Gram Schmidt vector

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. 

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. 

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... 

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. 

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. 

To construct the projection operators corresponding to the constraints, the subspace unit vectors representing different constraints must be independent. As shown by Miller et ah, this can be affected by Gram-Schmidt orthogonalization that yields a set of orthogonal unit vectors ... 

Then use Gram-Schmidt orthonormalisation of either result to construct a set of basis vectors. 

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... 

The Gram-Schmidt orthogonalization of the frequency independent vectors (j), A(j) produces the orthonormal basis qi, , qm by the Lanczos process so that... 

Nonorthogonal vectors, qj,..., q are transformed into r orthogonal vectors Pj, Pj,..., Pr- The remaining vectors p are preserved as they are orthogonal to each other, but also orthogonal with respect to the new vectors, since they do not appear in the definition q. This transformation can be carried out by various different techniques, such as the Gram-Schmidt transformation (Golub and Van Loan, 1983). 

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

This situation may be met when aiming to describe dynamic electron correlation starting from a single, multi-determinantal reference vector. Correction schemes based on perturbation theory (PT) have been applying successive Gram-Schmidt orthogonahzation in such circumstances [5-7], occasionally combined with Lowdin s symmetrical [8] or... 

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