Orthogonalization, Gram-Schmidt

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 

This procedure is important for producing orthogonal vectors and functions in numerical analysis. It is not unique since it can be started from any xk. 

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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. 

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. 

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 ... 

For the Hiickel calculations, the remainder of the Setup worksheet is devoted to the imposition of the orthonormality condition on the 5Hg[a] functions of Table ALL This condition in Hiickel theory requires only matrix multiplications between the matrix of coefficients and its transpose, with stepwise imposition of, for example, Gram-Schmidt orthogonalization until... 

Gram - Schmidt orthogonalization process Thus, equation (B.6) takes the form... 

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. 

Now we will examine the Gram-Schmidt orthogonalization (represented by Z—f2) in the framework of eqns. (6). The bases ft (= ), where for a predetermined sequence of vk s a sequence of ljk s is generated such that a vk has non-zero projections only on those d K s for which < k, satisfy the modified set of equations (6) with (vk, zK) 2=0 for all k > k together with the identity (7a). It is clear at the outset that none of the fi s (N in number) can coincide with the Symmetric basis 4>, since for any 12 at least cx is necessarily greater than ux 2. But, an 12 corresponding to a certain sequence of vk s can accidently coincide with the bases T or A. The possibility of the coincidence with T will however be precluded if the vks are normalized for then f coincides with as seen above. 

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

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