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THE TROUBLE WITH SPURIOUS EIGENVALUES

2007
*
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
*

Further analysis shows that

doi:10.1142/s0218127407017872
fatcat:chicgacmgrggtk6lhsjoif4w5q
*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*...##
###
BASIS-SET EXPANSION OF THE DIRAC EQUATION FOR ATOMS

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. ...

##
###
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

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. ...

##
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Calculating the Singular Values and Pseudo-Inverse of a Matrix

1965
*
Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis
*

Some applications are mentioned, in particular

doi:10.1137/0702016
fatcat:cgiayyu6hjgqtna6o4yx6rc7a4
*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. ...##
###
Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems $Ax=\lambda Bx$ with singular $B$

2007
*
Mathematics of Computation
*

In this paper, a new approach based on implicitly restarted Arnoldi will be presented that avoids most of

doi:10.1090/s0025-5718-07-02040-6
fatcat:6ba5air24zdo5pxrd34ouphfxy
*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. ...##
###
Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems

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 ...

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

2009
*
Mathematics of Computation
*

We shall prove that

doi:10.1090/s0025-5718-09-02227-3
fatcat:3dlm7ywihfgmbnmsns5s4brthq
*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. ...##
###
Elimination of spurious solutions from eight-bandk⋅ptheory

1997
*
Physical Review B (Condensed Matter)
*

A method is developed for eliminating

doi:10.1103/physrevb.56.r12748
fatcat:6gm65zfynrdwjpjblouy5fdto4
*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. ...##
###
Eigenstates for billiards of arbitrary shapes
[article]

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 ...

##
###
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

2017
*
Journal of Computational And Graphical Statistics
*

Controlling

doi:10.1080/10618600.2017.1390469
fatcat:vxvku2v7cfambemmtwy2olyzkm
*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. ...##
###
The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors

2004
*
Mathematics of Computation
*

To this end, we propose to compute

doi:10.1090/s0025-5718-04-01684-9
fatcat:3czccl3ltbe3nbplcgvwurgnei
*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 ...##
###
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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