Tropical aspects of eigenvalue computation problems Stephane.Gaubert@inria.fr INRIA and CMAP, ´ Ecole Polytechnique S´ eminaire Algo Lundi 11 Janvier 2010 Synthesis of: Akian, Bapat, SG CRAS 2004, arXiv:0402090; SG, Sharify POSTA 09; and current work. . . Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 1 / 51
Tropical / max-plus algebra R max := R ∪ {−∞} equipped with “ a + b ” = max( a , b ) “ ab ” = a + b Tropical algebra is hidden in the three following problems . . . Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 2 / 51
1. Lidski˘ ı, Viˇ sik, Ljusternik perturbation theory Theorem (Lidski˘ ı 65; also Viˇ sik, Ljusternik 60) Let a ∈ C n × n be nilpotent, with m i Jordan blocks of size ℓ i . For a generic perturbation b ∈ C n × n , the matrix a + ǫ b has precisely m i ℓ i eigenvalues of order ǫ 1 /ℓ i as ǫ → 0 . Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 3 / 51
· 1 · · · · · · · · · 1 · · · · · · · · · · · · · · · � � · · · · 1 · · · · · · · · · 1 · · · a = · · · · · · · · · � � · · · · · · · 1 · · · · · · · · · · · · · · · · · · · 6 eigenvalues ∼ ωǫ 1 / 3 , ω 3 = λ , λ eigenvalue of � b 31 � b 34 b 61 b 64 Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 4 / 51
· 1 · · · · · · · · · 1 · · · · · · · · · · · · · · · � � � · · · · 1 · · · · a = · · · · · 1 · · · · · · · · · · · · � � � · · · · · · · 1 · · · · · · · · · · � � � · · · · · · · · · 2 eigenvalues ∼ ωǫ 1 / 2 , ω 2 = λ , � � b 31 � − 1 � b 37 � b 34 � λ = b 87 − b 81 b 84 b 61 b 64 b 67 Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 4 / 51
· 1 · · · · · · · · · 1 · · · · · · · · · · · · · · · � � � � · · · · 1 · · · · · · · · · 1 · · · a = � · · · � · · · � · · � · · · · · · · · 1 · · · · · · · · · · � � � � · · · · · · · · · � � � � 1 eigenvalue ∼ λǫ , − 1 b 31 b 34 b 37 b 39 � � λ = b 99 − b 91 b 94 b 97 b 61 b 64 b 67 b 69 b 81 b 84 b 87 b 89 Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 4 / 51
Lidski˘ ı’s approach does not give the correct orders in degenerate cases. . . If the matrix � b 31 b 34 � b 61 b 64 has a zero-eigenvalue, then, a + ǫ b has less than 6 eigenvalues of order ǫ 1 / 3 . Moreover, the Schur complement � � b 31 b 34 � − 1 � b 37 � � b 87 − b 81 b 84 b 61 b 64 b 67 is not defined, and there may be no eigenvalue of order ǫ 1 / 2 Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 5 / 51
Finding, in general, the correct order of magnitude of all eigenvalues (Puiseux series) ⇐ ⇒ characterizing (combinatorially) the Newton polygon of the curve { ( λ, ǫ ) | det( a + ǫ b − λ I ) = 0 } long standing open problem (see survey Moro, Burke, Overton, SIMAX 97) This talk: tropical algebra yields the correct order of magnitudes, in degenerate cases (new degenerate cases appear but of a higher order). Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 6 / 51
2. Computing the roots of matrix pencils � 1 2 � � − 3 10 � � 12 15 � P ( λ ) = λ 2 10 − 18 +10 − 18 + λ 3 4 16 45 34 28 Apply the QZ algorithmb to the companion form of P ( λ ) Matlab (7.3.0) [similar in Scilab] We get: − Inf , − 7 . 731 e − 19 , Inf , 3 . 588 e − 19 Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 7 / 51
2. Computing the roots of matrix pencils � 1 2 � � − 3 10 � � 12 15 � P ( λ ) = λ 2 10 − 18 +10 − 18 + λ 3 4 16 45 34 28 Apply the QZ algorithmb to the companion form of P ( λ ) Matlab (7.3.0) [similar in Scilab] We get: − Inf , − 7 . 731 e − 19 , Inf , 3 . 588 e − 19 Scaling of Fan, Lin and Van Dooren (2004): − Inf , Inf , − 3 . 250 e − 19 , 3 . 588 e − 19 Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 7 / 51
