On the Extreme Eigenvalues of Certain Gram Matrices of Hermite - PowerPoint PPT Presentation

On the Extreme Eigenvalues of Certain Gram Matrices of Hermite Polynomials q Martin Pleˇ singer & Ivana Pultarov´ a KMD TU Liberec, KO-MIX Lectures — March 18, 2019

I. Hermite Polynomials

Motivation & Introduction Monic Hermite polynomials (MHP) h 0 ( x ) = 1 , h 1 ( x ) = x, h 2 ( x ) = x 2 − 1 , h 3 ( x ) = x 3 − 3 x, h 4 ( x ) = x 4 − 6 x 2 + 3 , h 5 ( x ) = x 5 − 10 x 3 + 15 x, . . . given by recursive schemes h m +1 = xh m ( x ) − h ′ h 0 ( x ) = 1 , m ( x ) , m = 0 , 1 , 2 , . . . , or h 0 ( x ) = 1 , h 1 ( x ) = x, h m +1 = xh m ( x ) − mh m − 1 ( x ) , m = 1 , 2 , . . . , are orthogonal w.r.t. inner-product � ̺ ( x ) = e − x 2 � 2 . � f, g � = f ( x ) g ( x ) ̺ ( x )d x, � f � = � f, f � . R Sometimes h j ( x ) are called probabilists’ HP’s; the physicists’ HP’s use ̺ ( x ) = e − x 2 .

The first six Hermite polynomials 10 8 6 4 2 0 -2 -4 -6 -8 -10 -3 -2 -1 0 1 2 3

Since √ � h m , h m � = � h m � 2 = 2 π m ! , m = 0 , 1 , . . . , then h m ( x ) h m ( x ) p m ( x ) = � h m ( x ) � = √ , � p m , p k � = δ m,k , m, k = 0 , 1 , . . . , √ 4 2 π m ! are normalized Hermite polynomials (NHP). q Note that HP’s have symmetric distribution of roots: 0 = h m ( x ) = p m ( x ) ⇐ ⇒ p m ( − x ) = h m ( − x ) = 0 .

Shifted inner-product Let us consider � f ( x ) g ( x )e − x 2 2 + αx d x, � f, g � α = α ∈ R . R Since 2 + αx = − x 2 − 2 αx + α 2 − α 2 − x 2 = − ( x − α ) 2 + α 2 2 2 2 then � f ( x ) g ( x )e − ( x − α ) 2 α 2 � f, g � α = e d x 2 2 R � α 2 f ( x + α ) g ( x + α )e − x 2 α 2 2 d x = e 2 � f ( x + α ) , g ( x + α ) � = e 2 R Trivially � f, g � 0 ≡ � f, g � . Shifted inner-product of HP’s ≡ the standard inner-product of shifted HP’s (up to the scaling factor). Note that �· , ·� and �· , ·� α live on different spaces, but P’s are subspaces of both; � 1 2( x 2 2 − αx ) e x ≥ 0 let α > 0, then try f ( x ) = g ( x ) = 0 x < 0

Shifted Hermite polynomials We are interested in spectral properties of Gram matrices of NHP’s w.r.t. shifted inner-product. Shifted MHP as linear combination of MHP’s h 0 ( x + α ) = 1 = h 0 ( x ) , h 1 ( x + α ) = x + α = h 1 ( x ) + αh 0 ( x ) , h 2 ( x + α ) = x 2 + 2 αx + α 2 − 1 = h 1 ( x ) + 2 αh 1 ( x ) + α 2 h 0 ( x ) , h 3 ( x + α ) = x 3 + 3 αx 2 + 3 α 2 x + α 3 − 3 x − 3 α = h 3 ( x ) + 3 αh 2 ( x ) + 3 α 2 h 1 ( x ) + α 3 h 0 ( x ) . There is a clear pattern m � m � � α m − ℓ h ℓ ( x ) . h m ( x + α ) = ℓ ℓ =0 Proof by induction employs recurrent formula h s +1 = xh s ( x ) − h ′ s ( x ).

