Gram Schmidt vector orthogonalization

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 ... 

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. 

Gram-Schmidt orthogonalization is general and is performed as follows. The first step is to form a unit vector Ui from an interferogram vector Ii. Vector 11 is an interferogram that was recorded when no compound is present in the light-pipe. 

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. 

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... 

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

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