Majorization inequalities for valuations of eigenvalues Marianne - PowerPoint PPT Presentation

Majorization inequalities for valuations of eigenvalues Marianne Akian INRIA Saclay - ˆ Ile-de-France and CMAP, ´ Ecole polytechnique CNRS SYMBIONT ANR/DFG Kick-off Meeting Bonn, Jul 4-5, 2018 Based on works with Ravindra Bapat, St´ ephane Gaubert, Andrea Marchesini, Meisam Sharify, and Fran¸ coise Tisseur

Majorizations of scalar polynomial roots Let ζ 1 , . . . , ζ d , | ζ 1 | ≥ · · · ≥ | ζ d | , denote the roots of the polynomial P = a 0 + a 1 z + · · · + a k z k + · · · + a d z d , a i ∈ C . Let α 1 ≥ · · · ≥ α d denote the inclinaisons 10 num´ eriques of P , that is the exponential 8 6 of the opposites of the slopes of the New- 4 ton polygon of P , concave( k �→ log | a k | ). 2 0 0 2 4 6 8 10 12 14 16 For each k ≥ 1, there exists L k , U k > 0 independent of d such that L k α 1 · · · α k ≤ | ζ 1 · · · ζ k | ≤ U k α 1 · · · α k , � d with L − 1 � � ( k + 1) k +1 / k k . = and U k ≤ k k (Ostrowski, “Recherches sur la m´ ethode de Graeffe et les z´ eros des polynomes et des s´ eries de Laurent”, Acta Math 72, 99-257, 1940)

Majorizations of scalar polynomial roots Let ζ 1 , . . . , ζ d , | ζ 1 | ≥ · · · ≥ | ζ d | , denote the roots of the polynomial P = a 0 + a 1 z + · · · + a k z k + · · · + a d z d , a i ∈ C . Let α 1 ≥ · · · ≥ α d denote the inclinaisons 10 num´ eriques of P , that is the exponential 8 6 of the opposites of the slopes of the New- 4 ton polygon of P , concave( k �→ log | a k | ). 2 0 0 2 4 6 8 10 12 14 16 For each k ≥ 1, there exists L k , U k > 0 independent of d such that L k α 1 · · · α k ≤ | ζ 1 · · · ζ k | ≤ U k α 1 · · · α k , � d with L − 1 � � ( k + 1) k +1 / k k . = and U k ≤ k k Ostrowski proved L k (which is optimal) and reproduced the proof of U k given by P´ olya

Majorizations of scalar polynomial roots Let ζ 1 , . . . , ζ d , | ζ 1 | ≥ · · · ≥ | ζ d | , denote the roots of the polynomial P = a 0 + a 1 z + · · · + a k z k + · · · + a d z d , a i ∈ C . Let α 1 ≥ · · · ≥ α d denote the inclinaisons 10 num´ eriques of P , that is the exponential 8 6 of the opposites of the slopes of the New- 4 ton polygon of P , concave( k �→ log | a k | ). 2 0 0 2 4 6 8 10 12 14 16 For each k ≥ 1, there exists L k , U k > 0 independent of d such that L k α 1 · · · α k ≤ | ζ 1 · · · ζ k | ≤ U k α 1 · · · α k , � d with L − 1 � � ( k + 1) k +1 / k k . = and U k ≤ (Ostrowski, P´ olya, 1940) k k Previous inequalities include: U k ≤ k + 1 (Hadamard, 1891), | ζ 1 | ≤ 2 α 1 (Fujiwara, 1916), | ζ 1 · · · ζ k | ≤ ( k + 1) α k 1 (Specht, 1938), U k ≤ 2 k + 1 in (Ostrowski, 1940).

