Structured matrix methods for the computation of multiple roots of a polynomial

This paper considers the application of structured matrix methods for the computation of multiple roots of a polynomial. In particular, the given polynomial f(y) is formed by the addition of noise to the coefficients of its exact form [email protected]?(y), and the noise causes multiple roots of [email protected]?(y) to break up into simple roots. It is shown that structured matrix methods enable the simple roots of f(y) that originate from the same multiple root of [email protected]?(y) to be 'sewn' together, which therefore allows the multiple roots of [email protected]?(y) to be computed. The algorithm that achieves these results involves several greatest common divisor computations and polynomial deconvolutions, and special care is required for the implementation of these operations because they are ill-posed. Computational examples that demonstrate the theory are included, and the results are compared with the results from MultRoot, which is a suite of Matlab programs for the computation of multiple roots of a polynomial.

[1]  T. A. Brown,et al.  Theory of Equations. , 1950, The Mathematical Gazette.

[2]  W. Burnside,et al.  Theory of equations , 1886 .

[3]  Joab R. Winkler,et al.  The computation of multiple roots of a polynomial , 2012, J. Comput. Appl. Math..

[4]  Giuseppe Fiorentino,et al.  Design, analysis, and implementation of a multiprecision polynomial rootfinder , 2000, Numerical Algorithms.

[6]  D. Hough.  Explaining and ameliorating the ill condition of zeros of polynomials. , 1977 .

[7]  W. Kahan.  Conserving Confluence Curbs Ill-Condition , 1972 .

[8]  L. Foster.  Generalizations of Laguerre’s Method: Higher Order Methods , 1981 .

[9]  Zhonggang Zeng.  Computing multiple roots of inexact polynomials , 2005, Math. Comput..

[10]  Donna Kastle Dunaway.  A composite algorithm for finding zeros of real polynomials , 1972 .

[11]  G. Arfken.  Mathematical Methods for Physicists , 1967 .

[12]  S. Klasa,et al.  Polynomial scaling , 1990 .

[13]  Joab R. Winkler,et al.  A non-linear structure preserving matrix method for the low rank approximation of the Sylvester resultant matrix , 2010, J. Comput. Appl. Math..

[14]  Jim Euchner.  Design , 2014, Catalysis from A to Z.

[15]  M. A. Jenkins,et al.  A three-stage variable-shift iteration for polynomial zeros and its relation to generalized rayleigh iteration , 1970 .

[16]  K. Madsen.  A root-finding algorithm based on Newton's method , 1973 .

[17]  Joab R. Winkler,et al.  High Order Terms for Condition Estimation of Univariate Polynomials , 2006, SIAM J. Sci. Comput..

[18]  Joab R. Winkler,et al.  Two methods for the calculation of the degree of an approximate greatest common divisor of two inexact polynomials , 2012 .

[19]  Wei-Hua Chieng,et al.  Method for finding multiple roots of polynomials , 2006, Comput. Math. Appl..

[20]  Joab R. Winkler,et al.  An improved non-linear method for the computation of a structured low rank approximation of the Sylvester resultant matrix , 2013, J. Comput. Appl. Math..

[21]  Joab R. Winkler,et al.  The calculation of the degree of an approximate greatest common divisor of two polynomials , 2011, J. Comput. Appl. Math..

[22]  W. G. Horner,et al.  A new method of solving numerical equations of all orders, by continuous approximation , 1815 .

[24]  Zhonggang Zeng.  Algorithm 835: MultRoot---a Matlab package for computing polynomial roots and multiplicities , 2004, TOMS.

[25]  Curtis F. Gerald,et al.  APPLIED NUMERICAL ANALYSIS , 1972, The Mathematical Gazette.

[26]  E. Hansen,et al.  Some modifications of Laguerre's method , 1977 .