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2007 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering  
Further analysis shows that the spurious eigenvalues -those that are farthest from the mean value -correspond to eigenfunctions which produce the largest residual in the eigenvalue problem used to compute  ...  the eigenvalues and eigenfunctions.  ...  This mechanism is consistent with the highly erratic behavior of the spurious eigenvalues as a function of basis function truncation number used to discretize the invariant circle solution; however, the  ... 
doi:10.1142/s0218127407017872 fatcat:chicgacmgrggtk6lhsjoif4w5q


1987 Le Journal de Physique Colloques  
The energy eigenvalues are in good agreement with those obtained from the numerical solution of the Dirac equation except for a spurious root for K > 0.  ...  Despite this spurious root, the present method is found to give good approximate atomic energy eigenvalues and wave functions.  ...  The authors also wish to acknowledge support from the US-Japan NSF/JSPS cooperative research program.  ... 
doi:10.1051/jphyscol:1987985 fatcat:kvwbkzqpqjaffnstzhifbpqx2y

Page 487 of None Vol. 108, Issue 2 [page]

1957 None  
In calculating matrix elements, one can completely rely on the operator U to remove spurious gtates without taking the trouble to recognize them explicitly. V.  ...  However, using our prescrip tion one would take the functions V, =U, to be the eigenfunctions of H with eigenvalues Z,; that is, we have AV ,= EV», (32) which implies directly the equation, (H—P*/2M)®,  ... 

Harmonic projection methods for large non-symmetric eigenvalue problems

Ronald B. Morgan, Min Zeng
1998 Numerical Linear Algebra with Applications  
The subject of estimating the boundary of the entire spectrum is briefly discussed, and the importance of preconditioning for interior eigenvalue problems is mentioned.  ...  It is related to the Rayleigh-Ritz procedure, but is designed for finding interior eigenvalues. Harmonic Ritz values and other approximate eigenvalues are generated.  ...  The research of the first author was supported by the National Science Foundation under contract CCR-9102221.  ... 
doi:10.1002/(sici)1099-1506(199801/02)5:1<33::aid-nla125>;2-1 fatcat:at5v3wxoqnbc5mlz3imrehtqfy

Calculating the Singular Values and Pseudo-Inverse of a Matrix

G. Golub, W. Kahan
1965 Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis  
Some applications are mentioned, in particular the use of the pseudo-inverse A VZxU to solve least squares problems in a way which dampens spurious oscillation and cancellation.  ...  The scheme described here is complicated but does not suffer from the computational difficulties which occasionally afflict some previously known methods.  ...  Our scheme is based upon an idea exploited by Lanczos [20] ; the matrix has for its eigenvalues the singular values of A, each appearing with both a positive and a negative sign.  ... 
doi:10.1137/0702016 fatcat:cgiayyu6hjgqtna6o4yx6rc7a4

Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems $Ax=\lambda Bx$ with singular $B$

Joost Rommes
2007 Mathematics of Computation  
In this paper, a new approach based on implicitly restarted Arnoldi will be presented that avoids most of the problems due to the singularity of B.  ...  If exact linear solves with A − σB are available, implicitly restarted Arnoldi with purification is a common approach for problems where B is positive semidefinite.  ...  I am grateful to the anonymous referee, whose suggestions and remarks helped me to improve the paper.  ... 
doi:10.1090/s0025-5718-07-02040-6 fatcat:6ba5air24zdo5pxrd34ouphfxy

Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems

J.J. Dongarra, B. Straughan, D.W. Walker
1996 Applied Numerical Mathematics  
The method employing a Chebyshev representation of the fourth derivative operator, D 4 ; is compared with those involving the second and rst derivative operators, D 2 ; D , respectively.  ...  The latter two representations require use of the QZ algorithm in the resolution of the singular generalised matrix eigenvalue problem which arises.  ...  This research w as supported in part by an appointment to the post-Chebyshev tau -QZ algorithm methods Page 44 graduate research program at the Oak Ridge National Laboratory ORNL administered by the Oak  ... 
doi:10.1016/s0168-9274(96)00049-9 fatcat:gfa22qyimfanpleilz7ypv72ya

