Various methods of constructing an orthonomal set out of a given set of linearly independent vectors are discussed. Particular attention is paid to the Gram-Schmidt and the Schweinler-Wigner orthogonalization procedures. A new orthogonalization procedure which, like the SchweinlerWigner procedure, is democratic and is endowed with an extremal property is suggested. PACS No: 02.10.Sp, 31.15.+q e...

This paper describes modeling discrete event systems in Metropolis using one of the key concepts employed by Metropolis, orthogonalization of design aspects, which in this particular case, is the orthogonalization between capability and cost. To support the orthogonalization, quantity annotation mechanism is introduced. This paper formally analyzes simulation strategies for quantity annotation ...

two symmetric matrices, where one matrix has a restricted rank. A comparison of the two cases reveals a remarkable similarity between Eckhart-Young theorem and Ky Fan’s maximum principle. The extremum principles of Ky Fan consider sums and products of eigenvalues of Hermitian matrices. The third part of the talk derives extended rules for rectangular matrices. In this case singular values take ...

All-electron calculations play an important role in density functional theory, which improving computational efficiency is one of the most needed and challenging tasks. In model formulations, both nonlinear eigenvalue problem total energy minimization pursue orthogonal solutions. Most existing algorithms for solving these two models invoke orthogonalization process either explicitly or implicit...

Model reduction techniques are needed to directly solve the chemical master equation (CME) due to its enormous size. We recently described an algorithm that solved the CME by combining the finite state projection, stochastic simulation algorithm and Krylov subspace approximations. In this work, we add further improvements that consist of an incomplete orthogonalization process with Krylov subsp...

This paper is devoted to the study of some preconditioners for the conjugate gradient algorithm used to solve large sparse linear and symmetric positive definite systems. The construction of a preconditioner based on the Gram–Schmidt orthogonalization process and the least squares method is presented. Some results on the condition number of the preconditioned system are provided. Finally, numer...