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Submitted on 3 Mar 2011
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Uncertainty Quantification for
Eigenvalue-Realization-Algorithm, A Class of Subspace
System Identification
Xuan-Binh Lam, Laurent Mevel
To cite this version:
Xuan-Binh Lam, Laurent Mevel. Uncertainty Quantification for Eigenvalue-Realization-Algorithm,
A Class of Subspace System Identification. [Research Report] RR-7462, INRIA. 2010, pp.16.
�inria-00538941�
a p p o r t
d e r e c h e r c h e
0249-6399
ISRN
INRIA/RR--7462--FR+ENG
Stochastic Methods and Models
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Uncertainty Quantification for
Eigensystem-Realization-Algorithm,
A Class of Subspace System Identification
Xuan-Binh Lam — Laurent Mevel
N° 7462
Centre de recherche INRIA Rennes – Bretagne Atlantique
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❘és✉♠é ✿ ❊♥ ❆♥❛❧②s❡ ♠♦❞❛❧❡✱ ❧❡s ♣❛r❛♠ètr❡s ♠♦❞❛✉① ✭❢réq✉❡♥❝❡s✱ ❛♠♦rt✐ss❡♠❡♥ts ❡t ❞é❢♦r♠é❡s✮✱ ♦❜t❡♥✉s à ♣❛rt✐r ❞❡s ♠❡t❤♦❞❡s st♦❝❤❛st✐q✉❡s s♦♥t s✉❥❡t à ❞❡s ❡rr❡✉rs ❞✬✐♥❝❡rt✐t✉❞❡✳ ■❧ ❡st ♥é❝❡ss❛✐r❡ ❞✬é✈❛❧✉❡r ❧❡s ✐♥t❡r✈❛❧❧❡s ❞❡ ❝♦♥✜❛♥❝❡ ❝♦rr❡s♣♦♥❞❛♥ts✳ ❈❡ ❝❛❧❝✉❧ ❡st ❢❛✐t ♣♦✉r ✉♥❡ ❝❧❛ss❡ ❞✬❛❧❣♦r✐t❤♠❡s s♦✉s ❡s♣❛❝❡s ❛♣♣❡❧és ❊❘❆ ✭❊✐❣❡♥s②st❡♠✲❘❡❛❧✐③❛t✐♦♥✲❆❧❣♦r✐t❤♠✮✳ ▼♦ts✲❝❧és ✿ ❆♥❛❧②s❡ ♠♦❞❛❧❡✱ ✐❞❡♥t✐✜❝❛t✐♦♥ ❞❡s s②stè♠❡s st♦❝❤❛st✐q✉❡s✱ ❝❛❧❝✉❧ ❞✬✐♥❝❡rt✐t✉❞❡s✱ ♠é❝❛♥✐q✉❡ ❡t ❛❡r♦s♣❛t✐❛❧❡❯♥❝❡rt❛✐♥t② q✉❛♥t✐✜❝❛t✐♦♥ ✸
✶ ■♥tr♦❞✉❝t✐♦♥
❚❤❡ ❞❡s✐❣♥ ❛♥❞ ♠❛✐♥t❡♥❛♥❝❡ ♦❢ ♠❡❝❤❛♥✐❝❛❧ str✉❝t✉r❡s s✉❜❥❡❝t t♦ ♥♦✐s❡ ❛♥❞ ✈✐✲ ❜r❛t✐♦♥s ✐s ❛♥ ✐♠♣♦rt❛♥t t♦♣✐❝ ✐♥ ♠❡❝❤❛♥✐❝❛❧ ❡♥❣✐♥❡❡r✐♥❣✳ ■t ✐s ❛♥ ✐♠♣♦rt❛♥t ❝♦♠♣♦♥❡♥t ♦❢ ❝♦♠❢♦rt ✭❝❛rs ❛♥❞ ❜✉✐❧❞✐♥❣s✮ ❛♥❞ ❝♦♥tr✐❜✉t❡s s✐❣♥✐❝❛♥t❧② t♦ t❤❡ s❛❢❡t② r❡❧❛t❡❞ ❛s♣❡❝ts ♦❢ ❞❡s✐❣♥ ❛♥❞ ♠❛✐♥t❡♥❛♥❝❡ ✭❛✐r❝r❛❢ts✱ ❛❡r♦s♣❛❝❡ ✈❡❤✐❝❧❡s ❛♥❞ ♣❛②❧♦❛❞s✱ ❝✐✈✐❧ str✉❝t✉r❡s✮✳ ❘❡q✉✐r❡♠❡♥ts ❢r♦♠ t❤❡s❡ ❛♣♣❧✐❝❛t✐♦♥ ❛r❡❛s ❛r❡ ♥✉♠❡r♦✉s ❛♥❞ ❞❡♠❛♥❞✐♥❣✳ ▲❛❜♦r❛t♦r② ❛♥❞ ✐♥✲♦♣❡r❛t✐♦♥ t❡sts ❛r❡ ♣❡r❢♦r♠❡❞ ♦♥ t❤❡ ♣r♦t♦t②♣❡ str✉❝t✉r❡✱ ✐♥ ♦r❞❡r t♦ ❣❡t s♦✲❝❛❧❧❡❞ ♠♦❞❛❧ ♠♦❞❡❧s✱ ✐✳❡✳✱ t♦ ❡①tr❛❝t t❤❡ ♠♦❞❡s ❛♥❞ ❞❛♠♣✐♥❣ ❢❛❝t♦rs ✭t❤❡s❡ ❝♦rr❡s♣♦♥❞ t♦ s②st❡♠ ♣♦❧❡s✮✱ t❤❡ ♠♦❞❡ s❤❛♣❡s ✭❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❡❝t♦rs✮✱ ❛♥❞ ❧♦❛❞s✳ ❚❤❡s❡ r❡s✉❧ts ❛r❡ ✉s❡❞ ❢♦r ✉♣✲ ❞❛t✐♥❣ t❤❡ ❞❡s✐❣♥ ♠♦❞❡❧ ❢♦r ❛ ❜❡tt❡r ✜t t♦ ❞❛t❛✱ ❛♥❞ s♦♠❡t✐♠❡s ❢♦r ❝❡rt✐✜❝❛t✐♦♥ ♣✉r♣♦s❡s ✭❡✳❣✳✱ ✐♥ ✢✐❣❤t ❞♦♠❛✐♥ ♦♣❡♥✐♥❣ ❢♦r ♥❡✇ ❛✐r❝r❛❢ts✮✳ ❚❤❡ ❡st✐♠❛t✐♦♥ ♦❢ ♠♦❞❛❧ ♣❛r❛♠❡t❡rs ♦❢ str✉❝t✉r❡s ❝❛♥ ❡❛s✐❧② ❜❡ ❝❛rr✐❡❞ ♦✉t ❜② ✉s✐♥❣ ❙t♦❝❤❛st✐❝ ❙②st❡♠ ■❞❡♥t✐✜❝❛t✐♦♥ ♠❡t❤♦❞s ♦♥ s❡♥s♦r ♠❡❛s✉r❡♠❡♥ts✳ ❬✸❪ ♣r♦✈❡❞ t❤❛t t❤❡ ■♥str✉♠❡♥t❛❧ ❱❛r✐❛❜❧❡ ♠❡t❤♦❞ ❛♥❞ ✇❤❛t ✇❛s ❝❛❧❧❡❞ t❤❡ ❇❛❧❛♥❝❡❞ ❘❡❛❧✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ❧✐♥❡❛r ❡✐❣❡♥str✉❝t✉r❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ❛r❡ ❝♦♥s✐st❡♥t ✐♥ ❛ ♥♦♥st❛t✐♦♥❛r② ❝♦♥t❡①t✳ ❋r♦♠ t❤❛t ♦♥✱ t❤❡ ❢❛♠✐❧② ♦❢ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠s ❤❛s ❜❡❡♥ ❡①t❡♥s✐✈❡❧② st✉❞✐❡❞ ✭s❡❡ ✐♥ ❬✾✱ ✶✹❪✮ ❛♥❞ ❤❛s ❡①♣❛♥❞❡❞ r❛♣✐❞❧②✳ ❚❤❡r❡ ❛r❡ ❛ ♥✉♠❜❡r ♦❢ ❝♦♥✈❡r❣❡♥❝❡ st✉❞✐❡s ♦♥ s✉❜s♣❛❝❡ ♠❡t❤♦❞s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ✭s❡❡ ❬✻✱ ✷✱ ✶✱ ✺❪✮ t♦ ♠❡♥t✐♦♥ ❥✉st ❛ ❢❡✇ ♦❢ t❤❡♠✳ ❚❤❡s❡ ♣❛♣❡rs ♣r♦✈✐❞❡ ❞❡❡♣ ❛♥❞ t❡❝❤♥✐❝❛❧❧② ❞✐✣❝✉❧t r❡s✉❧ts ✐♥❝❧✉❞✐♥❣ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡s✳ ❖✉r ♦❜❥❡❝t✐✈❡ ✐s t♦ ❞❡r✐✈❡ s✐♠♣❧❡ ❢♦r♠✉❧❛ ❢♦r s✉❝❤ s❡♥s✐t✐✈✐t✐❡s✳ ❙❡♥s✐t✐✈✐t✐❡s ❢♦r t❤❡ ❛❧❣♦r✐t❤♠s ❝♦♥s✐❞❡r❡❞ ✐♥ t❤✐s ♣❛♣❡r✱ ❊❘❆✱ ❛r❡ ♥♦t ❛❞❞r❡ss❡❞ ❜② t❤♦s❡ ♣❛♣❡rs✳ ❚❤❡ ✉♥❝❡rt❛✐♥t② ♦♥ ♠♦❞❛❧ ♣❛r❛♠❡t❡rs ❛♣♣❡❛rs ❢♦r ♠❛♥② r❡❛s♦♥s✱ ❡✳❣✳ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❞❛t❛ s❛♠♣❧❡s✱ ✉♥❞❡✜♥❡❞ ♠❡❛s✉r❡♠❡♥t ♥♦✐s❡s✱ ♥♦♥st❛t✐♦♥❛r② ❡①❝✐✲ t❛t✐♦♥s✱ ♥♦♥❧✐♥❡❛r str✉❝t✉r❡✱ ♠♦❞❡❧ ♦r❞❡r r❡❞✉❝t✐♦♥✱✳✳✳✱ t❤❡♥ t❤❡ s②st❡♠ ✐❞❡♥✲ t✐✜❝❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❞♦ ♥♦t ②✐❡❧❞ t❤❡ ❡①❛❝t s②st❡♠ ♠❛tr✐❝❡s✳ Pr❛❝t✐❝❛❧❧②✱ t❤❡ st❛t✐st✐❝❛❧ ✉♥❝❡rt❛✐♥t② ♦❢ t❤❡ ♦❜t❛✐♥❡❞ ♠♦❞❛❧ ♣❛r❛♠❡t❡rs ❛t ❛ ❝❤♦s❡♥ s②st❡♠ ♦r❞❡r ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ✉♥❝❡rt❛✐♥t② ♦❢ t❤❡ s②st❡♠ ♠❛tr✐❝❡s✱ ✇❤✐❝❤ ❞❡✲ ♣❡♥❞s ♦♥ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✉❜s♣❛❝❡ ♠❛tr✐①✳ ◆♦t ❦♥♦✇✐♥❣ t❤❡ ♠♦❞❡❧ ♦r❞❡r ②✐❡❧❞s t♦ ✉s❡ ❡♠♣✐r✐❝❛❧ ♠✉❧t✐✲♦r❞❡r ♣r♦❝❡❞✉r❡ s✉❝❤ ❛s t❤❡ st❛✲ ❜✐❧✐③❛t✐♦♥ ❞✐❛❣r❛♠ ✭❬✶✵❪✮✱ ✇❤❡r❡ ♠♦❞❡s ♦❢ t❤❡ s②st❡♠ ❛r❡ ❛ss✉♠❡❞ t♦ st❛❜✐❧✐③❡ ✇❤❡♥ t❤❡ ♠♦❞❡❧ ♦r❞❡r ✐♥❝r❡❛s❡s✳ ■♥ ❬✶✷❪✱ ✐t ❤❛s ❜❡❡♥ s❤♦✇♥ ❤♦✇ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ♦❢ ♠♦❞❛❧ ♣❛r❛♠❡t❡rs ❝❛♥ ❜❡ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ t❤❡ ❝♦✈❛r✐❛♥❝❡s ♦❢ t❤❡ s②st❡♠ ♠❛tr✐❝❡s ❛♥❞ t❤❡ ❝♦✈❛r✐❛♥❝❡s ♦❢ s✉❜s♣❛❝❡ ♠❛tr✐❝❡s✳ ❚❤❡ ❝✉rr❡♥t ♣❛♣❡r ✇✐❧❧ ❡①♣❛♥❞ ♦♥ t❤✐s ❛♥❞ ❝♦♠♣❛r❡ s❡♥✲ s✐t✐✈✐t✐❡s ❢♦r t✇♦ ♦✉t♣✉t✲♦♥❧② s②st❡♠ ✐❞❡♥t✐✜❝❛t✐♦♥ ♠❡t❤♦❞s✱ ♥❛♠❡❧② ♦✉t♣✉t✲♦♥❧② ❙t♦❝❤❛st✐❝ ❙✉❜s♣❛❝❡ ■❞❡♥t✐✜❝❛t✐♦♥ ✭❙❙■✮ ❛♥❞ ❊✐❣❡♥s②st❡♠✲❘❡❛❧✐③❛t✐♦♥✲❆❧❣♦r✐t❤♠ ✭❊❘❆✮ ❙②st❡♠ ■❞❡♥t✐✜❝❛t✐♦♥✳ ❙✉❜s♣❛❝❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❝♦♠♣✉✲ t❛t✐♦♥ ♦❢ ♦♥❡ s✉❜s♣❛❝❡ ♠❛tr✐① ❢r♦♠ t❤❡ ❝♦rr❡❧❛t✐♦♥ t❛✐❧✳ ❯♥❧✐❦❡ s✉❜s♣❛❝❡ ❛❧❣♦✲ r✐t❤♠✱ ❊❘❆ ❝♦♠♣✉t❡s t❤❡ s②st❡♠ ♠❛tr✐❝❡s ✉s✐♥❣ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ♦❢ ❜♦t❤ ✭❦✮✲ ❛♥❞ ✭❦✰✶✮✲❧❛❣ ♦❢ s❤✐❢t❡❞ ❝♦rr❡❧❛t✐♦♥ t❛✐❧s✳ ■♥ t❤✐s ♣❛♣❡r✱ ❢♦❧❧♦✇✐♥❣ t❤❡ ❧✐♥❡s ♦❢ ✭❬✶✷❪✮✱ ❛♥ ❛❧❣♦r✐t❤♠ ✇✐❧❧ ❜❡ ❞❡✈❡❧♦♣❡❞ ❢♦r ❡st✐♠❛t✐♥❣ t❤❡ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ✐♥ ❊❘❆ s②st❡♠ ✐❞❡♥t✐✜❝❛t✐♦♥✳ ❚❤❡ ✉♥❝❡r✲ t❛✐♥t② ♦♥ st❛t❡ tr❛♥s✐t✐♦♥ ♠❛tr✐① ✐s ❞❡r✐✈❡❞✱ ❜❛s❡❞ ♦♥ t❤❡ ✉♥❝❡rt❛✐♥t✐❡s ♦❢ ✭❦✮✲ ❛♥❞ ✭❦✰✶✮✲❧❛❣ s✉❜s♣❛❝❡ ♠❛tr✐❝❡s✳ ❆ r❡❧❡✈❛♥t ✐♥❞✉str✐❛❧ ❡①❛♠♣❧❡ ✐s ❛♣♣❧✐❡❞ t♦ ❊❘❆ ❡st✐♠❛t❡s✳ ❚❤❡ ❡✣❝✐❡♥❝② ♦❢ t❤❡s❡ ❛❧❣♦r✐t❤♠s ❛♥❞ ❧❛❣ ❡✛❡❝t ❛r❡ ❛❧s♦ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t✳ ❈♦♠♣❛r✐s♦♥ ✇✐t❤ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠ ❡st✐♠❛t❡s ✐s ❛❧s♦ ♣❡r❢♦r♠❡❞✳❯♥❝❡rt❛✐♥t② q✉❛♥t✐✜❝❛t✐♦♥ ✹
✷ ❙t♦❝❤❛st✐❝ ❙②st❡♠ ■❞❡♥t✐✜❝❛t✐♦♥
✷✳✶ ❚❤❡ ●❡♥❡r❛❧ ❙❙■ ❆❧❣♦r✐t❤♠
❚❤❡ ❞✐s❝r❡t❡ t✐♠❡ ♠♦❞❡❧ ✐♥ st❛t❡✲s♣❛❝❡ ❢♦r♠ ✐s✿ Xk+1 = AXk+ Vk+1 Yk = CXk ✭✶✮ ✇✐t❤ t❤❡ st❛t❡ X ∈ Rn✱ t❤❡ ♦✉t♣✉t Y ∈ Rr✱ t❤❡ st❛t❡ tr❛♥s✐t✐♦♥ ♠❛tr✐① A ∈ Rn×n ❛♥❞ t❤❡ ♦❜s❡r✈❛t✐♦♥ ♠❛tr✐① C ∈ Rr×n✳ ❚❤❡ st❛t❡ ♥♦✐s❡ V ✐s ✉♥♠❡❛s✉r❡❞ ❛♥❞ ❛ss✉♠❡❞ t♦ ❜❡ ●❛✉ss✐❛♥✱ ③❡r♦✲♠❡❛♥✱ ✇❤✐t❡✳ ▲❡t r ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ s❡♥s♦rs✱ p ❛♥❞ q ❜❡ ❝❤♦s❡♥ ♣❛r❛♠❡t❡rs ✇✐t❤ (p+1)r ≥ qr≥ n✳ ❋r♦♠ t❤❡ ♦✉t♣✉t ❞❛t❛✱ ❛ ♠❛tr✐① Hp+1,q ∈ R(p+1)r×qr ✐s ❜✉✐❧t ❛❝❝♦r❞✐♥❣ t♦ ❛ ❝❤♦s❡♥ ❙❙■ ❛❧❣♦r✐t❤♠✱ s❡❡ ❡✳❣✳ ❬✹❪ ❢♦r ❛♥ ♦✈❡r✈✐❡✇✳ ❚❤❡ ♠❛tr✐① Hp+1,q ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ✏s✉❜s♣❛❝❡ ♠❛tr✐①✑ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ❛♥❞ t❤❡ ❙❙■ ❛❧❣♦r✐t❤♠ ✐s ❝❤♦s❡♥ s✉❝❤ t❤❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✉❜s♣❛❝❡ ♠❛tr✐① ❡♥❥♦②s ✭❛s②♠♣t♦t✐❝❛❧❧② ❢♦r ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ s❛♠♣❧❡s✮ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♣r♦♣❡rt② Hp+1,q = Op+1 Zq ✭✷✮ ✐♥t♦ t❤❡ ♠❛tr✐① ♦❢ ♦❜s❡r✈❛❜✐❧✐t② Op+1 def = C CA ✳✳✳ CAp ✭✸✮ ❛♥❞ ❛ ♠❛tr✐① Zq ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ s❡❧❡❝t❡❞ ❙❙■ ❛❧❣♦r✐t❤♠✳ ▲❡t N ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❛✈❛✐❧❛❜❧❡ s❛♠♣❧❡s ❛♥❞ Yk ∈ Rr✱ {k ∈ 1, . . . , N} t❤❡ ✈❡❝t♦r ❝♦♥t❛✐♥✐♥❣ t❤❡ s❡♥s♦r ❞❛t❛✳ ❚❤❡♥✱ t❤❡ ✏❢♦r✇❛r❞✑ ❛♥❞ ✏❜❛❝❦✇❛r❞✑ ❞❛t❛ ♠❛tr✐❝❡s Yp+1+ = 1 √ N− p − q Yq+1 Yq+2 . . . YN −p Yq+2 Yq+3 . . . YN −p+1 ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ Yq+p+1 Yq+p+2 . . . YN , Y− q = 1 √ N− p − q Yq Yq+1 . . . YN −p−1 Yq−1 Yq . . . YN −p−2 ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ Y1 Y2 . . . YN −p−q ✭✹✮ ❛r❡ ❜✉✐❧t✳ ❋♦r t❤❡ ❝♦✈❛r✐❛♥❝❡✲❞r✐✈❡♥ ❙❙■ ✭s❡❡ ❛❧s♦ ❬✸❪✱ ❬✶✵❪✮✱ t❤❡ s✉❜s♣❛❝❡ ♠❛tr✐① H✭❝♦✈✮p+1,q = Yp+1+ Yq− T ✐s ❜✉✐❧t✱ ✇❤✐❝❤ ❡♥❥♦②s t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♣r♦♣❡rt② ✭✷✮✱ ✇❤❡r❡ Zq ✐s t❤❡ ❝♦♥tr♦❧❧❛❜✐❧✐t② ♠❛tr✐①✳ ❋♦r s✐♠♣❧✐❝✐t②✱ ❧❡t p ❛♥❞ q ❜❡ ❣✐✈❡♥✱ s❦✐♣ t❤❡ s✉❜s❝r✐♣ts ♦❢ Hp+1,q✱ Op+1 ❛♥❞ Zq✳ ❚❤❡ ❡✐❣❡♥str✉❝t✉r❡ ♦❢ t❤❡ s②st❡♠ ✭✶✮ ✐s r❡tr✐❡✈❡❞ ❢r♦♠ ❛ ❣✐✈❡♥ ♠❛tr✐① H✳ ❚❤❡ ♦❜s❡r✈❛❜✐❧✐t② ♠❛tr✐① O ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ ❛ t❤✐♥ ❙✐♥❣✉❧❛r ❱❛❧✉❡ ❉❡❝♦♠✲ ♣♦s✐t✐♦♥ ✭❙❱❉✮ ♦❢ t❤❡ ♠❛tr✐① H ❛♥❞ ✐ts tr✉♥❝❛t✐♦♥ ❛t t❤❡ ❞❡s✐r❡❞ ♠♦❞❡❧ ♦r❞❡r ❘❘ ♥➦ ✼✹✻✷❯♥❝❡rt❛✐♥t② q✉❛♥t✐✜❝❛t✐♦♥ ✺ n✿ H = UΣVT = U1 U0 Σ1 0 0 Σ0 VT, ✭✺✮ O = U1Σ1/21 . ✭✻✮ ◆♦t❡ t❤❛t t❤❡ s✐♥❣✉❧❛r ✈❛❧✉❡s ✐♥ Σ1∈ Rd×d♠✉st ❜❡ ♥♦♥✲③❡r♦ ❛♥❞ ❤❡♥❝❡ O ✐s ♦❢ ❢✉❧❧ ❝♦❧✉♠♥ r❛♥❦✳ ❚❤❡ ♦❜s❡r✈❛t✐♦♥ ♠❛tr✐① C ✐s t❤❡♥ ❢♦✉♥❞ ✐♥ t❤❡ ✜rst ❜❧♦❝❦✲r♦✇ ♦❢ t❤❡ ♦❜s❡r✈❛❜✐❧✐t② ♠❛tr✐① O✳ ❚❤❡ st❛t❡ tr❛♥s✐t✐♦♥ ♠❛tr✐① A ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ s❤✐❢t✐♥❣ ✐♥✈❛r✐❛♥❝❡ ♣r♦♣❡rt② ♦❢ O✱ ♥❛♠❡❧② ❛s t❤❡ ❧❡❛st sq✉❛r❡s s♦❧✉t✐♦♥ ♦❢
O↑A = O↓, ✇❤❡r❡ O↑def= C CA ✳✳✳ CAp−1 , O↓def= CA CA2 ✳✳✳ CAp . ✭✼✮ ❚❤❡ ❡✐❣❡♥str✉❝t✉r❡ (λ, ϕλ)r❡s✉❧ts ❢r♦♠
det(A − λI) = 0, Aφλ= λφλ, ϕλ= Cφλ, ✭✽✮
✇❤❡r❡ λ r❛♥❣❡s ♦✈❡r t❤❡ s❡t ♦❢ ❡✐❣❡♥✈❛❧✉❡s ♦❢ A✳ ❋r♦♠ λ✱ t❤❡ ♥❛t✉r❛❧ ❢r❡q✉❡♥❝② ❛♥❞ ❞❛♠♣✐♥❣ r❛t✐♦ ❛r❡ ♦❜t❛✐♥❡❞✱ ❛♥❞ ϕλ ✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠♦❞❡ s❤❛♣❡✳ ❚❤❡r❡ ❛r❡ ♠❛♥② ♣❛♣❡rs ♦♥ t❤❡ ✉s❡❞ ✐❞❡♥t✐✜❝❛t✐♦♥ t❡❝❤♥✐q✉❡s✳ ❆ ❝♦♠♣❧❡t❡ ❞❡s❝r✐♣t✐♦♥ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✸❪✱ ❬✶✹❪✱ ❬✶✵❪✱ ❬✹❪✱ ❛♥❞ t❤❡ r❡❧❛t❡❞ r❡❢❡r❡♥❝❡s✳ ❆ ♣r♦♦❢ ♦❢ ♥♦♥✲st❛t✐♦♥❛r② ❝♦♥s✐st❡♥❝② ♦❢ t❤❡s❡ s✉❜s♣❛❝❡ ♠❡t❤♦❞s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✹❪✳
✷✳✷ ❊❘❆ ✭❊✐❣❡♥s②st❡♠✲❘❡❛❧✐③❛t✐♦♥✲❆❧❣♦r✐t❤♠✮
❆♥♦t❤❡r ✈❛r✐❛♥t ♦❢ r❡❛❧✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❜❛s❡❞ ♦♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ s✉❜✲ s♣❛❝❡ ♠❛tr✐❝❡s ✐s ❝❛❧❧❡❞ ❊❘❆ ✭❊✐❣❡♥s②st❡♠✲❘❡❛❧✐③❛t✐♦♥✲❆❧❣♦r✐t❤♠✮ ✭s❡❡ ✐♥ ❬✽❪✮✳ ■t ✐s ❜❛s❡❞ ♦♥ t❤❡ ❣❡♥❡r❛❧ r❡♠❛r❦ t❤❛t ♦♥❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ s✉❜s♣❛❝❡ ♠❛tr✐① H ♥♦t ✉s✐♥❣ t❤❡ ✜rst ❧❛❣s ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ t❛✐❧✳ ❉❡✜♥✐♥❣ H(k)❛s H(k)= Rk+1 Rk+2 . . . Rk+q Rk+2 Rk+3 . . . Rk+q+1 ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ Rk+p+1 . . . Rk+p+q , ✭✾✮ ✐♥ ✇❤✐❝❤ t❤❡ ❝♦rr❡❧❛t✐♦♥s ❛r❡ r❡❧❛t❡❞ t♦ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ Rj def = E Yl+jYlT = CAjG ✭✶✵✮ ✇✐t❤ t❤❡ ❝r♦ss✲❝♦✈❛r✐❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ st❛t❡ ❛♥❞ t❤❡ ♦❜s❡r✈❡❞ ♦✉t♣✉ts G = E [Xl YlT]✳ ❚❤❡♥✱ ❛ ❙✐♥❣✉❧❛r ❱❛❧✉❡ ❉❡❝♦♠♣♦s✐t✐♦♥ ✐s ♣❡r❢♦r♠❡❞ ♦♥ H(k) ❛s H(k)= U1 U0 Σ1 0 0 Σ0 V1T V0T ✭✶✶✮ ❚❤❡ st❛t❡ tr❛♥s✐t✐♦♥ ♠❛tr✐① ✇✐❧❧ ❜❡ ❞❡✜♥❡❞ ❛s A = O†1 H(k+1)Z1† , ✭✶✷✮❯♥❝❡rt❛✐♥t② q✉❛♥t✐✜❝❛t✐♦♥ ✻ ✇❤❡r❡† ♠❡❛♥s ▼♦♦r❡✲P❡♥r♦s❡ ♣s❡✉❞♦✲✐♥✈❡rs❡✱ ❛♥❞ O†1 = (Σ1)− 1 2 UT 1, ✭✶✸✮ Z1† = V1(Σ1)− 1 2. ✭✶✹✮ ■❢ t❤❡ ❝♦rr❡❧❛t✐♦♥s ❛r❡ ❝♦♠♣✉t❡❞ ❢r♦♠ ❝r♦ss