Gram-Schmidt

The weakly occupied pre-NAOs on each centre are made orthogonal to the strongly occupied NAOs on the same centre by a standard Gram-Schmidt orthogonalization. 

Gram-Schmidt 986 integrated absor) nee 986 maximum absorban 986 data management 986 interfaces 987... 

KBr) databases. Quantitative analysis by GC-FUR is complicated by many uncertainties associated with both the chromatography and spectroscopy [196]. Bulk property detectors (e.g. TCD, FID, etc.) can be used for quantitative analysis when mixture components are known, but provide little structural information for unknown mixture components. Both integrated absorbance and Gram-Schmidt vector methods have been used for the quantitative analysis of mixture components in GC-FTIR. 

Figure 7.27 Constructed Gram-Schmidt chromatogram of a temperature-programmed packed-capillary LC separation of 4.8 ig Irgafos P-EPQ dissolved in DMF temperature programme 50°C for 8min, 4°Cmin-1 up to 140°C. Legend 1, mono-P-EPQ 2, Irgafos 168 3, 4,3 -P-EPQ 4, oxidised 4,4 -P-EPQ 5, 4,4 -P-EPQ. After Bruheim et al. [511]. From I. Bruheim et al., Journal of High Resolution Chromatography, 23, 525-530 (2000). Wiley-VCH, 2000. Reproduced by permission of Wiley-VCH.

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. 

Figure 18 (a) GC-FTIR Gram-Schmidt chromatograms from evolved gases from PHB at... 

Gram, Christian, 77 7 Gram-Schmidt chromatogram, 74 233 Grand canonical ensemble, 7 33 Grand Composite Curve (GCC), 73 221, 20 737-738... 

Given a set of independent vectors x1 x2,. .., xn, it is requested to form a new set yk,y2, y , of orthogonal vectors spanning the same space. The Gram-Schmidt orthogonalization scheme sets... 

Because of a different normalization , the coefficients of the parentheses are not identical for the Gram-Schmidt orthogonalization and for the recursion formula. 

The vectors generated by the Lanczos recursion differ from the Krylov vectors in that the former are mutually orthogonal and properly normalized, at least in exact arithmetic. In fact, the Lanczos vectors can be considered as the Gram-Schmidt orthogonalized Krylov vectors.27 Because the orthogonalization is performed implicitly along the recursion, the numerical costs are minimal. 

The Gram-Schmidt orthonormalization method is applied using the following steps. 

From Eq.(73) the Gram-Schmidt orthonormalization is applied to obtain the normalized values [Sxin, Sx2n, Sx2n, Sx2n]- The following integration step is carried out with the values ... 

The addition of more terms does not influence the values of the already calculated terms. In this aspect, orthogonal polynomials are superior to other polynomials calculation of the coefficients is simple and fast. Moreover, according to the Gram-Schmidt theory every function can be expressed as a series of orthogonal polynomials, using the weighting function w(t). 

Exercise 6.14 Use the Gram-Schmidt technique of orthogonalization to find a recursive formula for an orthogonal basis ofC[—l, 1] with the property that the kth basis vector is a polynomial of degree n (for n = 0, 1, 2,. . J. Show (from general principles) that the nth basis element is precisely the character of the representation ofSU(T) on P". Use the recursive formula to calculate and /4. 

GC-FTIR requires a large number of spectra to be stored over a very short time interval. The gas phase exiting the column is sampled at very short time intervals throughout the elution. The resolution of the spectra is lowered in order to occupy less memory. The data are then used to reconstruct a pseudo-chromato-gram called Gram-Schmidt after the mathematicians whose method is used to treat the data. 

Furthermore, the basis set is easily transformed to yield a new set 6, with the property that (6, 6j) = Si Such a set is called orthonormal. The standard procedure is the Gram-Schmidt ON-algorithm ... 

The simple Gram-Schmidt ON-process could have been applied to any initial basis set to find a set of solutions to (3.1.15). The equations above are all equivalent. Any two solutions to these equations are related by some orthogonal matrix V ... 

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