2. Computing the roots of matrix pencils � 1 2 � � − 3 10 � � 12 15 � P ( λ ) = λ 2 10 − 18 +10 − 18 + λ 3 4 16 45 34 28 Apply the QZ algorithmb to the companion form of P ( λ ) Matlab (7.3.0) [similar in Scilab] We get: − Inf , − 7 . 731 e − 19 , Inf , 3 . 588 e − 19 Scaling of Fan, Lin and Van Dooren (2004): − Inf , Inf , − 3 . 250 e − 19 , 3 . 588 e − 19 tropical scaling (this talk) : − 7 . 250 E − 18 ± 9 . 744 E − 18 i , − 2 . 102 E + 17 ± 7 . 387 E + 17 i the correct answer (agrees with Pari ). Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 7 / 51
3. Location of roots of polynomials Given f ( z ) = a 0 + a 1 z + · · · + a k z k + · · · + a n z n , a i ∈ C Let ζ 1 , . . . , ζ n be the solutions of f ( z ) = 0, ordered by | ζ 1 | ≥ · · · ≥ | ζ n | . Bound | ζ i | ? E.g., Cauchy (1829) | a k | | ζ 1 | ≤ 1 + max | a n | . 0 ≤ k ≤ n − 1 Fujiwara (1916) � | a k | | ζ 1 | ≤ 2 max n − k | a n | . 0 ≤ k ≤ n − 1 Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 8 / 51
This talk: Fujiwara’s inequality is of a tropical nature the tropical point of view yields other inequalities Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 9 / 51
Tropical polynomial functions. . . are convex piecewise-linear with nonnegative integer slopes p ( x ) = “( − 1) x 2 + 1 x + 2” = max( − 1 + 2 x , 1 + x , 2) Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 10 / 51
“Fondamental theorem of algebra” A tropical polynomial function � b k x k ” = max p ( x ) = “ 0 ≤ k ≤ n b k + kx . 0 ≤ k ≤ n can be factored uniquely (Cuninghame-Green & Meijer, 80) as � p ( x ) = “ b n ( x + α k )” 1 ≤ k ≤ n � = b n + max( x , α k ) . 1 ≤ k ≤ n The points α 1 , . . . , α n are the tropical roots: the maximum is attained twice. Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 11 / 51
The Newton polygon ∆ is the concave hull of the points ( k , b k ), k = 0 , . . . , n . Proposition Two formal (tropical) polynomials yield the same polynomial function iff their Newton polygons coincide Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 12 / 51
The Newton polygon ∆ is the concave hull of the points ( k , b k ), k = 0 , . . . , n . Proposition Two formal (tropical) polynomials yield the same polynomial function iff their Newton polygons coincide Indeed, the function x �→ max 0 ≤ k ≤ n b k + kx is the Legendre- Fenchel transform of k �→ − b k . Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 12 / 51
The Newton polygon ∆ is the concave hull of the points ( k , b k ), k = 0 , . . . , n . Proposition Two formal (tropical) polynomials yield the same polynomial function iff their Newton polygons coincide Indeed, the function x �→ max 0 ≤ k ≤ n b k + kx is the Legendre- Fenchel transform of k �→ − b k . The tropical roots α 1 , . . . , α k are the opposite of the slopes of ∆. They can be computed in O ( n ) time. Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 12 / 51
p ( x ) = max(2 + 7 x , 6 + 4 x , 5 + 2 x , 2 + x , 3) = 2 + 2 max( − 1 , x ) + 2 max( − 1 / 2 , x ) + max(4 / 3 , x ) Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 12 / 51
Associate to f = a 0 + · · · + a n z n , a i ∈ C , the tropical polynomial p ( x ) = max 0 ≤ k ≤ n log | a k | + kx . The maximal tropical root is log | a k | − log | a n | α 1 = max n − k 1 ≤ k ≤ n − 1 Fujiwara’s bound readsa � | a k | | ζ 1 | ≤ 2 max n − k | a n | . 0 ≤ k ≤ n − 1 Stephane Gaubert (INRIA and CMAP) Tropical aspects of eigenvalue problems S´ eminaire Algo 13 / 51
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