s♣❡❝tr❛✱ t❤❡ ♠❡t❤♦❞ ✐s ❝❛❧❧❡❞ ◆❊❳❚✲❊❘❆❀ ❜✉t ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ ✐t ✐s ❥✉st ❛ss✉♠❡❞ t❤❛t t❤❡ ❝♦rr❡✲ ❧❛t✐♦♥s ❛r❡ ❝♦♠♣✉t❡❞ ❢r♦♠ t✐♠❡ s❛♠♣❧❡s✳ ❚❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ A r❡❧❛t❡s t♦ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ U1,Σ1, V1✳ ❆♥❞ ❛s s✉❝❤✱ ❛ st❛❜✐❧✐③❛t✐♦♥ ❞✐❛❣r❛♠ ✐s ♦❜t❛✐♥❡❞ ❜② ♣❡r❢♦r♠✐♥❣ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ A ❢♦r ♠✉❧t✐♣❧❡ ♠♦❞❡❧ ♦r❞❡rs ❛♥❞ ❦❡❡♣✐♥❣ ❛s st❛❜❧❡ ♣♦❧❡s t❤❡ ♠♦❞❡s ✇❤✐❝❤ r❡♣❡❛t ♦✈❡r ♠✉❧t✐♣❧❡ ♠♦❞❡❧ ♦r❞❡rs✳
✸ ❈♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s
✸✳✶ ❉❡s❝r✐♣t✐♦♥s ♦❢ ❙❙■ ❈♦♥✜❞❡♥❝❡ ■♥t❡r✈❛❧s ❛❧❣♦r✐t❤♠
❚❤❡ st❛t✐st✐❝❛❧ ✉♥❝❡rt❛✐♥t② ♦❢ t❤❡ ♦❜t❛✐♥❡❞ ♠♦❞❛❧ ♣❛r❛♠❡t❡rs ❛t ❛ ❝❤♦s❡♥ s②st❡♠ ♦r❞❡r ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ✉♥❝❡rt❛✐♥t② ♦❢ t❤❡ s②st❡♠ ♠❛tr✐❝❡s✱ ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✉❜s♣❛❝❡ ♠❛tr✐① H✳ ❚❤❡ ❧❛tt❡r ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞ ❜② ❝✉tt✐♥❣ t❤❡ s❡♥s♦r ❞❛t❛ ✐♥t♦ ❜❧♦❝❦s ♦♥ ✇❤✐❝❤ ✐♥st❛♥❝❡s ♦❢ t❤❡ s✉❜s♣❛❝❡ ♠❛tr✐① ❛r❡ ❝♦♠♣✉t❡❞✳ ❙♦✱ t❤✐s ♦✛❡rs ❛ ♣♦ss✐❜✐❧✐t② t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ♦❢ t❤❡ ♠♦❞❛❧ ♣❛r❛♠❡t❡rs ❛t ❛ ❝❡rt❛✐♥ s②st❡♠ ♦r❞❡r ✇✐t❤♦✉t r❡♣❡❛t✐♥❣ t❤❡ s②st❡♠ ✐❞❡♥t✐✜❝❛t✐♦♥✳ ■♥ ❬✶✷❪✱ t❤✐s ❛❧❣♦r✐t❤♠ ✇❛s ❞❡s❝r✐❜❡❞ ✐♥ ❞❡t❛✐❧ ❢♦r t❤❡ ❝♦✈❛r✐❛♥❝❡✲❞r✐✈❡♥ ❙❙■✳ ❚❤❡ ✉♥❝❡rt❛✐♥t② ∆A ❛♥❞ ∆C ♦❢ t❤❡ s②st❡♠ ♠❛tr✐❝❡s A ❛♥❞ C ❛r❡ ❝♦♥♥❡❝t❡❞ t♦ t❤❡ ✉♥❝❡rt❛✐♥t② ♦❢ t❤❡ s✉❜s♣❛❝❡ ♠❛tr✐① t❤r♦✉❣❤ ❛ ❏❛❝♦❜✐❛♥ ♠❛tr✐① vec∆A vec∆C = JA,C vec∆H, ✭✶✺✮ ✇❤❡r❡ vec ✐s t❤❡ ✈❡❝t♦r✐③❛t✐♦♥ ♦♣❡r❛t♦r✳ ❚❤❡♥✱ t❤❡ ✉♥❝❡rt❛✐♥t② ♦❢ t❤❡ ♠♦❞❛❧ ♣❛r❛♠❡t❡rs ✭♥❛t✉r❛❧ ❢r❡q✉❡♥❝② f✱ ❞❛♠♣✐♥❣ r❛t✐♦ d ❛♥❞ ♠♦❞❡ s❤❛♣❡ φ✮ ✐s ❞❡r✐✈❡❞ ❢r♦♠ ∆fµ= Jfµ vec∆A vec∆C , ∆dµ= Jdµ vec∆A vec∆C , ✭✶✻✮ ❛♥❞ ∆φµ = Jφµ vec∆A vec∆C . ✭✶✼✮ ❚❤❡ ❏❛❝♦❜✐❛♥s Jfµ✱ Jdµ ❛♥❞ Jφµ ❛r❡ ❝♦♠♣✉t❡❞ ❢♦r ❡❛❝❤ ♠♦❞❡ µ✳ ❋✐♥❛❧❧②✱ t❤❡ ❝♦✈❛r✐❛♥❝❡s ♦❢ t❤❡ ♠♦❞❛❧ ♣❛r❛♠❡t❡rs ❛r❡ ♦❜t❛✐♥❡❞ ❛s cov(fµ) = Jfµ JA,C cov(vec H) JT A,C JfTµ
cov(dµ) = Jdµ JA,C cov(vec H) J
T A,C JdTµ
cov(φµ) = Jφµ JA,C cov(vec H) J
T
A,C JφTµ ✭✶✽✮
❯♥❝❡rt❛✐♥t② q✉❛♥t✐✜❝❛t✐♦♥ ✼ ✇❤❡r❡ cov(vecH) ✐s t❤❡ ❝♦✈❛r✐❛♥❝❡ ♦❢ t❤❡ ✈❡❝t♦r✐③❡❞ s✉❜s♣❛❝❡ ♠❛tr✐①✳ ❆❢t❡r r❡tr✐❡✈✐♥❣ t❤❡ ✉♥❝❡rt❛✐♥t✐❡s ♦♥ t❤❡ s②st❡♠ ♠❛tr✐❝❡s A ❛♥❞ C✱ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ t❤❡ ✉♥❝❡rt❛✐♥t✐❡s ♦♥ t❤❡ ❢r❡q✉❡♥❝② ❛♥❞ ❞❛♠♣✐♥❣ ✐s str❛✐❣❤t❢♦r✇❛r❞✳ ❍♦✇❡✈❡r✱ ❢♦r t❤❡ ♠♦❞❡ s❤❛♣❡✱ t❤❡r❡ ✐s ❛♥ ✐ss✉❡ ♦❢ ♥♦r♠❛❧✐③❛t✐♦♥ ❛s ❡❛❝❤ ♦♥❡ ✐s ❞❡✜♥❡❞ ✉♣ t♦ ❛♥ ✉♥❦♥♦✇♥ ❝♦♥st❛♥t✳ ❚❤✐s ✇❛s ❛❞❞r❡ss❡❞ ✐♥ ❬✼❪✳
✸✳✷ ❉❡r✐✈❛t✐♦♥ ♦❢ ❊❘❆ ❈♦♥✜❞❡♥❝❡ ■♥t❡r✈❛❧s
■♥ t❤✐s s❡❝t✐♦♥✱ ❢♦r ❊❘❆✱ ✐t ✐s ✐♥✈❡st✐❣❛t❡❞ ❤♦✇ t❤❡ ❝♦✈❛r✐❛♥❝❡s ♦❢ ♠♦❞❛❧ ♣❛r❛♠✲ ❡t❡rs ❝❛♥ ❜❡ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♦❢ s✉❜s♣❛❝❡ ♠❛tr✐❝❡s t❛❦✐♥❣ ❝❛r❡ ♦❢ t❤❡ ✉♥❝❡rt❛✐♥t✐❡s ♦❢ ♦❜s❡r✈❛❜✐❧✐t②✱ ❝♦♥tr♦❧❧❛❜✐❧✐t② ❛♥❞ s②st❡♠ ♠❛tr✐❝❡s✳ ❋✐rst❧②✱ t❤❡ ✉♥❝❡rt❛✐♥t② ♦♥ t❤❡ s②st❡♠ ♠❛tr✐① A ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s❡♥s✐t✐✈✲ ✐t✐❡s ♦❢ H(k+1)✱ O† 1 ❛♥❞ Z † 1✿ ∆A = ∆h O†1 H(k+1)Z1† i = h∆O†1 i H(k+1)Z1† +O†1 h ∆H(k+1)i Z1† +O†1 H(k+1)h∆Z1† i . ✭✶✾✮ ❚❤❡ ✉♥❝❡rt❛✐♥t② ♦♥ t❤❡ ✈❡❝t♦r✐③❡❞ s②st❡♠ ♠❛tr✐① A ✐s r❡✇r✐tt❡♥ ❛s vec∆A = Z1† T H(k+1)T⊗ Id vec∆O†1 +Z1† T ⊗ O1† vec∆H(k+1) +Id⊗ O1†H(k+1) vec∆Z1† , ✭✷✵✮ ✇❤❡r❡ Id ✐s ✐❞❡♥t✐t② ♠❛tr✐① ✇✐t❤ ❞✐♠❡♥s✐♦♥ d✳ ⊗ ✐s t❤❡ ❑r♦♥❡❝❦❡r ♣r♦❞✉❝t✳ ❚❤❡ ✉♥❝❡rt❛✐♥t② ♦❢ H(k+1) ❝❛♥ s✐♠♣❧② ❜❡ ❡st✐♠❛t❡❞ ❜② ❝✉tt✐♥❣ t❤❡ s✐❣♥❛❧s✳ ❚❤❡ ✉♥❝❡rt❛✐♥t② ♦♥ t❤❡ ♣s❡✉❞♦✲✐♥✈❡rs❡ ♦❢ ♦❜s❡r✈❛❜✐❧✐t② O1 ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❞✐r❡❝t❧② ❢r♦♠ t❤❡ s✐♥❣✉❧❛r ✈❛❧✉❡s ❛♥❞ s✐♥❣✉❧❛r ✈❡❝t♦rs ❜② ∆O1† = ∆Σ−12 1 U1T = h∆Σ−12 1 i U1T + Σ−12 1 ∆ U1T = −1 2Σ −3 2 1 (∆Σ1) U1T + Σ −1 2 1 ∆ U1T ✭✷✶✮ ❚❤❡ ✉♥❝❡rt❛✐♥t② ♦❢ O† 1 ✐s ♥♦✇ ✈❡❝t♦r✐③❡❞ ❛s vec∆O1† = U1⊗ −12Σ−32 1 vec∆Σ1 +I(p+1)r⊗ Σ −1 2 1 vec ∆ U1T = U1⊗ −12Σ−32 1 vec∆Σ1 +I(p+1)r⊗ Σ −1 2 1 PU1 vec∆U1 ✭✷✷✮❯♥❝❡rt❛✐♥t② q✉❛♥t✐✜❝❛t✐♦♥ ✽ ✇❤❡r❡ PU1 ✐s ❛ ♠❛tr✐① t❤❛t ❝❛♥ ♣❡r♠✉t❛t❡ vec∆U1t♦ vec ∆ U T 1 ✳ ❙✐♠✐❧❛r❧②✱ t❤❡ ✉♥❝❡rt❛✐♥t② ♦♥ t❤❡ ♣s❡✉❞♦✲✐♥✈❡rs❡ ♦❢ ❝♦♥tr♦❧❧❛❜✐❧✐t② Z1❝❛♥ ❜❡ ❞❡s❝✐❜❡❞ ❛s ∆Z1† = ∆V1Σ− 1 2 1 = (∆V1) Σ −1 2 1 + V1∆ Σ−12 1 = (∆V1) Σ −1 2 1 + V1 −1 2 Σ−32 1 ∆Σ1 ✭✷✸✮ ❛♥❞ r❡❝♦♥str✉❝t✐♥❣ ✐t ✐♥ ✈❡❝t♦r✐③❡❞ ❢♦r♠ ❧❡❛❞s t♦ vec∆Z1† = Σ−12 1 ⊗ Iqr vec∆V1 + Id⊗ −12V1Σ −3 2 1 vec∆Σ1. ✭✷✹✮ ❚❤❡ s❡♥s✐t✐✈✐t② ♦❢ t❤❡ ❧❡❢t s✐♥❣✉❧❛r ✈❡❝t♦rs ❝❛♥ ❜❡ r❡❧❛t❡❞ t♦ t❤❡ ✉♥❝❡rt❛✐♥t② ♦❢ s✉❜s♣❛❝❡ ♠❛tr✐① H(k) ✭s❡❡ ✐♥ ❬✶✶❪✮ vec∆U1 = L1d B†1C1 ✳✳✳ Bd†Cd vec∆H (k) ✭✷✺✮ ✇✐t❤ ❛ s❡❧❡❝t✐♦♥ ♠❛tr✐① ❞❡✜♥❡❞ ❜② L1d = Id⊗ I(p+1)r O(p+1)r×qr ✭✷✻✮ ❛♥❞ Bj= " I(p+1)r −H (k) σj −(H(k)σj)T Iqr # , ✭✷✼✮ Cj = 1 σj vT j ⊗ (I(p+1)r− ujuTj) (uT j ⊗ (Iqr− vjvjT))P , ✭✷✽✮ P = (p+1)r X k1=1 qr X k2=1 E(p+1)r×qrk1k2 ⊗ Ekqr×(p+1)r2k1 , ✭✷✾✮ ✇❤❡r❡ σj ✐s t❤❡ ❡✐❣❡♥✈❛❧✉❡ ❛t s②st❡♠ ♦r❞❡r j {j ∈ 1, . . . , d}✱ uj ✭r❡s♣✳ vj✮ ✐s ❝♦❧✉♠♥ ♥✉♠❜❡r j ♦❢ U ✭r❡s♣✳ V ✮✳ E(p+1)r×qr k1k2 ✐s ❛ (p + 1)r × qr ♠❛tr✐① ✇❤♦s❡ ❡❧❡♠❡♥t ✐s ✶ ❛t ♣♦s✐t✐♦♥ (k1, k2)❛♥❞ ③❡r♦ ❡❧s❡✇❤❡r❡✳ ❚❤❡ s❡♥s✐t✐✈✐t② ♦❢ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛❞❞r❡ss❡❞ ❛s ✭s❡❡ ✐♥ ❬✶✶❪✮ vec(∆Σ1) = S3d (v1⊗ u1)T ✳✳✳ (vd⊗ ud)T vec∆H (k), ✭✸✵✮ ❘❘ ♥➦ ✼✹✻✷
❯♥❝❡rt❛✐♥t② q✉❛♥t✐✜❝❛t✐♦♥ ✾ ✐♥ ✇❤✐❝❤ S3d✐s ❛ s❡❧❡❝t✐♦♥ ♠❛tr✐① S3d = d X s=1 E(s−1)d+s,sd2×d ✭✸✶✮ ❚❤❡ s❡♥s✐t✐✈✐t② ♦❢ r✐❣❤t ❡✐❣❡♥✈❡❝t♦rs ✭s❡❡ ✐♥ ❬✶✶❪✮ ✐s t❤❡♥ s♣❡❝✐✜❡❞ ❜② vec∆V1 = L2d B1†C1 ✳✳✳ B†dCd vec∆H (k) ✭✸✷✮ ✇✐t❤ s❡❧❡❝t✐♦♥ ♠❛tr✐① L2d = Id⊗ Oqr×(p+1)r Iqr . ✭✸✸✮ ❊s♣❡❝✐❛❧❧②✱ vec∆H(k) ❝❛♥ ❜❡ s✐♠♣❧✐✜❡❞ ❜② ♠❛❦✐♥❣ ✉s❡ ♦❢ ❛ ❜❧♦❝❦✲st♦r✐♥❣ ♠❛tr✐① M(k) vec∆H(k) = S4(k)vec∆M(k) ✭✸✹✮ ✇❤❡r❡ M(k) = R1 R2 ✳✳✳ Rp+q+k ✭✸✺✮ S(k)4 = Ir⊗ S(k)5,1 Ir⊗ S(k)5,2 ✳✳✳ Ir⊗ S(k)5,q ✭✸✻✮ S5,t(k) = O(p+1)r×(t−1+k)r I(p+1)r O(p+1)r×(q−t)r ✭✸✼✮ ❋✐♥❛❧❧②✱ t❤❡ ✉♥❝❡rt❛✐♥t② ♦❢ s②st❡♠ ♠❛tr✐① A ❝❛♥ ❜❡ s❤♦✇♥ ✐♥ ✈❡❝t♦r✐③❛t✐♦♥ ❢♦r♠ vec∆A = JAvec∆M(k), ✭✸✽✮ ✐♥ ✇❤✐❝❤ JA✐s ❛ ❏❛❝♦❜✐❛♥ ♠❛tr✐① JA = N1 (v1⊗ u1)T ✳✳✳ (vd⊗ ud)T S (k) 4 +N2 B1†C1 ✳✳✳ Bd†Cd S (k) 4 +Z1† T ⊗ O1† S4(k+1) ✭✸✾✮
❯♥❝❡rt❛✐♥t② q✉❛♥t✐✜❝❛t✐♦♥ ✶✵ ✇✐t❤ t❤❡ ♠❛tr✐❝❡s N1 = Z1† T H(k+1)T⊗ Id U1⊗ −12Σ−32 1 S3d +Id⊗ O†1H(k+1) Id⊗ −12V1Σ −3 2 1 S3d, ✭✹✵✮ N2 = Z1† T H(k+1)T⊗ Id I(p+1)r⊗ Σ −1 2 1 PU1 L1d +Id⊗ O†1H (k+1) Σ−1 2 1 ⊗ Iqr L2d. ✭✹✶✮ ▲✐❦❡✇✐s❡✱ t❤❡ ✉♥❝❡rt❛✐♥t② ♦❢ t❤❡ ✈❡❝t♦r✐③❡❞ s②st❡♠ ♠❛tr✐① C ✐s vec∆C = JC vec∆M(k) ✭✹✷✮ ✇✐t❤ ❏❛❝♦❜✐❛♥ ♠❛tr✐① JC = (Id⊗ SC) (Bd+ Cd) S4(k), ✭✹✸✮ ✇❤❡r❡ SC = Ir Or×pr , ✭✹✹✮ Bd = Id⊗ 1 2U1Σ −1 2 1 S3d (v1⊗ u1)T ✳✳✳ (vd⊗ ud)T ✭✹✺✮ Cd = Σ12 1 ⊗ I(p+1)r L1d B1†C1 ✳✳✳ Bd†Cd . ✭✹✻✮ ❋✐♥❛❧❧②✱ t❤❡ ✉♥❝❡rt❛✐♥t② ♦❢ s②st❡♠ ♠❛tr✐❝❡s ❝❛♥ ❜❡ ❥♦✐♥❡❞ t♦❣❡t❤❡r vec∆A vec∆C = JA JC vec∆M(k) = JA,C vec∆M(k) ✭✹✼✮ ❚❤❡♥✱ t❤❡ ❝♦✈❛r✐❛♥❝❡s ♦❢ t❤❡ ♠♦❞❛❧ ♣❛r❛♠❡t❡rs ❛r❡ ♦❜t❛✐♥❡❞ ❛s cov(fµ) = Jfµ JA,C cov(vec M
(k)) JT A,C JfTµ
cov(dµ) = Jdµ JA,C cov(vec M
(k)) JT A,C JdTµ
cov(φµ) = Jφµ JA,C cov(vec M
(k)) JT
A,C JφTµ ✭✹✽✮
❯♥❝❡rt❛✐♥t② q✉❛♥t✐✜❝❛t✐♦♥ ✶✶
✹ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s
