The aim of POD is to find a set of optimal

orthogonal basis [PHI] = {[phi].sub.i = 1,2,..., m} to provide a best approximation to the behavior of the full-order system dynamics.

Instead of building a block

orthogonal basis of [K.sub.m](A,[R.sub.0]), we look for a block

orthogonal basis [V.sub.m] = [[V.sub.1], ..., [V.sub.m]] of A[K.sub.m](A, [R.sub.0]).

It must be highlighted that even in the more elaborated formulation in (2), it is not possible to guarantee the existence of a complete

orthogonal basis for the Maxwell's equations [6, 7].

This project is to insert the mean value between the eigenvectors and get a series of nonstandard

orthogonal basis vectors.

Watanabe obtains the unique closest point in any closed gyrovector subspace, by using the ordinary orthogonal decomposition and shows that each element has the orthogonal gyroexpansion with respect to any

orthogonal basis in a Mobius gyrovector space.

Gauss matrix is not only related to most fixed

orthogonal basis but also satisfies the restricted isometry property, so the Gauss matrix can be used as the projection observation matrix [18, 19].

FOM/GMRES use an

orthogonal basis, BiCG/QMR use a biorthogonal basis, and Hessenberg/CMRH use a basis originating from the LU factorization with pivoting of the Krylov matrix.

Let {[e.sup.j.sub.i]} be an

orthogonal basis of [m.sub.j] with respect to B, where j = 1,2,3 and 1 [less than or equal to] i [less than or equal to] [d.sub.j].

Shape functions for the element [[OMEGA].sup.e] are built based on an

orthogonal basis (N is the order of a spectral element):

[3] put forward a new method based on weighted Laguerre polynomials

orthogonal basis, which firstly expands the time-domain Maxwell equations using weighted Laguerre polynomials

orthogonal basis and then eliminates every orthogonal term by Galerkin method so as to obtain the coefficients matrix equation.