, S. (eds.) Proc. Third Int. Workshop on Symbolic–Numeric Computation (SNC 2007), London, Ontario, Canada, pp. 161–169. ACM Press, New York (2007)
Pan, V.Y., Zheng, A.: New progress in real and complex polynomial root-finding. Computers and Math (Also in Proc. ISSAC 2010) 61, 1305–1334 (2010)
Pan, V.Y., Qian, G., Zheng, A.: Randomized Matrix Computations II. Tech. Report TR 2012006, Ph.D. Program in Computer Science, Graduate Center, the City University of New York (2012), http://www.cs.gc.cuny.edu/tr/techreport.php?id=433
Stewart, G.W.: Matrix Algorithms, Vol II: Eigensystems, 2nd edn. SIAM, Philadelphia (2001)
Yap, C., Sagraloff, M.: A simple but exact and efficient algorithm for complex root isolation. In: Proc. ISSAC 2011, pp. 353–360 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pan, V.Y., Qian, G., Zheng, AL. (2012). Real and Complex Polynomial Root-Finding by Means of Eigen-Solving. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2012. Lecture Notes in Computer Science, vol 7442. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32973-9_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-32973-9_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32972-2
Online ISBN: 978-3-642-32973-9
eBook Packages: Computer ScienceComputer Science (R0)