Euclidian norm

For realizing (10), we need an adequate norm for measuring the error. It obviously ma,kes no sense to ise an Euclidian norm of z indiscriminately of... 

The determination of eigenvalues and eigenvectors of the matrix A is based on a routine by Grad and Brebner (1968). The matrix is first scaled by a sequence of similarity transformations and then normalized to have the Euclidian norm equal to one. The matrix is reduced to an upper Hessenberg form by Householder s method. Then the QR double-step iterative process is performed on the Hessenberg matrix to compute the eigenvalues. The eigenvectors are obtained by inverse iteration. 

We know that a PDE is stable as a linear approximation (see Sect. 2). Whence from eqns. (137) and (138) we establish that, at sufficiently low um and vout and t - oo, a solution of the kinetic equations for homogeneous systems tends to a unique steady-state point localized inside the reaction polyhedron with balance relationships (138) in a small vicinity of a positive PDE. If b(c(0)) = 6(c,n) vinjvout, then at low v,n and eout the function c(t) is close to the time dependence of concentrations for a corresponding closed system. To be more precise, if vm -> 0, uout -> 0, vmjvOM, c(0), cin are constant and c(O) is not a boundary PDE, then we obtain max c(t) — cc](t) -> 0, where ccl(t) is the solution of the kinetic equations for closed systems, ccl(0) = c(0),and is the Euclidian norm in the concentration space. 

Starting from the hyper vector described in Eq. (16), one can compute the Euclidian norms first ... 

The idea of a vector space is usefully extended to an infinite number of dimensions for continuous functions. Given a function /(e.g.,/ = sinx) and a definition domain (e.g., 0 to In), the coordinates of / = sin x will be the infinite number of values of the function over the definition domain. This definition is consistent with that of Euclidian spaces if a metric is defined. In about the same way as the squared norm of the n-vector x(xux2,. .., x ) is... 

A code example of an imperative language (C-h-) is shown following. This is a function that normalizes a vector according to the Euclidian L -Norm — that is, it divides each component of a vector by the square root of the sum of products of all vector components. 

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