Forward Stable Computation of Roots of Real Polynomials with Real Simple Roots

Authors

Nevena Jakovcevic Stor, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Rudjera Boˇskovi´ca 32, 21000 Split, CroatiaFollow
Ivan Slapnicar, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Rudjera Boˇskovi´ca 32, 21000 Split, CroatiaFollow

Author Country (or Countries)

Croatia

Abstract

As showed in (Fiedler, 1990), any polynomial can be expressed as a characteristic polynomial of a complex symmetric arrowhead matrix. This expression is not unique. If the polynomial is real with only real distinct roots, the matrix can be chosen real. By using the accurate forward stable algorithm for computing eigenvalues of the real symmetric arrowhead matrices from (Jakovˇcevi´c Stor, Slapniˇcar, Barlow, 2015), we derive a new forward stable algorithm for computation of roots of such polynomials in O(n2) operations. The algorithm computes each root to almost full accuracy. In some cases, the algorithm invokes extended precision routines, but only in the non-iterative part. Our examples include numerically difficult problems, like the well-known Wilkinson’s polynomials. Our algorithm compares favorably to other method for polynomial root-finding, like MPSolve or Newton’s method.

Digital Object Identifier (DOI)

http://dx.doi.org/10.18576/amis/110105

Recommended Citation

Jakovcevic Stor, Nevena and Slapnicar, Ivan (2017) "Forward Stable Computation of Roots of Real Polynomials with Real Simple Roots," Applied Mathematics & Information Sciences: Vol. 11: Iss. 1, Article 5.
DOI: http://dx.doi.org/10.18576/amis/110105
Available at: https://digitalcommons.aaru.edu.jo/amis/vol11/iss1/5