Max-plus or tropical algebra Let T = R max := ( R ∪ {−∞} , ⊕ , ⊗ ) be the additive tropical or max-plus idempotent semifield, where a ⊕ b = max( a , b ) and a ⊗ b = a + b , with neutral elements 0 = −∞ and 1 = 0. On can define vectors, matrices, polynomials, eigenvalues, eigenvectors, permanents,... T is the limit of the logarithmic deformation of R + semiring: log( ε − a + ε − b ) max( a , b ) = lim ε → 0 + − log( ε ) log( ε − a ε − b ) a + b = . − log( ε ) And it gives bounds: ε log( e a /ε + e b /ε ) ≤ ε log(2) + max( a , b ) . max( a , b ) ≤ It is isomorphic to the multiplicative tropical semifield T m := ( R ≥ 0 , max , × ), by x �→ exp( x ).

Valuations Let K be a field, a map v : K → R ∪ {−∞} is a non- a map v : K → R + is an archimedean archimedean valuation if valuation if v( s 1 + s 2 ) ≤ max(v( s 1 ) , v( s 2 )) , v( z 1 + z 2 ) ≤ v( z 1 ) + v( z 2 ) , v( s 1 s 2 ) = v( s 1 ) + v( s 2 ) v( z 1 z 2 ) = v( z 1 ) v( z 2 ) , v( s ) = −∞ ⇔ s = 0 . v( z ) = 0 ⇔ z = 0 . Then Then v( s 1 + s 2 ) = max(v( s 1 ) , v( s 2 )) if v( s 1 ) � = v( s 2 ). v( z 1 + z 2 ) ≤ 2 max(v( z 1 ) , v( z 2 )). Ex: K = C , v( z ) = | z | . Ex: K = C {{ ǫ }} , the field of (general- ized) Puiseux series, v( s ) = − val( s ), e.g. v( ǫ − 1 / 3 + 3 − 8 ǫ 2 + · · · ) = 1 / 3. Given a (archimedean or non-archimedean) valuation v on K , one apply v entrywise on K n , K n × n , K [ X ], e.g. v( z 1 , . . . , z n ) := (v( z 1 ) , . . . , v( z n )). Given a variety V on K , its image by v is called the amoeba of V .

Let T m = ( R ≥ 0 , max , × ) be the multiplicative tropical semifield eriques” α i of P = a 0 + a 1 z + · · · + a k z k + · · · + a d z d are The “inclinaisons num´ the tropical roots (nondifferentiability locus) of the multiplicative tropical polynomial v( P ) ∈ T m [ X ]: 0 ≤ j ≤ d ( | a j | x j ) v( P )( x ) = max ⇔ the log α i are the tropical roots of p ( x ) = max 0 ≤ j ≤ d (log | a j | + jx ). Here v is the archimedean valuation : v : C → R ≥ 0 , x �→ | x | .

Let T m = ( R ≥ 0 , max , × ) be the multiplicative tropical semifield eriques” α i of P = a 0 + a 1 z + · · · + a k z k + · · · + a d z d are The “inclinaisons num´ the tropical roots (nondifferentiability locus) of the multiplicative tropical polynomial v( P ) ∈ T m [ X ]: 0 ≤ j ≤ d ( | a j | x j ) v( P )( x ) = max ⇔ the log α i are the tropical roots of p ( x ) = max 0 ≤ j ≤ d (log | a j | + jx ). Since the Newton polygon of P is the graph of the convave hull of the map j �→ log | a j | and p is its Legendre-Fenchel transform. 10 35 30 8 25 6 20 4 15 10 2 5 0 0 0 2 4 6 8 10 12 14 16 −6 −5 −4 −3 −2 −1 0 1 2 Newton polygon of P Graph of p Example: P = 1 + eX + e 6 X 2 + e 4 X 4 + e 9 X 8 + e 5 X 10 + eX 16. Tropical roots: − 3, − 1 / 2, and 1, with multiplicities 2, 6 and 8 resp.

Theorem ( See Kapranov, 2006 ) Let K be an algebraically closed field with a (non-archimedean) valuation v , and k f k X k ∈ K [ X ] , then the valuations of the roots of f (counted with f = � multiplicities) coincide with the tropical roots of the tropical polynomial v( f ) := � n k =0 v( f k ) X k . Follows from Puiseux theorem in dimension 1, and is derived from Puiseux in other dimensions.

