Orthogonal Gram-Schmidt method

A sensitive technique used for real time reconstruction of chromatograms from the interferogram is the Gram Schmidt vector orthogonalization method. The Gram Schmidt method relies on the fact that the interferogram contains information on absorbing samples at all optical retardations less than the reciprocal of the width of each band in the spectrum. 

The Gram-Schmidt method to orthogonalize a set of vectors is considered obsolete because of its poor stability. Moreover, it requires 0(wy) evaluations. 

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,... 

Figure 1.4 Gram-Schmidt method makes vectors orthogonal through projection operations.

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. 

It is evident from these discussions that population balance equations are important in the description of dispersed-phase systems. However, they are still of limited use because of difficulties in obtaining solutions. In addition to the numerical approaches, solution of the scalar problem has been via the generation of moment equations directly from the population balance equation (H2, H17, R6, S23, S24). This approach has limitations. Ramkrishna and co-workers (H2, R2, R6) presented solutions of the population balance equation using the method of weighted residuals. Trial functions used were problem-specific polynomials generated by the Gram-Schmidt orthogonalization process. Their approach shows promise for future applications. 

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... 

B. Results from TG-FTIR and STA-FTIR. It is possible to display a number of features of the decomposition pattern in real time and for each of these to be on the same time base for strict comparison purposes. Thus in addition to the weight profile (and its derivative) and the DSC signal, various gas evolution profiles can be plotted. The first of these is the Evolved Gas Profile (EGP) using the Gram-Schmidt orthogonalization method (8). This plots the mean of the integrated IR absorbance in the spectra as a function of time. It should be noted that the shape of the EGP... 

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

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