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Numerical Methods for Eigenvalue Problems
Published in Federico Milano, Ioannis Dassios, Muyang Liu, Georgios Tzounas, Eigenvalue Problems in Power Systems, 2020
Federico Milano, Ioannis Dassios, Muyang Liu, Georgios Tzounas
A coarse taxonomy of algorithms for the solution of non-symmetric eigenvalue problems is as follows: Vector iteration methods, which, in turn, are separated to single and simultaneous vector iteration methods. Single vector iteration methods include the power method and its variants, such as the Rayleigh quotient iteration. Simultaneous vector iteration methods include the subspace iteration method and its variants, such as the inverse subspace method.Schur decomposition methods, which mainly include the QR algorithm, the QZ algorithm and their variants, such as the QR algorithm with shifts.Krylov subspace methods, which basically include the Arnoldi iteration and its variants, such as the implicitly restarted Arnoldi iteration and the Krylov-Schur method. In this category belong also preconditioned extensions of the Lanczos algorithm, such as the non-symmetric versions of the Generalized Davidson and Jacobi-Davidson methods.Contour integration methods, which basically include a moment-based Hankel method and a Rayleigh-Ritz-based projection method proposed by Sakurai and Sugiura; and the FEAST algorithm.
Evaluating the performance of an Inexact Newton method with a preconditioner for dynamic building system simulation
Published in Journal of Building Performance Simulation, 2022
Zhelun Chen, Jin Wen, Anthony J. Kearsley, Amanda Pertzborn
Specifically, the least squares problem in Equation (7) is solved by constructing an orthonormal basis for using an Arnoldi iteration1 (Arnoldi 1951; Sorensen 1992). Starting with a normalized vector as a basis for , Arnoldi iteration constructs an orthonormal basis for from an orthonormal basis for the previous Krylov subspace . That is, where .