70,989 research outputs found
Efficient Method for Computing Lower Bounds on the -radius of Switched Linear Systems
This paper proposes lower bounds on a quantity called -norm joint
spectral radius, or in short, -radius, of a finite set of matrices. Despite
its wide range of applications to, for example, stability analysis of switched
linear systems and the equilibrium analysis of switched linear economical
models, algorithms for computing the -radius are only available in a very
limited number of particular cases. The proposed lower bounds are given as the
spectral radius of an average of the given matrices weighted via Kronecker
products and do not place any requirements on the set of matrices. We show that
the proposed lower bounds theoretically extend and also can practically improve
the existing lower bounds. A Markovian extension of the proposed lower bounds
is also presented
Scroll waves in isotropic excitable media : linear instabilities, bifurcations and restabilized states
Scroll waves are three-dimensional analogs of spiral waves. The linear
stability spectrum of untwisted and twisted scroll waves is computed for a
two-variable reaction-diffusion model of an excitable medium. Different bands
of modes are seen to be unstable in different regions of parameter space. The
corresponding bifurcations and bifurcated states are characterized by
performing direct numerical simulations. In addition, computations of the
adjoint linear stability operator eigenmodes are also performed and serve to
obtain a number of matrix elements characterizing the long-wavelength
deformations of scroll waves.Comment: 30 pages 16 figures, submitted to Phys. Rev.
Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals
High-order derivatives of analytic functions are expressible as Cauchy
integrals over circular contours, which can very effectively be approximated,
e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius
of convergence is equal, numerical stability strongly depends on r. We give a
comprehensive study of this effect; in particular we show that there is a
unique radius that minimizes the loss of accuracy caused by round-off errors.
For large classes of functions, though not for all, this radius actually gives
about full accuracy; a remarkable fact that we explain by the theory of Hardy
spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and
by the saddle-point method of asymptotic analysis. Many examples and
non-trivial applications are discussed in detail.Comment: Version 4 has some references and a discussion of other quadrature
rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Mat
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