Gram-Schmidt orthonormalization

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. 

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 ... 

When this method is applied to the polynomial approximation of mechanisms, the function G is the response of the kinetic model calculated using the original detailed reaction mechanism, and (/> is a series of orthonormal polynomials constructed by a Gram-Schmidt orthonormalization process using the data set. The function F, defining the final algebraic model, is constructed in such a way that only the significant members of the summation are considered. 

R in the DMRG representation. This allows terminating the Gram-Schmidt orthonormalization process when the number of orthonormal functions obttiined is Ir- In prax tice, we first obtain the matrix representation of in the direct product DMRG basis in a way which is analogous to the setting-up of the full Hamiltonian matrix from the blocks. The linear dependencies discussed above manifest as linearly dependent rows of the matrix of Pr. 

The matrix U was eliminated from equations (see the text before Eq. 12.6) by using Gram-Schmidt orthonormalization. This orthonormalization can be performed in R supposing that the operator V is positive definite and that the norm i V j) exists. If we keep the norm of radial projectors fixed and just V-ortogonalize the basis (and so diagonalize U), we obtain the solution also for non-positive definite pseudopotential. Using the coefficients... 

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 ... 

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... 

The Lanczos method is based on generating the orthonormal basis in Krylov space Ki =span c, Ac, A c by applying the Gram-Schmidt orthogonaliza-tion process, described in Appendix A. In matrix notations this approach is associated with the reduction of the symmetric matrix A to a tridiagonal matrix and also with the special properties of T/,. This reduction (called also QT decomposition) is described by the formula... 

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... 

As we have not found a proof, this is now demonstrated. Property (2) implies that the eigenvectors o) of X form an orthonormal set if the nondegenerate ones are unity normed and if the Gram-Schmidt procedure is applied to the degenerate ones. The formal representation of X in this set is then Y = a>A a. This relation and... 

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

Appendix. First, the Kohn-Sham equations are converged for the metal slab without the solvent. Thereafter the DR procedure is applied simultaneously to the KS-DFT equations for the electron wave functions, and to the 3D-RISM/KH equations for the solvent site correlation functions. On each DR step for tpj k), the new wave functions of M states are orthonormalized by the Gram-Schmidt process. Then the exchange-correlation potential Vxc( ) is recomputed, the metal Hartree potential Vjj(r) is obtained from the Poisson equation by using the 3D-FFT, and the metal-solvent site potential is calculated. Next, several... 

Gram-Schmidt procedure An algorithm for obtaining a set of orthonormal functions from a set of signals. 

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... 

The application of Fourier expansion (7.108) requires orthonormal functions, which can be generated from independent functions using the Gram Schmidt orthonormahsation process. We denote fj, j, .I to be a set of linearly independent functions. Using these functions, orthonormal functions can be generated as follows ... 

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