Generic Asymptotics of Eigenvalues and Min-Plus Algebra
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Generic Asymptotics of Eigenvalues and Min-Plus Algebra Marianne Akian — Ravindra Bapat — Stéphane Gaubert. N° 5104 February 2004. ISSN 0249-6399. ISRN INRIA/RR--5104--FR+ENG. THÈME 4. apport de recherche.
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(7) . . . Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France) Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30.
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(37) H. B
(38) @$ 9
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(48) > / 2 ! c ! H c0 ∩ N = c c0 G(A) &tycdo/T®¯_X°hvo w ½¶] \^\h _ ¥. ®y8X° . ¨ ¡ ]r]"j. . ρmin (Schur(C, λ, A)) = =. VU. . 0. ®y8P°. . . CC. |c |A − λ|c |C |c0 | − |c0 |C c0 G(A) # >! . 9 $ U 0 C Schur(C, c * λ, A). ρmin (Schur(C, λ, A)) = min. . L×L. c. min . c. . N. . . |c|Schur(C,λ,A) |c| min 0. . |c0 |A − λ|c0 |C N c |c| G(A), c0 ∩N =c 0 0 |c |A − λ|c |C min , |c0 | − |c0 |C G(A), c0 ∩N 6=∅. . min. ut cdoz"] |c ∩ N | = |c | − |c | £¤mr}fhrgng z" cd}l z e/c j t c ¨ YZ/citxEcd] g w t®HP° ¨ .9£ w c cithpz"}lcnjuciz hvgz"cd}lz e/c jmr£ hvo Ex½¶] \^\hT ¨ _ ¥ aEjuZ/cdo]8z"}u]] Schur(C, ρ (Schur(C, )/|c| w ] {scdtyjlthpz"λ,cd} z"A)e/cnj ¥ jucZ] mro £ G(A) tye@z Z¦jlZhkλ,j cA))∩ N= (|c| r h o cdo jlZhkjz hPty] ¥ |c| = |c ∩ N | = |c | − |c | ¥ ¡ =] w c] w e@z"]|c|jlwZhkj c \^cno/cd\^=cn²8]8|ctp|®HP−° ¨ λ|c mr| oE¢P¨ ] } ty]8gnx ¥ cn£ c \^cnocn\^cd² ]8t®HP° >¥ mPjuZ/}u]8]8mko ¢r¥ ]8c} =Exc½∩]8\^N\^hcdt o/8mr_$o] hr\^Phvqscdjo x hvo Xx ½¶] \^\h ¨ _ ¥ c cit$h¦z"cd} z"e/cnjmr£ ¥ ¥ G(Schur(C, λ, A)) ¨ ¨ =. 0. 0. 0. c0. 0. C. min. 0. 0. 0. 0. 0. 0. ρmin (Schur(C, λ, A)) ≤. Schur(C,λ,A). 0. 0. Schur(C,λ,A) 0. C. A. . 0. C. |c|Schur(C,λ,A) |c0 |A − λ|c0 |C ≤ = ρmin (Schur(C, λ, A)) , |c| |c0 | − |c0 |C. jlZEet c cdthz }ucnjuciz hrgz"cd}lz e/c jmr£ Schur(C, λ, A) ¨ ¸¬mvju]µjlZhkjc £ cdthz"cd}lz e/cnjªcdo jlZhkjcitc £jlZ/] w ] omr\^cno@hkjumP}cdo®y8X°«cit² ]8}um ¥ jlZ/]oXe\$]8}lhvjumP} cito/]Hz"]8tltlhv}lcngdxo/mrcoo/] Xhkjucd¢r] ¥ tycdoz"] Cλ¥ ≤ ρ (A ) ¨ *,"-sp'"( S'±f "%'" -/.0("( < %f) ! " º"!$#µ%'"( 0. min. . . CC.
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