Real and Complex Polynomial Root-Finding by Means of Eigen-Solving

  • Victor Y. Pan
  • Guoliang Qian
  • Ai-Long Zheng
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)


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.


Root-finding Eigen-solving Randomization Matrix sign function 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Victor Y. Pan
    • 1
    • 2
  • Guoliang Qian
    • 2
  • Ai-Long Zheng
    • 2
  1. 1.Department of Mathematics and Computer ScienceLehman College of the City University of New YorkBronxUSA
  2. 2.Ph.D. Programs in Mathematics and Computer ScienceThe Graduate Center of the City University of New YorkNew YorkUSA

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