❚❤❡ ❙✶✵✶ ❜r✐❞❣❡ ✭❬✶✸❪✮ ❝♦♥♥❡❝t❡❞ ❙❛❧③❜✉r❣ ✲ ❱✐❡♥♥❛ ❝❛rr✐❛❣❡ ✇❛② ✐♥ ❆✉str✐❛✳ ❚❤❛t ✐s ❛ ♣♦st✲t❡♥s✐♦♥❡❞ ❝♦♥❝r❡t❡ ❜r✐❞❣❡ ✇✐t❤ ♠❛✐♥ s♣❛♥ ♦❢ ✸✷ ♠✱ s✐❞❡ s♣❛♥s ♦❢ ✶✷ ♠✱ ❛♥❞ t❤❡ ✇✐❞t❤ ♦❢ ✻✳✻ ♠✳ ❚❤❡ ❞❡❝❦ ✐s ❝♦♥t✐♥✉♦✉s ♦✈❡r t❤❡ ♣✐❡rs✳ ❚❤✐s ❜r✐❞❣❡✱ ❝♦♥tr✉❝t❡❞ ✐♥ ✶✾✻✵✱ ❤❛s ❜❡❡♥ ❛ t②♣✐❝❛❧ ♦✈❡r♣❛ss ❜r✐❞❣❡ ✐♥ ❆✉str✐❛ ♥❛t✐♦♥❛❧ ❤✐❣❤✇❛②✳ ■♥ t❤❡ ❝✉rr❡♥t ♣❛♣❡r✱ t❤❡ ❛♠❜✐❡♥t ✈✐❜r❛t✐♦♥ ❞❛t❛ ✐s ❝♦❧❧❡❝t❡❞ ♦♥ ✶✺ s❡♥s♦rs✳ ❚❤❡ ♦r✐❣✐♥❛❧ s❛♠♣❧✐♥❣ ❢r❡q✉❡♥❝② ✐s ✺✵✵ ❍③ ✇✐t❤ ✶✻✺✵✵✵ t✐♠❡ s❛♠♣❧❡s ❛✈❛✐❧❛❜❧❡✳ ❚❤❡ ❞❛t❛ ✐s ❞❡❝✐♠❛t❡❞ t♦ ✸✺✳✼ ❍③ ❛♥❞ ♦♥❧② ✜✈❡ ♠♦❞❡s ❛r❡ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t✳ ❋✐❣✉r❡ ✶✿ ❙✶✵✶ ❜r✐❞❣❡ ✐♥ ❘❡✐❜❡rs❞♦r❢✱ ❆✉str✐❛✹✳✶ ▼♦❞❛❧ ❛♥❛❧②s✐s
❋♦r t❤❡ ♦✉t♣✉t✲♦♥❧② ♠♦❞❛❧ ❛♥❛❧②s✐s ♦❢ t❤❡ ❛♠❜✐❡♥t ✈✐❜r❛t✐♦♥ ❞❛t❛ ♦❢ t❤❡ ❙✶✵✶ ❜r✐❞❣❡✱ s✐♠✐❧❛r ♣❛r❛♠❡t❡rs ❢♦r ❜♦t❤ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠ ❛♥❞ ❊❘❆ ❛r❡ ❡♠♣❧♦②❡❞✳ ✻✹ ❝♦rr❡❧❛t✐♦♥s ✭p + 1 = q = 32✮ ❛r❡ ✉s❡❞✱ ❧❡❛❞✐♥❣ t♦ ❍❛♥❦❡❧ ♠❛tr✐❝❡s ✇✐t❤ ✸✷ ❜❧♦❝❦ r♦✇s ❛♥❞ ❝♦❧✉♠♥s✳ ❚❤❡ r❡s✉❧t✐♥❣ ♠✉❧t✐✲♦r❞❡r ❞✐❛❣r❛♠s ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❋✐❣✉r❡ ✷ ❛♥❞ ❋✐❣✉r❡ ✸✳❯♥❝❡rt❛✐♥t② q✉❛♥t✐✜❝❛t✐♦♥ ✶✷ ❋✐❣✉r❡ ✷✿ ❙t❛❜✐❧✐③❛t✐♦♥ ❞✐❛❣r❛♠ ✇✐t❤ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠ ✭♥❛t✉r❛❧ ❢r❡q✉❡♥❝② ✈s✳ ♠♦❞❡❧ ♦r❞❡r✮ ❋✐❣✉r❡ ✸✿ ❙t❛❜✐❧✐③❛t✐♦♥ ❞✐❛❣r❛♠ ✇✐t❤ ❊❘❆ ✭♥❛t✉r❛❧ ❢r❡q✉❡♥❝② ✈s✳ ♠♦❞❡❧ ♦r❞❡r✮ ❚❤❡ s✉♠♠❛r② ♦❢ t❤❡ ❢r❡q✉❡♥❝✐❡s ❛♥❞ ❞❛♠♣✐♥❣ r❛t✐♦s ♦❢ t❤❡ ✜✈❡ ✐❞❡♥t✐✜❡❞ ♠♦❞❡s ✐s ❣✐✈❡♥ ✐♥ ❚❛❜❧❡ ✶ ❛♥❞ ❚❛❜❧❡ ✷✱ ❢♦r ❜♦t❤ s✉❜s♣❛❝❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ❛♥❞ ❊❘❆✳ ❊❘❆✲✢ ✐s t❤❡ ❊❘❆ ✇❤✐❝❤ ✉s❡s ✜rst✲❧❛❣ ❛♥❞ s❡❝♦♥❞✲❧❛❣ s✉❜s♣❛❝❡ ♠❛tr✐❝❡s✱ ❊❘❆✲s❧ ✐s t❤❡ ❊❘❆ ✇❤✐❝❤ ✉t✐❧✐③❡s s❡❝♦♥❞✲❧❛❣ ❛♥❞ t❤✐r❞✲❧❛❣ s✉❜s♣❛❝❡ ♠❛tr✐❝❡s✱ ❊❘❆✲t❧ ✐s t❤❡ ❊❘❆ ✇❤✐❝❤ ❡♠♣❧♦②s t❤✐r❞✲❧❛❣ ❛♥❞ ❢♦✉rt❤✲❧❛❣ s✉❜s♣❛❝❡ ♠❛tr✐❝❡s✳ ❚❤❡ ❞✐✛❡r❡♥❝❡s ✐♥ t❤❡ ♦❜t❛✐♥❡❞ ❢r❡q✉❡♥❝✐❡s ❜❡t✇❡❡♥ s✉❜s♣❛❝❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ❛♥❞ ❊❘❆ ❛r❡ s♠❛❧❧✱ ❧❡ss t❤❛♥ ✵✳✺%✱ ❢♦r ❛❧❧ ✜✈❡ ♠♦❞❡s✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ❞❛♠♣✐♥❣ r❛t✐♦s✱ t❤❡ ❞✐✛❡r❡♥❝❡s ❛r❡ ❜✐❣❣❡r ❜❡❝❛✉s❡ ♦❢ ❤✐❣❤❡r ✉♥❝❡rt❛✐♥t② ✐♥ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ ❞❛♠♣✐♥❣ r❛t✐♦s✳ ❘❘ ♥➦ ✼✹✻✷
❯♥❝❡rt❛✐♥t② q✉❛♥t✐✜❝❛t✐♦♥ ✶✸ ▼♦❞❡ ❋r❡q✉❡♥❝② ❢ ✭❍③✮ ❙✉❜s♣❛❝❡ ❊❘❆✲✢ ❊❘❆✲s❧ ❊❘❆✲t❧ ✶ ✹✳✵✸✾ ✹✳✵✸✽ ✹✳✵✸✼ ✹✳✵✸✼ ✷ ✻✳✷✽✷ ✻✳✷✽✸ ✻✳✷✽✹ ✻✳✷✽✸ ✸ ✾✳✻✽✷ ✾✳✻✽✹ ✾✳✻✽✸ ✾✳✻✽✹ ✹ ✶✸✳✷✽✹ ✶✸✳✷✽✸ ✶✸✳✷✾✵ ✶✸✳✸✵✼ ✺ ✶✺✳✼✷✶ ✶✺✳✼✷✵ ✶✺✳✼✻✸ ✶✺✳✻✸✵ ❚❛❜❧❡ ✶✿ ■❞❡♥t✐✜❡❞ ❢r❡q✉❡♥❝✐❡s ✇✐t❤ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠ ❛♥❞ ❊❘❆ ▼♦❞❡ ❉❛♠♣✐♥❣ ❘❛t✐♦ ❞ ✭✪✮ ❙✉❜s♣❛❝❡ ❊❘❆✲✢ ❊❘❆✲s❧ ❊❘❆✲t❧ ✶ ✵✳✼✺✾ ✵✳✼✺✹ ✵✳✼✻✷ ✵✳✼✺✻ ✷ ✵✳✻✶✼ ✵✳✻✵✽ ✵✳✺✽✽ ✵✳✺✼✸ ✸ ✶✳✶✾✸ ✶✳✶✽✾ ✶✳✶✽✺ ✶✳✷✷✽ ✹ ✶✳✹✸✻ ✶✳✹✷✹ ✶✳✸✵✾ ✶✳✶✸✺ ✺ ✶✳✻✸✽ ✶✳✾✻✻ ✷✳✸✻✵ ✷✳✹✼✺ ❚❛❜❧❡ ✷✿ ■❞❡♥t✐✜❡❞ ❞❛♠♣✐♥❣ r❛t✐♦s ✇✐t❤ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠ ❛♥❞ ❊❘❆
✹✳✷ ❈♦♥✜❞❡♥❝❡ ■♥t❡r✈❛❧s
❋♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ♦♥ ♠♦❞❛❧ ♣❛r❛♠❡t❡rs✱ ✷✹ t✐♠❡ ❧❛❣s✱ ❧❡❛❞✐♥❣ t♦ p+1 = q = 12✱ ❛♥❞ ✹✵ ♠♦❞❡❧ ♦r❞❡rs ❛r❡ ✉t✐❧✐③❡❞ ❞✉❡ t♦ t❤❡ ❧✐♠✐t❛t✐♦♥ ✐♥ ❝♦♠♣✉t❡r ♠❡♠♦r②✳ ▼♦❞❡ ❋r❡q✉❡♥❝② ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ✭✪✮ ❙✉❜s♣❛❝❡ ❊❘❆✲✢ ❊❘❆✲s❧ ❊❘❆✲t❧ ✶ ✵✳✶✶✺ ✵✳✶✶✵ ✵✳✶✷✷ ✵✳✶✵✵ ✷ ✵✳✵✾✷ ✵✳✵✾✸ ✵✳✵✾✷ ✵✳✵✽✾ ✸ ✵✳✶✸✸ ✵✳✶✸✹ ✵✳✶✺✻ ✵✳✶✺✾ ✹ ✵✳✹✼✶ ✵✳✸✵✵ ✵✳✷✹✼ ✵✳✺✼✸ ✺ ✶✳✶✹✽ ✶✳✽✷✺ ✷✳✹✽✺ ✶✳✹✹✷ ❚❛❜❧❡ ✸✿ ❋r❡q✉❡♥❝② ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ✇✐t❤ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠ ❛♥❞ ❊❘❆ ▼♦❞❡ ❉❛♠♣✐♥❣✲r❛t✐♦ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ✭✪✮ ❙✉❜s♣❛❝❡ ❊❘❆✲✢ ❊❘❆✲s❧ ❊❘❆✲t❧ ✶ ✶✼✳✼✾✶ ✶✼✳✽✶✾ ✶✼✳✼✺✼ ✶✻✳✸✺✷ ✷ ✶✾✳✸✽✸ ✶✾✳✷✸✾ ✷✵✳✹✵✶ ✶✾✳✾✵✵ ✸ ✶✻✳✽✹✺ ✶✸✳✸✵✷ ✶✶✳✸✸✷ ✶✶✳✾✻✵ ✹ ✷✽✳✾✶✾ ✶✺✳✵✶✷ ✺✻✳✼✼✼ ✻✺✳✵✶✾ ✺ ✻✵✳✺✷✷ ✼✵✳✽✵✵ ✶✻✶✳✼✹✾ ✶✸✵✳✻✾✶ ❚❛❜❧❡ ✹✿ ❉❛♠♣✐♥❣✲r❛t✐♦ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ✇✐t❤ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠ ❛♥❞ ❊❘❆ ■♥ ❚❛❜❧❡ ✸ ❛♥❞ ❚❛❜❧❡ ✹✱ t❤❡ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ♦❢ t❤❡ ♥❛t✉r❛❧ ❢r❡q✉❡♥❝✐❡s ❛♥❞ ❞❛♠♣✐♥❣ r❛t✐♦s ♦❢ t❤❡ ✜✈❡ ♠♦❞❡s ❛r❡ ♣r❡s❡♥t❡❞✱ r❡s♣❡❝t✐✈❡❧②✳ ❈♦♥✜❞❡♥❝❡❯♥❝❡rt❛✐♥t② q✉❛♥t✐✜❝❛t✐♦♥ ✶✹ ✐♥t❡r✈❛❧s ♦❢ t❤❡ ❢r❡q✉❡♥❝✐❡s ❛r❡ ♠✉❝❤ s♠❛❧❧❡r t❤❛♥ t❤♦s❡ ♦❢ ❞❛♠♣✐♥❣ r❛t✐♦s✳ ❚❤✐s ✐s ❝♦❤❡r❡♥t ✇✐t❤ st❛t✐st✐❝❛❧ t❤❡♦r②✱ s✐♥❝❡ t❤❡ ❧♦✇❡r ❜♦✉♥❞ ♦❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ❣✐✈❡♥ ❜② ❋✐s❤❡r ✐♥❢♦r♠❛t✐♦♥ ♠❛tr✐① ✐s s♠❛❧❧❡r ❢♦r t❤❡ ❢r❡q✉❡♥❝✐❡s t❤❛♥ ❢♦r t❤❡ ❞❛♠♣✐♥❣ r❛t✐♦s✳ ❇❡s✐❞❡s✱ ❢♦r t❤✐s ❛♣♣❧✐❝❛t✐♦♥✱ ❝♦♥✜❞❡♥❝❡ ❜♦✉♥❞s ♦♥ ♠♦❞❛❧ ♣❛r❛♠❡t❡rs ♦❢ ❊❘❆ ❛r❡ r❡❧❛t✐✈❡❧② s✐♠✐❧❛r ❛s t❤♦s❡ ♦❜t❛✐♥❡❞ ✇✐t❤ t❤❡ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠✳ ❲❤✐❧❡ s❤✐❢t✐♥❣ t❤❡ ❧❛❣s ♦❢ ❊❘❆✱ t❤❡ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧ ✢✉❝t✉❛t✐♦♥s s❡❡♠ t♦ ❜❡ st❛❜❧❡✱ ❛♥❞ t❤❡ ❊❘❆✲✢ ♠❛② s✉♣♣❧② t❤❡ ♠♦st ❝♦♠♣❛r❛❜❧❡ r❡s✉❧ts t♦ s✉❜s♣❛❝❡ ✐❞❡♥t✐✜❝❛t✐♦♥✳
✺ ❈♦♥❝❧✉s✐♦♥s
■♥ t❤✐s ♣❛♣❡r✱ t❤❡ ♦✉t♣✉t✲♦♥❧② s②st❡♠ ✐❞❡♥t✐✜❝❛t✐♦♥ ❛♥❞ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ♦♥ ♠♦❞❛❧ ♣❛r❛♠❡t❡rs ❛r❡ ❞❡r✐✈❡❞ ❛♥❞ ✐♠♣❧❡♠❡♥t❡❞ ❢♦r ❜♦t❤ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠ ❛♥❞ ❊❘❆✳ ❆❧❧ t❤❡ ♠❡t❤♦❞s ✇❡r❡ s✉❝❝❡ss❢✉❧❧② ❛♣♣❧✐❡❞ ❛♥❞ t❡st❡❞ ♦♥ t❤❡ ❛♠❜✐❡♥t ✈✐❜r❛t✐♦♥ ❞❛t❛ ♦❢ t❤❡ ❙✶✵✶ ♦✈❡r♣❛ss ❜r✐❞❣❡✳ ❚❤❡ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠ ❛♥❞ ❊❘❆ ❣✐✈❡ ❝♦♠♣❛r❛❜❧❡ r❡s✉❧ts✳ ❚❤❡ q✉❛❧✐t② ♦❢ st❛❜✐❧✐③❛t✐♦♥ ❞✐❛❣r❛♠s ❛s ✇❡❧❧ ❛s ❢r❡q✉❡♥❝✐❡s ♦❢ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠ ❛♥❞ ❊❘❆ ❛r❡ ❛❧♠♦st s✐♠✐❧❛r✳ ❚❤❡ ❞❛♠♣✐♥❣ r❛t✐♦s ❛r❡ s❧✐❣❤t❧② ❞✐✛❡r❡♥t ❞✉❡ t♦ ❛♥ ❡①♣❡❝t❡❞❧② ❤✐❣❤❡r ✉♥❝❡rt❛✐♥t② ♦♥ ❡st✐♠❛t✐♦♥✳ ❚❤❡ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ♦♥ ♠♦❞❛❧ ♣❛r❛♠❡t❡rs ❛r❡ ❛❧s♦ ❝♦♠♣✉t❡❞✳ ■t ✐s ♦❜✲ s❡r✈❡❞ t❤❛t t❤❡ ✉♥❝❡rt❛✐♥t② ❢♦r ❊❘❆ ✐s r❡❧❛t✐✈❡❧② s✐♠✐❧❛r ✇✐t❤ t❤❛t ❛ss♦❝✐❛t❡❞ t♦ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠✳ ❲❤❡♥ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❧❛❣ ❡✛❡❝t ❢♦r ❊❘❆✱ ❊❘❆✲✢ s❡❡♠s t♦ ❜❡ t❤❡ ♠♦st r❡❛s♦♥❛❜❧❡ s❡❧❡❝t✐♦♥ ❢♦r t❤❡ ❞❡s✐❣♥❡rs ❞❡❛❧✐♥❣ ✇✐t❤ ❊❘❆✳ ❘❘ ♥➦ ✼✹✻✷❯♥❝❡rt❛✐♥t② q✉❛♥t✐✜❝❛t✐♦♥ ✶✺
❈♦♥t❡♥ts
✶ ■♥tr♦❞✉❝t✐♦♥ ✸ ✷ ❙t♦❝❤❛st✐❝ ❙②st❡♠ ■❞❡♥t✐✜❝❛t✐♦♥ ✹ ✷✳✶ ❚❤❡ ●❡♥❡r❛❧ ❙❙■ ❆❧❣♦r✐t❤♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✷ ❊❘❆ ✭❊✐❣❡♥s②st❡♠✲❘❡❛❧✐③❛t✐♦♥✲❆❧❣♦r✐t❤♠✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✸ ❈♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ✻ ✸✳✶ ❉❡s❝r✐♣t✐♦♥s ♦❢ ❙❙■ ❈♦♥✜❞❡♥❝❡ ■♥t❡r✈❛❧s ❛❧❣♦r✐t❤♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✸✳✷ ❉❡r✐✈❛t✐♦♥ ♦❢ ❊❘❆ ❈♦♥✜❞❡♥❝❡ ■♥t❡r✈❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✹ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ✶✶ ✹✳✶ ▼♦❞❛❧ ❛♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✹✳✷ ❈♦♥✜❞❡♥❝❡ ■♥t❡r✈❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✺ ❈♦♥❝❧✉s✐♦♥s ✶✹❘❡❢❡r❡♥❝❡s
