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soderlind g logarithmic norms 2024

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Textbook in PDF format This book offers the first comprehensive account of how the logarithmic norm is used for matrices, nonlinear maps and linear differential operators, with a focus on initial and boundary value problems. Complementing the usual operator norm, the logarithmic norm is a versatile tool which provides unique additional information on the magnitude of an operator. It is instrumental in the stability theory of dynamical systems and in the theory of elliptic operator equations. The text adopts a unified approach to address a wide range of themes in applied mathematics. It explores the role of the logarithmic norm in scientific computing, compares the operator bounds with those of spectral theory, and illustrates the theory with classical models from science and engineering. Many previously unpublished results are presented alongside established material, supporting researchers in applied mathematics and computational engineering who seek a systematic approach to stability and perturbation bounds in initial value problems, boundary value problems and partial differential equations. Primarily intended as a reference text, the book can also serve as a graduate text for PhD students. Preface Acknowledgements Introduction to logarithmic norms Why logarithmic norms? The linear test equation Vector and operator norms The logarithmic norm Spectral radius and abscissa Differential inequalities Towards a general theory Matrix Theory Orientation Cartesian decomposition The numerical radius Polar decomposition Positive definite matrix products M¨obius transformations Spectral theory Determinant and trace Nonlinear maps Orientation Lipschitz algebra Coercivity, monotonicity and contractivity Circle conditions and M¨obius transformations Stability and perturbation bounds Nonlinear dynamics with applications Differential Operators Orientation Logarithmic norms of differential operators Applications Ellipticity and solvability Space discretization Spatial frequencies and iterative methods Time discretization Special problems Further reading Bibliography Index

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soderlind g logarithmic norms 2024

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