The computation of resonances in open systems using a perfectly matched layer

Seungil Kim, Joseph E. Pasciak
2009 Mathematics of Computation  
We shall prove that the first of these steps leads to eigenvalue convergence (to the desired resonance values) which is free from spurious computational eigenvalues provided that the size of computational  ...  Our goal will be to approximate the shifted operator first by replacing the infinite domain by a finite (computational) domain with a convenient boundary condition and second by applying finite elements  ...  This shows that there are no spurious eigenvalues associated with the domain truncation.  ... 
doi:10.1090/s0025-5718-09-02227-3 fatcat:3dlm7ywihfgmbnmsns5s4brthq

Elimination of spurious solutions from eight-bandk⋅ptheory

Bradley A. Foreman
1997 Physical Review B (Condensed Matter)  
A method is developed for eliminating spurious solutions from eight-band k•p theory.  ...  It reduces the bulk dispersion to a cubic equation ͑quadratic along ͗001͘ and ͗111͘), yet gives results virtually indistinguishable from ordinary k•p theory.  ...  13 than most people think, 8, 12 but it still runs into trouble with oscillatory modes ͑as shown below͒.  ... 
doi:10.1103/physrevb.56.r12748 fatcat:6gm65zfynrdwjpjblouy5fdto4

Eigenstates for billiards of arbitrary shapes [article]

Aurel Bulgac, Piotr Magierski
1999 arXiv   pre-print
The method is free from the limitations associated with the shape of the billiard and could be applied even for nonconvex geometries where other algorithms face difficulties.  ...  Moreover it does not suffer from the existence of eigenvalue degeneracies which is another serious shortcoming of many methods.  ...  One of the authors (PM) thanks the Nuclear Theory Group in the Department of Physics at the University of Washington for hospitality and acknowledge the financial support from the Swedish Institute and  ... 
arXiv:physics/9902057v1 fatcat:ni47ouhdzzcathoezracmyszea

Page 7438 of Mathematical Reviews Vol. , Issue 94m [page]

1994 Mathematical Reviews  
The method does not produce the spurious eigenvalues which generally occur when such problems are solved by the spectral tau method.  ...  A popular method is Baumgarte stabilization, but the choice of parameters to make it robust is unclear in practice. Here we explain why the Baumgarte method may run into trouble.  ... 

Finding the Number of Normal Groups in Model-Based Clustering via Constrained Likelihoods

Andrea Cerioli, Luis Angel García-Escudero, Agustín Mayo-Iscar, Marco Riani
2017 Journal of Computational And Graphical Statistics  
Controlling the maximal ratio between the eigenvalues of the scatter matrices to be smaller than a fixed constant c ≥ 1 is a sensible idea for setting such constraints.  ...  Constraints also prevent traditional EM and CEM algorithms from being trapped in (spurious) local maxima.  ...  The authors thank the editor, the associate editor, and three anonymous referees for constructive comments.  ... 
doi:10.1080/10618600.2017.1390469 fatcat:vxvku2v7cfambemmtwy2olyzkm

The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors

Zhongxiao Jia
2004 Mathematics of Computation  
To this end, we propose to compute the Rayleigh quotient ρ of A with respect tox and take it as a new approximate eigenvalue. ρ is shown to converge to λ oncex tends to x, no matter how τ is close to λ  ...  However, we show that the harmonic projection method can fail to find the desired eigenvalue λ-in other words, the method can miss λ if it is very close to τ .  ...  Acknowledgments I thank the anonymous referee very much for his/her valuable comments and suggestions, which helped me think over this topic in greater depth and improve on the presentation considerably  ... 
doi:10.1090/s0025-5718-04-01684-9 fatcat:3czccl3ltbe3nbplcgvwurgnei

Page 282 of American Society of Civil Engineers. Collected Journals Vol. 124, Issue 3 [page]

1998 American Society of Civil Engineers. Collected Journals  
Thus the situation is in a certain sense similar to the problems with spurious stress transfer. And again, switching to the scalar damage (SD) formulation can avoid the troubles.  ...  has a new negative eigenvalue.  ... 

Page 227 of Mathematical Reviews Vol. 12, Issue 3 [page]

1951 Mathematical Reviews  
The present author gives a new necessary condition on the sign of S’(Z) which eliminates the spurious eigenvalues in Ma’s example.  ...  ) of the S-matrix as func- tions of E, gives rise to redundant eigenvalues in addition to the actual ones.  ... 
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