Gram-Schmidt orthogonalization procedure

The inability to estimate the Volterra kernels in the general case of an infinite series prompted Wiener to suggest the orthogonalization of the Volterra series when a GWN test input is used. The functional terms of the Wiener series are constructed on the basis of a Gram-Schmidt orthogonalization procedure requiring that the covariance between any two Wiener functionals be zero. The resulting Wiener series expansion takes the form ... 

For this saitple problem a proper orthogonal basis can be determined by obtaining a set of polynomials that are orthogonal on the real line with respect to the wei t function w(x) = ejqp( px ). This task thou feasible throu Gram-Schmidt orthogonalization procedures, is extremely difficult to coirplete numerically. Further, unless the appropriate recurrence relation ips are establi ed the polynomials are of little practical use. 

We diagonalize the matrix U by choosing such functions 0 ) for which the matrix (0j v 0 ) (and hence the corresponding mauix U) is equal to identity. We can find such functions for example using the Gram-Schmidt orthogonalization procedure on 0i) with a norm (/ v g) (for functions / andg), more on that later. Then... 

This can be done using several methods, we chose the Gram-Schmidt orthogonal-ization procedure, which works according to the following scheme ... 

The basis vectors in Table 1 are complete but not unique. Besides trivial variations in the Gram-Schmidt orthogonalization, there is a substantive difference that depends on the choice of the weighting factors q these factors determine both the result of the orthogonalization procedure, as well as the back transformation from... 

This procedure is continued until there is a success and a failure connected with each of the independent variables. Then a new set of orthogonal directions is obtained. The first direction is obtained by connecting the initial point with the best point obtained. When there are many independent variables the Gram-Schmidt orthonormalization method should be used., The whole procedure is then repeated, with the best point obtained so far becoming the new origin. 

We never explicitly utilize the LMOs of the surroundings. They could be determined in a manner similar to the local space LMOs except that they must be orthogonal to the latter. In principle, the usual Gram-Schmidt procedure could be employed for that purpose. Finally, it is convenient to symmetrically orthogonalize the local space LMOs which involves only local space operations since, from Eq. (19), S M0 = BL Rl B ... 

In particular, the specification or modification of the d mole fraction variables, Xi, is done in the (d - l)-dimensional hyperplane orthogonal to the d-dimensional vector (1,1,..., l). lhis procedure ensures that the constraint Xi = 1 is maintained. This subspace is identified by a Gram-Schmidt procedure, which identifies a new set of basis vectors, u,, that span this hyperplane. Figure 3 illustrates the geometry for the case of three composi-... 

The remaining u,, 1 < i < d, are chosen to be orthogonal to u,/, so that they lie in the allowed hyperplane. Indeed, the u, form an orthonormal basis for the composition space. This orthonormal basis is identified by the Gram-Schmidt procedure. First, the original composition basis vectors are defined... 

First one can build up other effective Hamiltonians based on hierarchized orthogonalization procedures. The Gram-Schmidt procedure is recommended if one starts from the best projected wavefunctions of the bottom of the spectrum. Thus one can obtain a quite reliable effective Hamiltonian with well behaved wavefunctions and good transferability properties (see Section III.D.2). The main drawback of this approach is that the Gram-Schmidt method, which involves triangular matrices, does not lead to simple analytical expressions for perturbation expansions. A partial solution to these limitations is brought about by the new concept of intermediate Hamiltonian,... 

The first three states of the method of moments [8] characterize the space of the three relevant quasi-bound states (the model space). Since these states are non-orthogonal, the Gram-Schmidt procedure applied to 1), H l) and provide the orthonormalized states i), i — 1,2,3). In this basis the matrix representation of the exact energy-dependent Hamiltonian (13) may be written as... 

The Gram-Schmidt procedure provides a simpler ordu onalization scheme that leads to a hermitian effective Hamiltonian. Since and we already orthogonal to the other projections, we only have to worry about and This means that the coefficients of 4 are defined by that is, if I l = a ( afe + 6a ) -I- p ( aa -P bb )... 

The process of orthgonalization is somewhat analogous to the orthgonalization of vectors, known as the Gram-Schmidt procedure. Construction of orthogonal descriptors 2, 2, 2,... from given descriptors D, D, D,... is accomplished by calculating the residuals between the descriptors. " By definition, the residual of a correlation between two descriptors is the part of one descriptor that does not correlate with the other. 

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