Majorizations of scalar polynomial roots Let ζ 1 , . . . , ζ d , | ζ 1 | ≥ · · · ≥ | ζ d | , denote the roots of the polynomial P = a 0 + a 1 z + · · · + a k z k + · · · + a d z d , a i ∈ C . Let α 1 ≥ · · · ≥ α d denote the inclinaisons 10 num´ eriques of P , that is the exponential 8 6 of the opposites of the slopes of the New- 4 ton polygon of P , concave( k �→ log | a k | ). 2 0 0 2 4 6 8 10 12 14 16 For each k ≥ 1, there exists L k , U k > 0 independent of d such that L k α 1 · · · α k ≤ | ζ 1 · · · ζ k | ≤ U k α 1 · · · α k , with L − 1 � d � � = and U k ≤ ( k + 1) k +1 / k k . (Ostrowski, P´ olya, 1940) k k

Let Log : ( C ∗ ) d → R d , x �→ (log | x 1 | , . . . , log | x d | ). Ostrowski bounds give the distance between the archimedan amoeba A = Log ( V ) of the finite set V of C d of solutions ( ζ 1 , . . . , ζ d ) of � ζ i 1 · · · ζ i k = ( − 1) k a d − k / a d , k = 1 , . . . d , i 1 < ··· < i k up to permutations of coordinates, to its “tropical skeleton” T .

Let Log : ( C ∗ ) d → R d , x �→ (log | x 1 | , . . . , log | x d | ). Ostrowski bounds give the distance between the archimedan amoeba A = Log ( V ) of the finite set V of C d of solutions ( ζ 1 , . . . , ζ d ) of � ζ i 1 · · · ζ i k = ( − 1) k a d − k / a d , k = 1 , . . . d , i 1 < ··· < i k up to permutations of coordinates, to its “tropical skeleton” T . T is the set of (log α 1 , . . . , log α d ) solutions of � | a d − k / a d | ⊕ α i 1 · · · α i k “ = ” 0 , k = 1 , . . . d . i 1 < ··· < i k Ostrowski bounds implies d ( A , T ) ≤ 1 for the distance d ( x , y ) = min σ,σ ′ ∈ Σ d � ( x σ ( i ) ) − ( y σ ′ ( i ) ) � associated to the weak Minkowski (non symmetric) norm: (log U k ) − 1 ( x 1 + · · · + x k ) + − (log L k ) − 1 ( x 1 + · · · + x k ) − � � � x � = max . k =1 ,..., d

These are different from the bounds on the distance between the archimedan amoeba A = Log ( V ) of the finite set V of roots of P and its tropical skeleton: • such bounds can be obtained by using Rouch´ e theorem. • they are related to lopsided approximations. • see in particular the works of (Gaubert, Sharify, 2011), (Bini, Noferini, Sharify, 2013), (Erg¨ ur, Paouris, Rojas, 2014) . In the multivariate case (non zero dimension), such bounds appeared in: (Passare, R¨ ullgaard, 04); (Purbhoo, 06); (Avendano, Kogan, Nisse, Rojas, 13), (Forsg˚ ard, 16), ard, Matusevich, Mehlhop, de Wolff, 17) . (Forsg˚

This work: give majorizations/Ostrowski bounds for the eigen- values of matrix polynomials

This work: give majorizations/Ostrowski bounds for the eigen- values of matrix polynomials We are interested in • Non-archimedean case • Archimedean case • Eigenvectors • Condition numbers and applications to numerical computations of eigenvalues and eigenvectors

This work: give majorizations/Ostrowski bounds for the eigen- values of matrix polynomials We are interested in • Non-archimedean case • Archimedean case • Eigenvectors • Condition numbers and applications to numerical computations of eigenvalues and eigenvectors