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Algorithms for the matrix sector function. (English) Zbl 1190.65069
Summary: We consider algorithms for the matrix sector function, which is a generalization of the matrix sign function. We develop algorithms for computing the matrix sector function based on the (real) Schur decompositions, with and without reordering and the recurrence proposed by B. N. Parlett [Linear Algebra Appl. 14, 117–121 (1976; Zbl 0353.65027)]. We prove some results on the convergence regions for the specialized versions of Newton’s and Halley’s methods applied to the matrix sector function, using recent results of B. Iannazzo for the principal matrix \(p\)th root [SIAM J. Matrix Anal. Appl. 28, No. 2, 503–523 (2006; Zbl 1113.65054); ibid. 30, No. 4, 1445–1462 (2008; Zbl 1176.65054)]. Numerical experiments comparing the properties of algorithms developed in this paper illustrate the differences in the behaviour of the algorithms. We consider the conditioning of the matrix sector function and the stability of Newton’s and Halley’s methods. We also prove a characterization of the Fréchet derivative of the matrix sector function, which is a generalization of the result of C. Kenney and A. J. Laub [SIAM J. Sci. Stat. Comput. 12, No. 3, 488–504 (1991; Zbl 0725.65047)] for the Fréchet derivative of the matrix sign function, and we provide a way of computing it by Newton’s iteration.

MSC:
65F30 Other matrix algorithms (MSC2010)
15A24 Matrix equations and identities
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