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Perturbation Analysis of a Nonlinear Matrix Equation Arising in Tree-Like Stochastic Processes

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Advanced Computing in Industrial Mathematics (BGSIAM 2017)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 793))

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Abstract

The solution, obtained in the environment of finite precision machine arithmetic must be allays accompanied by an analysis of the conditioning of the problem solved. The perturbation analysis derives measures for the sensitivity of the solution to perturbations in the matrix coefficients. Motivated by these, in order to ascertain the accuracy of an iteratively calculated solution to a nonlinear matrix equation arising in Tree-like stochastic processes, in this paper norm-wise, mixed and component-wise condition numbers, as well as local perturbation bounds are formulated and norm-wise non-local residual bounds are derived using the methods of nonlinear perturbation analysis (Fréchet derivatives, Lyapunov majorants, fixed-point principles). The residual bounds are formulated in terms of the computed approximate solution to the equation and can be used as a stop criteria of the iterations, when solving the considered nonlinear matrix equation by a numerically stable iterative algorithm.

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Authors and Affiliations

  1. Institute of Information and Communication Technologies - BAS, Akad. G. Bonchev Str. Bl. 2, Sofia, Bulgaria

    Vera Angelova

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  1. Vera Angelova
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Correspondence to Vera Angelova .

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Editors and Affiliations

  1. Institute of Information and Communication Technology, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Krassimir Georgiev

  2. Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Sofia, Bulgaria

    Michail Todorov

  3. Institute of Mathematics and Informatics and Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Georgiev

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Angelova, V. (2019). Perturbation Analysis of a Nonlinear Matrix Equation Arising in Tree-Like Stochastic Processes. In: Georgiev, K., Todorov, M., Georgiev, I. (eds) Advanced Computing in Industrial Mathematics. BGSIAM 2017. Studies in Computational Intelligence, vol 793. Springer, Cham. https://doi.org/10.1007/978-3-319-97277-0_4

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