Schur vector

Ql is the submatrix of corresponding to the n - n c fastest variables, and so the slow manifold is defined by assuming that motion in the direction of the Schur vectors associated with the fast variables is zero. This means that the system is in local equilibrium with respect to its fastest time-scales. 

In Section 4.8.4 we introduced the Schur vector. We will now formalize this concept and show how the Schur matrix can be used as a lumping matrix for systems with time-scale separation. 

Check this out. Since the space of functions spanned by the single function ecx is a finite-dimensional vector space of dimension one, translation becomes an example of an irreducible representation on this space. The hypotheses of Schur s lemma are satisfied and we can conclude without computation that... 

From these equations Keller [7] suggests an exact algorithm for the solution of (4.4). It consists of successive steps for the solution of the Schur complement equation and finally a backward solution to determine Ax. Chan [5] further develops this algorithm using specific approximations of the vector p and the matrix C. To get an impression of his method we want to present his suggestion on the computation of p. Since... 

Pointwise multiplication of two vectors, also called the Schur or Hadamard product, and denoted in this work by Q (U + 2299), is defined as the multiplication of two vectors by taking each entry of the two vectors and multiplying them together, that is, Z). = Xkyk> where k are the index locations. 

The matrix W/consists of vectors w. In early applications of the method, it was found that the angles between vectors w can be small causing numerical problems or degenerate systems. Therefore, the Schur decomposition (Golub and Van Loan... 

For explaining the Schur complement, consider the stmcture with two subdomains in Fig. la. Accordingly, the stiffness matrix, displacement vector, and the force vector are partitioned, and the static equilibrium condition is written as... 

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