Infinity norm

Equation (9.173) is also ealled the small gain infinity-norm eontrol problem. 

Problem solution with INFINITY NORM OBJECTIVE... 

This integer specifies the kind of norm to be used in testing error vectors. Set Info(16)=0 to request the infinity-norm (maximum absolute element). 

The solver must stop when an acceptable accurate solution is found. A stop criterion must thus be defined by a sufficientfy small residual value or preferably a norm of the residual [166]. In the non-preconditioned CG-method it is natural to use the 2-norm since the euclidean inner product (r, r) is already calculated as part of the algorithm. This is not the case neither in the preconditioned version of the CG-solver nor the BCG-methods. In these methods extra computations are thus required to calculate the stop criterion. For that reason, the less expensive infinity-norm is frequently used as stop criterion for these solvers. One possible criterion is that the norm of the residual must fall below a specific value jrjm < e. However, this criterion is difficult to use when employing the p-norm since this norm is grid dependent. Besides, the tolerance e has to be fit to the system under consideration since the residuals... 

The most commonly used norms for constructing metrics are the special cases 1 -norm (p = 1), 2-norm (p = 2) and infinity norm. The formulas for the corresponding functional distance measures are listed in the first column of Table 1. In the second column the equivalent discrete distance measures for vectors and (g,...,gjy) are given. 

We define the infinity norm as the limit of v p as/ —> oo, which merely extracts from v the largest magnitude of any component,... 

Optimization terminated relative infinity-norm of gradient less than options.TolFun. x=l 2 F=0... 

A Euclidean norm declares two functions that differ only on a countable infinity of isolated points as being close. This is not too much of a difficulty for the problems we consider, but there is another difficulty. If we want to consider distributions that include one or more discrete components (a semicontinuous distribution), s(x) may well contain some delta functions. This implies, first, that all integrals have to be interpreted as a Stjieltjies ones but even so one has a problem with the right-hand side of Eq. (177), because the delta function is not Stjieltjies square-integrable. One could be a bit cavalier here and say that we agree that 5 (j ) = 5(x), but it is perhaps preferable to keep continuous and discrete components separate. Let, for instance, the mole distribution be i, ri2,.. ., /v, n(x) in a mixture with N discrete components and a distributed spectrum. One can now define the scalar product as the ordinary one over the discrete components, plus... 

The smaller the norm of the inverse operator, the bigger the resolution, Rmo aJid the closer to each other are models that can be resolved. If the inverse operator is not bounded, i.e. its norm goes to infinity, the resolution goes to zero, Rmo — 0, and the maximum possible errors in the determination of m are infinitely large. We have exactly this case for the ill-posed problem. 

Mike Hocker from Poughkeepsie, New York, suggested that the use of atomic weapons on Infinity World would be less of a concern. Mike believes that wars would be common as populations tried to destroy others before they were destroyed. Xenocide and paranoia would be the geopolitical norm. Nations would race to develop interworld ballistic missiles (IWBMs) as opposed to intercontinental ballistic missiles (ICBMs). Vast land areas would be radioactive or devastated by horrendous nuclear, chemical, and biological weaponry. 

Now consider an infinite sequence of x tending to a point y on the boundary of S. The corresponding sequence pj of unit vectors is bounded. According to the Bolzano Weierstrass Theorem (Section 9.17, p. 277), this sequence has a subsequence converging to a limit, say, p whose norm is also unity. Hence, as i tends to infinity, the above inequality becomes... 

If y is zero, d corresponds to the Newton vector. If y tends to infinity, d corresponds to the gradient direction with the null norm. 

Several stop criteria can be defined in terms of different norms of the residual [205]. The general p-norm of a vectoris defined as r p = (X =i When p tends to infinity, the vector norm... 

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