Shifted inner-product of HP’s √ √ 4 Recalling � h ℓ ( x ) � = 2 π ℓ !, the shifted inner-products of MHP & NHP are α 2 2 � h m ( x + α ) , h k ( x + α ) � � h m , h k � α = e � m � k � m � � k � α 2 � � α m − ℓ h ℓ ( x ) , α k − ℓ h ℓ ( x ) = e 2 ℓ ℓ ℓ =0 ℓ =0 min( m,k ) � m �� k � α 2 � α m + k − 2 ℓ � h ℓ ( x ) � 2 = e 2 ℓ ℓ ℓ =0 min( m,k ) α m + k − 2 ℓ √ � m �� k � α 2 � = e 2 π ℓ ! , 2 ℓ ℓ ℓ =0 q α 2 min( m,k ) � m �� k � � h m , h k � α e 2 � α m + k − 2 ℓ ℓ ! . � p m , p k � α = √ √ = √ √ √ ℓ ℓ 2 π m ! k ! m ! k ! ℓ =0

II. Gram Matrices

Gram matrix of NHP’s. Basic properties Let A α ∈ R ( N +1) × ( N +1) , α 2 min( m,k ) e � m �� k � 2 � α m + k − 2 ℓ ℓ ! , ( A α ) m +1 ,k +1 = � p m , p k � α = √ √ m, k = 0 , 1 , . . . , N. ℓ ℓ m ! k ! ℓ =0 • A α is symmetric positive definite.  α > 0 , A α > 0 ,  • For α = 0 , A α = I,  α < 0 , A α has nonzero entries with chess-board structure of sign pattern, i.e.,   + − + · · · − + − · · ·    .   + + − · · ·  . . . ... . . . . . . • Moreover | A α | = | A − α | . • Consider a “sign” matrix S = diag(1 , − 1 , 1 , . . . ), S = S T = S − 1 , then A α = SA − α S, i.e., sp( A α ) = sp( A − α ) .

Modified Gram matrix α 2 2 factor. Note that each entry contains e Let G α = e − α 2 2 A α ∈ R ( N +1) × ( N +1) , min( m,k ) 1 � m �� k � � α m + k − 2 ℓ ℓ ! , ( G α ) m +1 ,k +1 = √ √ m, k = 0 , 1 , . . . , N. ℓ ℓ m ! k ! ℓ =0 Observation: Denote χ N +1 ( λ ) = det( λI N +1 − G α ), for N = 0 , 1 , 2 , 3 we have χ 1 ( λ ) = λ − 1 , χ 2 ( λ ) = λ 2 − ( α 2 + 2) λ +1 , 2 + 3 α 2 + 3) λ 2 +( α 4 2 + 3 α 2 + 3) λ − 1 , χ 3 ( λ ) = λ 3 − ( α 4 6 + 2 α 4 + 6 α 2 + 4) λ 3 +( α 8 3 + 7 α 4 + 12 α 2 + 6) λ 2 χ 4 ( λ ) = λ 4 − ( α 6 12 + 4 α 6 6 + 2 α 4 + 6 α 2 + 4) λ +1 . − ( α 6 � � � � odd anti-palindromic The degree characteristic polynomial has coeff’s. even palindromic

Palindromic—anti-palindromic intermezzo Polynomial of degree n n � � � palindromic (PP) ⇐ ⇒ ϕ j = ϕ n − j ϕ j x j is f ( x ) = j = 0 , 1 , . . . , n. anti-palindromic (AP) ⇐ ⇒ ϕ j = − ϕ n − j j =0 Since n n � 1 � 1 ϕ j ± ϕ n − j � � x n f ( x ) = x n − j = = ± f , x n − j x j =0 j =0 it has reciprocal roots, i.e., � 1 � f ( x ) = 0 = 0 . ⇐ ⇒ f x A lot of interesting properties, e.g.: • AP has always root 1 (AP factor ( x − 1)), · PP AP • odd-degree PP has always root − 1 (P factor ( x + 1)), PP PP AP • PP-AP multiplication − → AP AP PP

Spectra of modified Gram matrices Observation: 5 6 10 10 4 10 2 10 0 0 10 10 −2 10 −4 10 −5 −6 10 10 −5 0 5 −5 0 5 α α

