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 Many previously unpublished results are presented alongside established material, supporting researchers in applied mathematics and computational engineering who seek a systematic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