❬✶❪ ❉✳ ❇❛✉❡r✱ ▼✳ ❉❡✐st❧❡r✱ ❛♥❞ ❲✳ ❙❝❤❡rr❡✳ ❈♦♥s✐st❡♥❝② ❛♥❞ ❛s②♠t♦t✐❝ ♥♦r✲ ♠❛❧✐t② ♦❢ s♦♠❡ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠s ❢♦r s②st❡♠s ✇✐t❤♦✉t ♦❜s❡r✈❡❞ ✐♥♣✉ts✳ ❆✉t♦♠❛t✐❝❛✱ ✸✺✿✶✷✹✸✕✶✷✺✹✱ ✶✾✾✾✳ ❬✷❪ ❉✳ ❇❛✉❡r ❛♥❞ ▼✳ ❏❛♥ss♦♥✳ ❆♥❛❧②s✐s ♦❢ t❤❡ ❛s②♠t♦t✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♠♦❡s♣ t②♣❡ ♦❢ s✉❜s♣❛❝❡ ❛❧❣♦r✐t❤♠s✳ ❆✉t♦♠❛t✐❝❛✱ ✸✻✿✹✾✼✕✺✵✾✱ ✷✵✵✵✳ ❬✸❪ ❆✳ ❇❡♥✈❡♥✐st❡ ❛♥❞ ❏✳✲❏✳ ❋✉❝❤s✳ ❙✐♥❣❧❡ s❛♠♣❧❡ ♠♦❞❛❧ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ❛ ♥♦♥✲ st❛t✐♦♥❛r② st♦❝❤❛st✐❝ ♣r♦❝❡ss✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❆✉t♦♠❛t✐❝ ❈♦♥tr♦❧✱ ❆❈✲✸✵✭✶✮✿✻✻✕✼✹✱ ✶✾✽✺✳ ❬✹❪ ❆✳ ❇❡♥✈❡♥✐st❡ ❛♥❞ ▲✳ ▼❡✈❡❧✳ ◆♦♥✲st❛t✐♦♥❛r② ❝♦♥s✐st❡♥❝② ♦❢ s✉❜s♣❛❝❡ ♠❡t❤✲ ♦❞s✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❆✉t♦♠❛t✐❝ ❈♦♥tr♦❧✱ ❆❈✲✺✷✭✻✮✿✾✼✹✕✾✽✹✱ ✷✵✵✼✳ ❬✺❪ ❆✳ ❈❤✐✉s♦ ❛♥❞ ●✳ P✐❝❝✐✳ ❆s②♠♣t♦t✐❝ ✈❛r✐❛♥❝❡ ♦❢ s✉❜s♣❛❝❡ ♠❡t❤♦❞s ❜② ❞❛t❛ ♦rt❤♦❣♦♥❛❧✐③❛t✐♦♥ ❛♥❞ ♠♦❞❡❧ ❞❡❝♦✉♣❧✐♥❣✿ ❛ ❝♦♠♣❛r❛t✐✈❡ ❛♥❛❧②s✐s✳ ❆✉t♦✲ ♠❛t✐❝❛✱ ✹✵✿✶✼✵✺✕✶✼✶✼✱ ✷✵✵✹✳ ❬✻❪ ▼✳ ❉❡✐st❧❡r✱ ❑✳ P❡t❡r♥❡❧❧✱ ❛♥❞ ❲✳ ❙❝❤❡rr❡r✳ ❈♦♥s✐st❡♥❝② ❛♥❞ r❡❧❛t✐✈❡ ❡✣✲ ❝✐❡♥❝② ♦❢ s✉❜s♣❛❝❡ ♠❡t❤♦❞s✳ ❆✉t♦♠❛t✐❝❛✱ ✸✶✿✶✽✻✺✕✶✽✼✺✱ ✶✾✾✺✳ ❬✼❪ ▼✳ ❉ö❤❧❡r✱ ❳✳✲❇✳ ▲❛♠✱ ❛♥❞ ▲✳ ▼❡✈❡❧✳ ❈♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ♦♥ ♠♦❞❛❧ ♣❛✲ r❛♠❡t❡rs ✐♥ st♦❝❤❛st✐❝ s✉❜s♣❛❝❡ ✐❞❡♥t✐✜❝❛t✐♦♥✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✸✹t❤ ■❆❇❙❊ ❙②♠♣♦s✐✉♠✱ ❱❡♥✐❝❡✱ ■t❛❧②✱ ❙❡♣t❡♠❜❡r ✷✵✶✵✳ ❬✽❪ ❏✳✲◆✳ ❏✉❛♥❣ ❛♥❞ ❘✳ ❙✳ P❛♣♣❛✳ ❆♥ ❡✐❣❡♥s②st❡♠ r❡❛❧✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r ♠♦❞❛❧ ♣❛r❛♠❡t❡r ✐❞❡♥t✐✜❝❛t✐♦♥ ❛♥❞ ♠♦❞❡❧ r❡❞✉❝t✐♦♥✳ ❏♦✉r♥❛❧ ♦❢ ●✉✐❞❛♥❝❡✱ ❈♦♥tr♦❧ ❛♥❞ ❉②♥❛♠✐❝s✱ ✽✭✺✮✿✻✷✵✕✻✷✼✱ ✶✾✽✺✳ ❬✾❪ ❲✳ ❊✳ ▲❛r✐♠♦r❡✳ ❙②st❡♠ ✐❞❡♥t✐✜❝❛t✐♦♥✱ r❡❞✉❝❡❞ ♦r❞❡r ✜❧t❡rs ❛♥❞ ♠♦❞❡❧❧✐♥❣ ✈✐❛ ❝❛♥♦♥✐❝❛❧ ✈❛r✐❛t❡ ❛♥❛❧②s✐s✳ ■♥ t❤❡ ❆♠❡r✐❝❛♥ ❈♦♥tr♦❧ ❈♦♥❢❡r❡♥❝❡✱ ♣❛❣❡s ✹✹✺✕✹✺✶✱ ✶✾✽✸✳❯♥❝❡rt❛✐♥t② q✉❛♥t✐✜❝❛t✐♦♥ ✶✻ ❬✶✵❪ ❇✳ P❡❡t❡rs ❛♥❞ ●✳ ❉❡ ❘♦❡❝❦✳ ❘❡❢❡r❡♥❝❡✲❜❛s❡❞ st♦❝❤❛st✐❝ s✉❜s♣❛❝❡ ✐❞❡♥✲ t✐✜❝❛t✐♦♥ ❢♦r ♦✉t♣✉t✲♦♥❧② ♠♦❞❛❧ ❛♥❛❧②s✐s✳ ▼❡❝❤❛♥✐❝❛❧ ❙②st❡♠s ❛♥❞ ❙✐❣♥❛❧ Pr♦❝❡ss✐♥❣✱ ✶✸✭✻✮✿✽✺✺✕✽✼✽✱ ◆♦✈❡♠❜❡r ✶✾✾✾✳ ❬✶✶❪ ❘✳ P✐♥t❡❧♦♥✱ P✳ ●✉✐❧❧❛✉♠❡✱ ❛♥❞ ❏✳ ❙❝❤♦✉❦❡♥s✳ ❯♥❝❡rt❛✐♥t② ❝❛❧❝✉❧❛t✐♦♥ ✐♥ ✭♦♣❡r❛t✐♦♥❛❧✮ ♠♦❞❛❧ ❛♥❛❧②s✐s✳ ▼❡❝❤❛♥✐❝❛❧ ❙②st❡♠s ❛♥❞ ❙✐❣♥❛❧ Pr♦❝❡ss✐♥❣✱ ✷✶✿✷✸✺✾✕✷✸✼✸✱ ✷✵✵✼✳ ❬✶✷❪ ❊✳ ❘❡②♥❞❡rs✱ ❘✳ P✐♥t❡❧♦♥✱ ❛♥❞ ●✳ ❉❡ ❘♦❡❝❦✳ ❯♥❝❡rt❛✐♥t② ❜♦✉♥❞s ♦♥ ♠♦❞❛❧ ♣❛r❛♠❡t❡rs ♦❜t❛✐♥❡❞ ❢r♦♠ st♦❝❤❛st✐❝ s✉❜s♣❛❝❡ ✐❞❡♥t✐✜❝❛t✐♦♥✳ ▼❡❝❤❛♥✐❝❛❧ ❙②st❡♠s ❛♥❞ ❙✐❣♥❛❧ Pr♦❝❡ss✐♥❣✱ ✷✷✭✹✮✿✾✹✽✕✾✻✾✱ ✷✵✵✽✳ ❬✶✸❪ ❉✳ ▼✳ ❙✐r✐♥❣♦r✐♥❣♦✱ ❚✳ ◆❛❣❛②❛♠❛✱ ❨✳ ❋✉❥✐♥♦✱ ❉✳ ❙✉✱ ❛♥❞ ❈✳ ❚❛♥❞✐❛♥✳ ❖❜✲ s❡r✈❡❞ ❞②♥❛♠✐❝ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❛♥ ♦✈❡r♣❛ss ❜r✐❞❣❡ ❞✉r✐♥❣ ❛ ❢✉❧❧✲s❝❛❧❡ ❞❡str✉❝t✐✈❡ t❡st✐♥❣✳ ■♥ ❚❤❡ ❋✐❢t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❇r✐❞❣❡ ▼❛✐♥✲ t❡♥❛♥❝❡✱ ❙❛❢❡t②✱ ▼❛♥❛❣❡♠❡♥t ❛♥❞ ▲✐❢❡✲❈②❝❧❡ ❖♣t✐♠✐③❛t✐♦♥✱ ✷✵✶✵✳ ❬✶✹❪ P✳ ❱❛♥ ❖✈❡rs❝❤❡❡ ❛♥❞ ❇✳ ❉❡ ▼♦♦r✳ ❙✉❜s♣❛❝❡ ■❞❡♥t✐✜❝❛t✐♦♥ ❢♦r ▲✐♥❡❛r ❙②st❡♠s✿ ❚❤❡♦r②✱ ■♠♣❧❡♠❡♥t❛t✐♦♥✱ ❆♣♣❧✐❝❛t✐♦♥s✳ ❑❧✉✇❡r✱ ✶✾✾✻✳ ❘❘ ♥➦ ✼✹✻✷
Centre de recherche INRIA Rennes – Bretagne Atlantique IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)
Centre de recherche INRIA Bordeaux – Sud Ouest : Domaine Universitaire - 351, cours de la Libération - 33405 Talence Cedex Centre de recherche INRIA Grenoble – Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier Centre de recherche INRIA Lille – Nord Europe : Parc Scientifique de la Haute Borne - 40, avenue Halley - 59650 Villeneuve d’Ascq
Centre de recherche INRIA Nancy – Grand Est : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex
Centre de recherche INRIA Paris – Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex Centre de recherche INRIA Saclay – Île-de-France : Parc Orsay Université - ZAC des Vignes : 4, rue Jacques Monod - 91893 Orsay Cedex
Centre de recherche INRIA Sophia Antipolis – Méditerranée : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex
Éditeur
INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)
❤tt♣✿✴✴✇✇✇✳✐♥r✐❛✳❢r ISSN 0249-6399
