By Pierre de la Harpe
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Extra resources for Algebres d'Operateurs
Otherwise, for any c and some sufficiently small r , m ||C£|| > |A|> > (1 + cr) > T r = exp ( ^ l n ( l + c r ) ) >exp^(cr-icV)^ >e ' . Tc 2 In the theory of automatic control systems there is the Zubov's stability criterion. Its essence is that it computes the quantities l | c j | , iiciii, H O i , w e l l . . Then at \\C \\ —> 0 asp —» oo it is proved that all eigenvalues of the matrix C belong to a unit circle. The substantiation of the Zubov's criterion was given by Zubov (1959). This criterion makes use of the following algebraic result: Let A , , .
For a hyperbolic system of first-order differential equations. The fundamentals of the differential approximation method were presented systematically for the first time in Shokin (1979). This method proved to be a convenient tool for the classification of difference schemes with respect to a number of features, which was demonstrated in a more recent work by Shokin et al. (1985). Using a simple example, we will elucidate the procedure of stability investigation by the differential approximation method.
We also consider the automated derivation of the differential approximations for the difference schemes with fractional steps. Once the differential approximation has been obtained, it can be used for determining the approximation order of a difference scheme. 102) has the order of approximation 0(h) + 0(r). We also present in Chapter 8 a number of theoretical results on the relation between the parabolicity properties of the differential approximation and the stability of the underlying difference scheme.