×

An iterative method to compute the dominant zero of a quaternionic unilateral polynomial. (English) Zbl 1397.65070

Summary: The aim of this paper is to propose an iterative method to compute the dominant zero of a quaternionic unilateral polynomial. We prove that the method is convergent in the sense that it generates a sequence of quaternions that converges to the dominant zero of the polynomial. The idea subjacent to this method is the well known Sebastião e Silva’s method, proposed in [J. Sebastião e Silva, Port. Math. 2, 271–279 (1941; Zbl 0026.05303)] to approximate the dominant zero of complex polynomials.

MSC:

65H04 Numerical computation of roots of polynomial equations
12E15 Skew fields, division rings

Citations:

Zbl 0026.05303
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bini, DA; Latouche, G; Meini, B, Solving matrix polynomial equations arising in queuing problems, Linear Algebra Appl., 340, 225-244, (2002) · Zbl 0994.65047 · doi:10.1016/S0024-3795(01)00426-8
[2] Cardinal, J.-P.: On Two Iterative Methods for Approximating the Roots of a Polynomial. The Mathematics of Numerical Analysis (Park City, UT, 1995), 165-188, Lectures in Appl. Math., vol. 32. Amer. Math. Soc., Providence (1996) · Zbl 0893.65031
[3] Chapman, A; Machen, C, Standard polynomial equations over division algebras, Adv. Appl. Clifford Algebras, 27, 1065-1072, (2016) · Zbl 1366.16014 · doi:10.1007/s00006-016-0740-4
[4] Chung, S.P.: Generalization and acceleration of an algorithm of Sebastião e Silva and its duals. Numer. Math. 25(4), 365-377 (1975/76)
[5] Damiano, A; Gentil, G; Struppa, D, Computations in the ring of quaternionic polynomials, J. Symb. Comput., 45, 38-45, (2010) · Zbl 1221.30104 · doi:10.1016/j.jsc.2009.06.003
[6] Dennis, JE; Traub, JF; Weber, RP, Algorithms for solvents of matrix polynomials, SIAM J. Numer. Anal., 15, 523-533, (1978) · Zbl 0386.65012 · doi:10.1137/0715034
[7] Dennis Jr., J.E., Traub, J.F., Weber, R.P.: On the matrix polynomial, lambda-matrix and block eigenvalue problems. Computer Science Department Technical Report, Cornell University, Ithaca, New York, and Carnigie-Mellon University. Pittsburgh, Pennsylvania (1971) · Zbl 1050.30037
[8] Falcão, MI, Newton method in the context of quaternion analysis, Appl. Math. Comput., 236, 458-470, (2014) · Zbl 1334.65085
[9] Falcão, MI; Miranda, F; Severino, R; Soares, MJ, Weierstrass method for quaternionic polynomial root-finding, Math. Methods Appl. Sci., 41, 423-437, (2018) · Zbl 1387.65043 · doi:10.1002/mma.4623
[10] Householder, AS, Generalization of an algorithm of sebastião e silva, Numer. Math., 16, 375-382, (1971) · Zbl 0197.43003 · doi:10.1007/BF02165009
[11] Householder, AS, Multigradients and the zeros of transcendental functions, Linear Algebra Appl., 4, 175-182, (1971) · Zbl 0213.16501 · doi:10.1016/0024-3795(71)90038-3
[12] Jacobson, N.: The Theory of Rings, Math. Surveys, vol. 2. AMS, New York (1943) · doi:10.1090/surv/002
[13] Janovská, D; Opfer, G, A note on the computation of all zeros of simple quaternionic polynomials, SIAM J. Numer. Anal., 48, 244-256, (2010) · Zbl 1247.65060 · doi:10.1137/090748871
[14] Kalantari, B, Algorithms for quaternion polynomial root-finding, J. Complex., 29, 302-322, (2013) · Zbl 1326.65060 · doi:10.1016/j.jco.2013.03.001
[15] Lam, T.Y.: A First Course in Noncommutative Rings. Springer, Berlin (1991) · Zbl 0728.16001 · doi:10.1007/978-1-4684-0406-7
[16] Lam, TY, A general theory of Vandermonde matrices, Expo. Math., 4, 193-215, (1986) · Zbl 0598.15015
[17] Lam, TY; Leroy, A, Vandermonde and Wronskian matrices over division rings, J. Algebra, 119, 308-336, (1988) · Zbl 0657.15015 · doi:10.1016/0021-8693(88)90063-4
[18] Leo, S; Ducati, G; Leonardi, V, Zeros of unilateral quaternionic polinomials, Electron. J. Linear Algebra, 15, 297-313, (2006) · Zbl 1151.15303
[19] Niven, I, Equations in quaternions, Am. Math. Mon., 48, 645-661, (1941) · Zbl 0060.08002 · doi:10.1080/00029890.1941.11991158
[20] Pan, VY, The amended dsesc power method for polynomial root-finding, Comput. Math. Appl., 49, 1515-1524, (2005) · Zbl 1077.65049 · doi:10.1016/j.camwa.2004.09.011
[21] Pogorui, A; Shapiro, M, On the structure of the set of zeros of quaternionic polynomials, Complex Var. Elliptic Equ., 49, 379-389, (2004) · Zbl 1160.30353
[22] Sebastião e Silva, J, Sur une méthode d’approximation semblable à celle de gräffe, Port. Math., 2, 271-279, (1941) · Zbl 0026.05303
[23] Serôdio, R; Pereira, E; Vitória, J, Computing the zeros of quaternion polynomials, Comput. Math. Appl., 42, 1229-1237, (2001) · Zbl 1050.30037 · doi:10.1016/S0898-1221(01)00235-8
[24] Serôdio, R; Siu, Lok-Shun, Zeros of quaternion polynomials, Appl. Math. Lett., 14, 237-239, (2001) · Zbl 0979.30030 · doi:10.1016/S0893-9659(00)00142-7
[25] Stewart, GW, On the convergence of sebastião e silva’s method for finding a zero of a polynomial, SIAM Rev., 12, 458-460, (1970) · Zbl 0198.20905 · doi:10.1137/1012085
[26] Topuridze, N, On roots of quaternion polynomials, J. Math. Sci., 160, 843-855, (2009) · Zbl 1243.16022 · doi:10.1007/s10958-009-9530-z
[27] Zhang, F, Quaternions and matrices of quaternions, Linear Algebra Appl., 251, 21-57, (1997) · Zbl 0873.15008 · doi:10.1016/0024-3795(95)00543-9
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.