Decomposition of the modified Gram matrix Recall the “sign” matrix S = S T = S − 1 . S = diag(1 , − 1 , 1 , . . . , ( − 1) N ) ∈ R ( N +1) × ( N +1) , Consider an upper triangular matrix   � 0 � � 1 � � 2 � � 3 � α 0 α 1 α 2 α 3   · · · α 2 α 3 1 α · · · 0 0 0 0 � 1 � � 2 � � 3 �  α 0 α 1 α 2  3 α 2 0 · · · 0 1 2 α · · ·     1 1 1 � 2 � � 3 �  ∈ R ( N +1) × ( N +1) .     α 0 α 1 U α = = 0 0 1 3 α · · · 0 0 · · ·     2 2 � 3 � 0 0 0 1    · · · α 0 0 0 0 · · ·   . . . 3 ... ... . . . . . . ... ... . . . . . . . . . Then U − 1 U α = SU − α S, U α U − α = I, so = U − α = SU α S. α Finally consider also a “factorial” matrix F = diag(0! , 1! , 2! , . . . , N !) ∈ R ( N +1) × ( N +1) , F = F T , then ...

The modified Gram matrix G α = e − α 2 2 A α ∈ R ( N +1) × ( N +1) with entries min( m,k ) � m �� k � 1 � α m + k − 2 ℓ ℓ ! , ( G α ) m +1 ,k +1 = √ √ m, k = 0 , 1 , . . . , N, ℓ ℓ m ! k ! ℓ =0 can be then factorized as � � T � � G α = F − 1 α FU α F − 1 1 2 U α F − 1 1 2 U α F − 1 2 U T 2 = F F . 2 2 � �� � Cholesky factorization Its inverse can be factorized as 1 1 G − 1 2 U − 1 α F − 1 U − T = F α F 2 α � � � � � � � �� � T 1 1 1 1 2 U α F − 1 1 1 2 U α F − 1 SF − 1 S U T 2 U α F − 1 U T = = SF 2 S = S F 2 S U α SF α F F F S, 2 2 2 α i.e., � �� � T � � T � � 2 U α F − 1 1 1 2 U α F − 1 2 U α F − 1 1 1 2 U α F − 1 G − 1 ∼ F F ∼ F F = G α . 2 2 2 2 α Consequently sp( G α ) = sp( G − 1 α ) .

Perron–Frobenius theory Entries of G α are positive and increasing for α ∈ (0 , ∞ ), i.e., ∀ ǫ > 0 , G α + ǫ > G α > 0 . Perron–Frobenius theory then gives λ max ( G α + ǫ ) > λ max ( G α ) . Since sp( G α ) = sp( G − 1 α ), then κ ( G α ) = λ max ( G α ) λ min ( G α ) = ( λ max ( G α )) − 1 λ min ( G α ) = λ max ( G α ) 2 , and and thus λ min ( G α + ǫ ) < λ min ( G α ) and κ ( G α + ǫ ) > κ ( G α ) . Since sp( G α ) = sp( G − α ), then analogous results can be obtained for α ∈ ( −∞ , 0).

Back to the original Gram matrix α 2 α 2 2 sp( G α ). Thus 2 G α is a scalar mutliple of G α , i.e., sp( A α ) = e Recall that A α = e λ max ( A α ) and κ ( A α ) are still increasing for α ∈ (0 , ∞ ). 10 12 10 10 10 10 8 10 8 10 6 10 6 10 4 10 4 10 2 10 2 10 0 10 0 10 −2 −2 10 10 −5 0 5 −5 0 5 α α

III. Optimal Diagonal Scaling

Optimal diagonal scaling Let B = ( b i,j ) ∈ R n × n be symmetric positive definite, and � � 1 1 D ∗ ≡ diag , . . . , . � � b 1 , 1 b n,n Then κ ( D ∗ BD ∗ ) ≤ n D diagonal κ ( DBD ) . min See [A van der Sluis, 1969] or [N Higham, 2002]. Note that b i,j ( D ∗ BD ∗ ) i,j = , ( D ∗ BD ∗ ) i,i = 1 . � � b i,i b j,j