Abstract
Nonlinear matrix equations arise in different scientific topics, such as applied statistics and control theory, among others. Standard approaches to solve them include and combine some variations of Newton’s method, matrix factorizations, and reduction to generalized eigenvalue problems. In this paper we explore the use of secant methods in the space of matrices, that represent a new approach with interesting features. For the special problem of computing the inverse or the pseudoinverse of a given matrix, we propose a specialized secant method for which we establish stability and q-superlinear convergence, and for which we also present some numerical results. In addition, for solving quadratic matrix equations, we discuss several issues, and present preliminary and encouraging numerical experiments.
Dedicated with friendship to Biswa Datta for his scientific contributions.
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Acknowledgements
Marlliny Monsalve was supported by the Scientific Computing Center at UCV, and CDCH-UCV project 03.00.6640.2008; and Marcos Raydan was partially supported by USB, the Scientific Computing Center at UCV, and CDCH-UCV project 03.00.6640.2008.
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Monsalve, M., Raydan, M. (2011). A Secant Method for Nonlinear Matrix Problems. In: Van Dooren, P., Bhattacharyya, S., Chan, R., Olshevsky, V., Routray, A. (eds) Numerical Linear Algebra in Signals, Systems and Control. Lecture Notes in Electrical Engineering, vol 80. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0602-6_18
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DOI: https://doi.org/10.1007/978-94-007-0602-6_18
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