Real Asymmetric Matrix Eigenvalue Analysis
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1 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Rel Asymmetri Mtrix Eigenvlue Anlysis Heewoo Lee omputtionl Menis Lbortory Deprtment of Menil Engineering n Applie Menis University of Miign Ann Arbor, MI Te University of Miign omputtionl Menis Lbortory
2 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis ontents of Presenttion Numeril Algoritms QR QZ Exmples Te University of Miign omputtionl Menis Lbortory
3 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Eigenvlue problem Ax λbx x λx, B A > Hessenberg form O(n ) > O(n ) QR lgoritm Te University of Miign omputtionl Menis Lbortory
4 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Hessenberg Mtrix Elementry similrity trnsformtion Ortogonl trnsformtion Te University of Miign omputtionl Menis Lbortory
5 Te University of Miign omputtionl Menis Lbortory Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Hessenberg Mtrix Mximum i+ m? row & olumn internge i m
6 Te University of Miign omputtionl Menis Lbortory Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Hessenberg Mtrix ,,, mx( ) i y i,,,, + y y ,,, mx( ) i y i,,,, + y y
7 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Hessenberg Mtrix Mximum i+ m? y, i mx( ) i y + y,,,,,,, Te University of Miign omputtionl Menis Lbortory
8 Te University of Miign omputtionl Menis Lbortory Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Single sift QR omplex version of two step QR Double sift QR QR lgoritm λ λ λ λ X λ λ λ λ X X
9 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Single sift QR lgoritm Q + H Q Q is unitry mtrix H Q R R is upper tringulr mtrix H Q ( ζ I) R Te University of Miign omputtionl Menis Lbortory
10 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis omplex version of two step QR Q + H Q H Q ( ζ I) R ζ ζ Q + H + Q + + Q + H + ( + ζ + I) R + Te University of Miign omputtionl Menis Lbortory
11 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Q Q Double sift QR lgoritm Q Q Q Q T T T T + Q ( ζ I)( ζ + I) R R + Q Q, R R + R, ( ζ I)( ζ + ) Γ + I T Q Q +, Q Γ R Te University of Miign omputtionl Menis Lbortory
12 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Double sift QR lgoritm If mtrix H is erive from su tt Qˆ QH ˆ Qˆ T or Qˆ H were Qˆ is ortogonl H is upper Hessenberg If Qˆ s te sme first olumn s Q, ten Te University of Miign Q ˆ Q n + H omputtionl Menis Lbortory
13 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Double sift QR lgoritm P I w r r w T r T T T T P P P P P P P P n n H ( γ γ, γ,,,) T, γ γ γ ( ( ζ + + ζ ) + ζ ζ ζ ζ ) + Te University of Miign omputtionl Menis Lbortory
14 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Double sift QR 5 n n n n nn P T P ' ' ' ' ' ' ' ' ' ' ' ' 5 n n n n nn Te University of Miign omputtionl Menis Lbortory
15 Te University of Miign omputtionl Menis Lbortory Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Double sift QR ( ) T r r r,,,,,,,,, χ β α
16 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Sifting Rule Before e itertion, lulte te roots of te lst prinipl submtrix η n η + Te oie of te origin sifts ζ n ζ + for te itertion epens on; η η η + ρ, ρ η η + η If tey re bot greter tn 5, ζ ζ + if tey re bot less tn 5, ζ η n ζ η + + oterwise we set bot ζ n ζ + to be te rel prt of eiter η or η +, wiever orrespons to te quntity less tn 5 Te University of Miign omputtionl Menis Lbortory
17 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Double sift QR Te University of Miign omputtionl Menis Lbortory
18 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Double sift QR E E Te University of Miign omputtionl Menis Lbortory
19 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Double sift QR lgoritm λ 57, x ± 76, 5 ± 65i i 5 ± 59i 68 ± 8i Te University of Miign omputtionl Menis Lbortory
20 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis MATLAB Eig(A,B) were A is te sme s n B is ientity mtrix Eigenvetors i i i i i i i i i i i i i i i i Eigenvlues E-7i i E-5i i Te University of Miign omputtionl Menis Lbortory
21 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis Nstrn upper Hessenberg meto λ i, x ± 76, i i i 6± 779i ± 65i 7 ± 8i i i i i Te University of Miign omputtionl Menis Lbortory
22 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis omplex Eigenvlue Solver MS/NASTRAN QZ lgoritm Rel Imginry Rel Imginry Te University of Miign omputtionl Menis Lbortory
23 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis QZ lgoritm Ax λbx QAZy λqbzy, x Zy A is reue to upper Hessenberg B is reue to upper tringulr A is reue to qusi-tringulr (generlition of QR) Te University of Miign omputtionl Menis Lbortory
24 Development of Optiml Design Meto for Bre Squel Noise Bse on omplex Eigenvlue Anlysis onlusions For rel symmetri eigenvlue problem, ouble sift QR or QZ lgoritm soul be use Oter metos proue fititious omplex eigenvlues Te University of Miign omputtionl Menis Lbortory
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