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The Jacobi matrix for functions in noncommutative algebras. (English) Zbl 1316.65049
Adv. Appl. Clifford Algebr. 24, No. 4, 1059-1073 (2014); erratum ibid. 24, No. 4, 1075 (2014).
The authors develop a general tool for constructing the exact Jacobi matrix for functions defined in noncommutative algebraic systems without using any partial derivative. The construction is applied to solving nonlinear problems of the form \(f(x) = 0\) with the aid of Newton’s method in algebras defined in \({\mathbb{R}^N}\).

MSC:
65F30 Other matrix algorithms (MSC2010)
65F60 Numerical computation of matrix exponential and similar matrix functions
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[1] H. Abou-Kandill, G. Freiling, V. Ionescu, G. Jank, Matrix Riccati equations in control and systems theory. Birkhäuser, Basel, Boston, Berlin, 2003, 572 pp. · Zbl 1027.93001
[2] D. A. Bini, B. Iannazzo, B. Meini, Numerical solution of algebraic Riccati equations. SIAM, Philadelphia, 2012, 250 pp. · Zbl 1244.65058
[3] D. C. Brody and E.-M. Graefe, On complexified mechanics and coquaternions. J. Phys. A: Math. Theory, 44 (2011) 072001, 9 pp., also: arXiv:1012.0757v2 [math-ph] 7 Jan 2011, 11 pp.
[4] J. Cockle, On systems of algebra involving more than one imaginary; and on equations of the fifth degree. Phil. Mag., (3) 35, 1849, pp. 434-437.
[5] J. Cockle, http://www.oocities.org/cocklebio/
[6] J. Dieudonné, Foundation of modern analysis. Academic Press, New York, London, 1960, 361 pp.
[7] M. I. Falcão, Newton method in the context of quaternionic analysis. Appl. Math. Comput. 236 (2014), pp. 458-470. · Zbl 1334.65085
[8] I. Frenkel, M. Libine, Split quaternionic analysis and separation of the series for\({SL(2, \mathbb{R})}\)and\({SL(2, \mathbb{C})/SL(2, \mathbb{R})}\) . Advances in Mathematics, 228, (2011), 678-763 pp., also:arXiv:1009.2532v2 [math.RT] 22 Jul 2011, 70 pp. · Zbl 1264.30037
[9] D. J. H. Garling, Clifford algebras: an introduction. Cambridge Univerity Press, Cambridge, 2011, 200 pp. · Zbl 1235.15025
[10] W. R. Hamilton, http://en.wikipedia.org/wiki/William_Rowan_Hamilton
[11] K. Gürlebeck, W. Sprössig, Quaternionic and Clifford calculus for physicists and engineers, Wiley, Chichester, 1997, 371 pp.
[12] N. J. Higham, Functions of matrices. Theory and computation. SIAM, Philadelphia, 2008, 425 pp.
[13] R. A. Horn and C. R. Johnson, Topics in matrix analysis. Cambridge University Press, Cambridge, 1991, 607 pp. · Zbl 0729.15001
[14] R. A. Horn and C. R. Johnson, Matrix analysis. Cambridge University Press, Cambridge, 1992, 561 pp.
[15] D. Janovská, G. Opfer, Zeros and singular points for one-sided, coquaternionic polynomials with an extension to other\({\mathbb{R}^4}\)algebras. ETNA 41 (2014), pp. 133- 158. · Zbl 1307.65060
[16] D. Janovská, G. Opfer, The algebraic Riccati equation for quaternions. Dedicated to Ivo Marek on the occasion of his 80th birthday, Adv. Appl. Clifford Algebras, 23 (2013), pp. 907-918.
[17] D. Janovská, G. Opfer, Linear equations and the Kronecker product in coquaternions. Mitt. Math. Ges. Hamburg, 33 (2013), pp. 181-196. · Zbl 1298.15006
[18] D. Janovská, G. Opfer, The classification and the computation of the zeros of quaternionic, two-sided polynomials. Numer. Math., 115 (2010), pp. 81-100. · Zbl 1190.65075
[19] D. Janovská, G. Opfer, A note on the computation of all zeros of simple quaternionic polynomials. SIAM J. Numer. Anal., 48 (2010), pp. 244-256. · Zbl 1247.65060
[20] D. Janovská, G. Opfer, Computing quaternionic roots by Newton’s method. Electron. Trans. Numer. Anal., 26 (2007), pp. 82-102. · Zbl 1160.65016
[21] P. Lancaster, L. Rodman, Algebraic Riccati equations. Clarendon Press, 1995, 498 pp. · Zbl 0836.15005
[22] B. Schmeikal, Tessarinen, Nektarinen und andere Vierheiten. Beweis einer Beobachtung von Gerhard Opfer. To appear in: Mitt. Math. Ges. Hamburg 34 (2014), 28 pp. · Zbl 1298.65083
[23] Simoncini, V.; Szyld, D.B.; Monsalve, M., On the numerical solution of largescale Riccati equations, IMA Journal on Numerical Analysis,, 34, 904-920, (2014) · Zbl 1298.65083
[24] A. E. Taylor, D. C. Lay, Introduction to functional analysis. 2nd ed., Wiley, New York, 1980, 467 pp. · Zbl 0501.46003
[25] F. Zhang, Quaternions and matrices of quaternions. Linear Algebra Appl., 251 (1997), pp. 21-57. · Zbl 0873.15008
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