Abstract
The paper describes some modifications of Newton’s method for refining the zeros of even-grade f(x)-twined (f(x)-egt) polynomials, defined as polynomials whose roots appear in pairs {x i ,f(x i )}. Particular attention is given to even-grade palindromic (egp) polynomials. The algorithms are derived from certain symmetric division processes for computing a symmetric quotient and a symmetric remainder of two given f(x)-egt polynomials. Numerical results indicate that the presented algorithms can be more accurate than other methods which do not take into consideration the symmetry of the coefficients.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bini D.A., Fiorentino, G.: Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numer. Algorithms 23(2–3), 127–173 (2000)
Bini, D.A., Noferini, V.: Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method. (2012). http://arxiv.org/pdf/1207.6292v1.pdf. Accessed 27 July 2012
da Fonseca, C.M., Petronilho, J.: Explicit inverses of some tridiagonal matrices. Linear Algebra Appl. 325(1–3), 7–21 (2001)
Gemignani, L., Noferini, V.: The Ehrlich-Aberth method for palindromic matrix polynomials represented in the Dickson basis. Linear Algebra Appl. (2012). doi:10.1016/j.laa.2011.10.035
Gohberg, I., Kaashoek, M.A., Lancaster, P.: General theory of regular matrix polynomials and band Toeplitz operators. Integr. Equ. Oper. Theory 11(6), 776–882 (1988)
Golberg, M.A.: The derivative of a determinant. Am. Math. Mon. 79, 1124–1126 (1972)
Goodman, T.N.T., Micchelli, C.A., Rodriguez, G., Seatzu, S.: Spectral factorization of Laurent polynomials. Adv. Comput. Math. 7(4), 429–454 (1997)
Kressner, D., Schröder, C., Watkins, D.S.: Implicit QR algorithms for palindromic and even eigenvalue problems. Numer. Algorithms 51(2), 209–238 (2009)
Lang, M., Frenzel, B.C.: Polynomial root finding. IEEE Signal Process. Lett. 1(10), 141–143 (1994)
Mackey, D.S., Mackey, N., Mehl, C., Mehrmann, V.: Smith Forms of Palindromic Matrix Polynomials. Electron. J. Linear Algebra 22, 53–91 (2011)
Mackey, D.S., Mackey, N., Mehl, C., Mehrmann, V.: Structured polynomial eigenvalue problems: good vibrations from good linearizations. SIAM J. Matrix Anal. Appl. 28(4), 1029–1051 (electronic), (2006)
Noferini, V.: The behaviour of the complete eigenstructure of a polynomial matrix under a generic rational transformation. Electron. J. Linear Algebra 23, 607–624 (2012)
Saux Picart, P., Brunie, C.: Symmetric subresultants and applications. J. Symb. Comput. 42(9), 884–919 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gemignani, L., Noferini, V. Modifications of Newton’s method for even-grade palindromic polynomials and other twined polynomials. Numer Algor 61, 315–329 (2012). https://doi.org/10.1007/s11075-012-9618-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-012-9618-2