Euclidean norm

CONTOL Calculates ratio of the difference of the Euclidean norm (Lapidus and Pinder, 1982) between successive iterations to the nonn of the solution, as... 

The formalism for computing Lyajmnov exponents for continuous dynamical systems that was introduced in the last section can also be used, with only minor modifications, for determining exponents for CA as well. The major modification involves replacing the Euclidean norm, V t) - used for measuring the divergence of two nearby trajectories (see equation 4.60) - by the Cantor-set metric, d t) ... 

The euclidean norm of a matrix considered as a vector in m2-space is a matrix norm that is consistent with the euclidean vector norm. This is perhaps the matrix norm that occurs most frequently in the literature. But the euclidean norm of I is n112 > 1 when n > 1, hence it is not a sup. In fact,... 

Let II II denote the Euclidean norm and define = gk+i gk- Table I provides a chronological list of some choices for the CG update parameter. If the objective function is a strongly convex quadratic, then in theory, with an exact line search, all seven choices for the update parameter in Table I are equivalent. For a nonquadratic objective functional J (the ordinary situation in optimal control calculations), each choice for the update parameter leads to a different performance. A detailed discussion of the various CG methods is beyond the scope of this chapter. The reader is referred to Ref. [194] for a survey of CG methods. Here we only mention briefly that despite the strong convergence theory that has been developed for the Fletcher-Reeves, [195],... 

In the above equation, the norm is usually the Euclidean norm. We have a linear convergence rate when 0 is equal to 1. Superlinear convergence rate refers to the case where 0=1 and the limit is equal to zero. When 0=2 the convergence rate is called quadratic. In general, the value of 0 depends on the algorithm while the value of the limit depends upon the function that is being minimized. 

Giddings (1990) presented a derivation applicable to both the planar format such as TLC that is distance-based and the comprehensive multidimensional separations that are time-based. The resolution was shown to be equal to the Euclidean norm of zone resolution components. This can be summarized as... 

In this formulation, the resolution is assumed to be independent of angle between zones. This assumption was checked by Schure (1997) as the equation for resolution as a function of angle between the two zones was previously available (Shi and Davis, 1993). This analysis shows that for a number of cases, the error is approximately less than 10% when using the Euclidean norm of resolutions, as shown in Fig. 2.3. 

FIGURE 2.3 The error inherent in using the Euclidean norm to calculate resolution as a function of the angle made between the two zones. The conditions used for curves A, B, C, and D are described in the original paper (Schure, 1997). Reprinted by permission from J. Micro. Sep. 

To compute the various cosine-like similarity indices it is necessary to evaluate, respectively, the inner product (vA,VB) and Euclidean norms 11 vx f I = V(vx,vx) (see also Eqs. 2.43 and 2.44) ... 

Because vA is normalized with respect to the Euclidean norm... 

Euclidean norm and condition number of a square matrix... 

X set of admissible state vectors y vector of measured output variables y set of admissible output vectors Euclidean norm... 

Change in the logarithm of the Euclidean norm of f.(x.i) as a function of the iteration number ... 

Besides Tikhonov regularization, there are numerous other regularization methods with properties appropriate to distinct problems [42, 53,73], For example, an iterated form of Tikhonov regularization was proposed in 1955 [77], Other situations include using different norms instead of the Euclidean norm in Equation 5.25 to obtain variable-selected models [53, 79, 80] and different basis sets such as wavelets [81],... 

We generally denote scalars by lowercase Greek letters (e.g., P), column vectors by boldface lowercase Roman letters (e.g., x), and matrices by capital italic Roman letters (e.g., H). A superscriptT denotes a vector or matrix transpose. Thus xT is a row vector, xTy is an inner product, and AT is the transpose of the matrix A. Unless stated otherwise, all vectors belong to R , the u-dimen-sional vector space. Components of a vector are typically written as italic letters with subscripts (e.g., xux2,.. . , ). The standard basis vectors in R" are the n vectors ei,e2,. . . , e , where e has the entry 1 in the th component and 0 in all others. Often, the associated vector norm is the standard Euclidean norm, j 2, defined as... 

Results for successive iterations of a typical run are shown in Table I. The composition norm is defined as the Euclidean norm of the composition errors... 

T Activity coefficient matrix for the liquid phase the ij element is the activity coefficient for component j in stage i kj Equal to zero when k j equal to 1.0 when k= j A Activity coefficient matrix for vapor phase the ij element is activity coefficient for component j in stage i v Subscript used to indicate iteration number P Euclidean norm of composition errors, Equation 33 a Euclidean norm of energy balance errors, Equation 34... 

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... 

Set Info(16) = l to request the Euclidean norm (root-mean-square value). InfodTl... 

The list of parameters x is found by minimizing the Euclidean norm II Ax - b II2. Finding x is straightforward assessing its validity less so. Consider the case of an LLS matrix A that is rank deficient. In other words, the dimension of the space spanned by the row and column vectors is less than n. The principal consequence of a rank-deficient LLS A matrix is the reduction in the number of parameters that may be used to describe the data vector b. 

There are two important differences between kriging and other basis function methods that made that kriging usually outperforms those methods. First, other methods usually do not have parameters in their basis functions, or if there is any parameter this is rarely optimized. Second, most of the methods use a Euclidean norm which makes them sensitive to the units of measurements. Kriging, however, capture all those effects in the 0 s parameters through a non-Euclidean norm. 

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