A given polynomial, possibly with multiple roots, is
factored into several lower-degree distinct-root polynomials with
natural-order-integer powers. All the roots, including multiplicities,
of the original polynomial may be obtained by solving these lowerdegree
distinct-root polynomials, instead of the original high-degree
multiple-root polynomial directly.
The approach requires polynomial Greatest Common Divisor
(GCD) computation. The very simple and effective process, “Monic
polynomial subtractions" converted trickily from “Longhand
polynomial divisions" of Euclidean algorithm is employed. It
requires only simple elementary arithmetic operations without any
advanced mathematics.
Amazingly, the derived routine gives the expected results for the
test polynomials of very high degree, such as p( x) =(x+1)1000.
[1] Z. Zeng, "Computing multiple roots of inexact polynomials," Math.
Comput, 74 (2005), pp. 869-903.
[2] F.C. Chang, "GCD of two univaiate polynomials by monic polynomial
subtractions," submitted to Appl. Math. Comput.
[3] W.S. Brown and J.F. Traub, "On Euclid-s algorithm and the theory of
subresultants," J. ACM 14 (1) (1967), pp. 128-142.
[4] I.S. Pace and S. Barnett, "Comparison of algorithm for calculation of
GCD of polynomials," Int. J. System Scien, 4 (1973), pp. 211-226.
[5] M. Mitrouli and N. Karcanias, "Comutation of the GCD of polynomials
using Gaussian transformation and shifting," Int. J. Control, 58 (1993),
pp. 211-228.
[6] A. Terui, "Recursive polynomial remainder sequence and its
subresultants," J. Algebra, (2008).
[7] C.D. Yan and W.H.Chieng, "Method for finding multiple roots of
polynomials," Int. J. Computers & Mathematics with Applications, 51
(2006), pp. 605-620.
[1] Z. Zeng, "Computing multiple roots of inexact polynomials," Math.
Comput, 74 (2005), pp. 869-903.
[2] F.C. Chang, "GCD of two univaiate polynomials by monic polynomial
subtractions," submitted to Appl. Math. Comput.
[3] W.S. Brown and J.F. Traub, "On Euclid-s algorithm and the theory of
subresultants," J. ACM 14 (1) (1967), pp. 128-142.
[4] I.S. Pace and S. Barnett, "Comparison of algorithm for calculation of
GCD of polynomials," Int. J. System Scien, 4 (1973), pp. 211-226.
[5] M. Mitrouli and N. Karcanias, "Comutation of the GCD of polynomials
using Gaussian transformation and shifting," Int. J. Control, 58 (1993),
pp. 211-228.
[6] A. Terui, "Recursive polynomial remainder sequence and its
subresultants," J. Algebra, (2008).
[7] C.D. Yan and W.H.Chieng, "Method for finding multiple roots of
polynomials," Int. J. Computers & Mathematics with Applications, 51
(2006), pp. 605-620.
@article{"International Journal of Information, Control and Computer Sciences:59149", author = "Feng Cheng Chang", title = "Factoring a Polynomial with Multiple-Roots", abstract = "A given polynomial, possibly with multiple roots, is
factored into several lower-degree distinct-root polynomials with
natural-order-integer powers. All the roots, including multiplicities,
of the original polynomial may be obtained by solving these lowerdegree
distinct-root polynomials, instead of the original high-degree
multiple-root polynomial directly.
The approach requires polynomial Greatest Common Divisor
(GCD) computation. The very simple and effective process, “Monic
polynomial subtractions" converted trickily from “Longhand
polynomial divisions" of Euclidean algorithm is employed. It
requires only simple elementary arithmetic operations without any
advanced mathematics.
Amazingly, the derived routine gives the expected results for the
test polynomials of very high degree, such as p( x) =(x+1)1000.", keywords = "Polynomial roots, greatest common divisor,
Longhand polynomial division, Euclidean GCD Algorithm.", volume = "2", number = "11", pages = "3852-4", }