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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.
Reviewer: Reviewer (Berlin)
##### MSC:
 65H04 Numerical computation of roots of polynomial equations 12E15 Skew fields, division rings
##### Keywords:
division algebra; polynomials; zeros of polynomials; quaternion
Full Text:
##### 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 [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 [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 [6] Dennis, JE; Traub, JF; Weber, RP, Algorithms for solvents of matrix polynomials, SIAM J. Numer. Anal., 15, 523-533, (1978) · Zbl 0386.65012 [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 [10] Householder, AS, Generalization of an algorithm of sebastião e silva, Numer. Math., 16, 375-382, (1971) · Zbl 0197.43003 [11] Householder, AS, Multigradients and the zeros of transcendental functions, Linear Algebra Appl., 4, 175-182, (1971) · Zbl 0213.16501 [12] Jacobson, N.: The Theory of Rings, Math. Surveys, vol. 2. AMS, New York (1943) [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 [14] Kalantari, B, Algorithms for quaternion polynomial root-finding, J. Complex., 29, 302-322, (2013) · Zbl 1326.65060 [15] Lam, T.Y.: A First Course in Noncommutative Rings. Springer, Berlin (1991) · Zbl 0728.16001 [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 [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 [20] Pan, VY, The amended dsesc power method for polynomial root-finding, Comput. Math. Appl., 49, 1515-1524, (2005) · Zbl 1077.65049 [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 [24] Serôdio, R; Siu, Lok-Shun, Zeros of quaternion polynomials, Appl. Math. Lett., 14, 237-239, (2001) · Zbl 0979.30030 [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 [26] Topuridze, N, On roots of quaternion polynomials, J. Math. Sci., 160, 843-855, (2009) · Zbl 1243.16022 [27] Zhang, F, Quaternions and matrices of quaternions, Linear Algebra Appl., 251, 21-57, (1997) · Zbl 0873.15008
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