Interpolation by Polynomials with Symmetries on the Imaginary Axis Izchak Lewkowicz ECE dept, Ben-Gurion University, Israel Ben-Gurion University, 24 May 2012 Joint work with D. Alpay, Math. dept BGU. – p. 1
Outline The function sets P GP GPE Polynomial Interpolation - Simple, no structure Nevanlinna-Pick interpolation - structured, underlying invertibility Convex (Invertible) Sets and Cones of Functions F ( s ) = P ( s ) + β Ψ( s ) Extensions – p. 2
(Initial) Problem Formulation Given: F a family of rational functions F = { F : C → C l × l } Data points Y 1 , . . . , Y m ∈ C l × l x 1 , . . . , x m ∈ C Find all minimal (McMillan) degree F ∈ F s.t. Y j = F ( x j ) j = 1 , . . . , m – p. 3
Interpolation - Convex Invertible Sets F a family of rational functions F b ( s ) in F F a ( s ) � Y j = F a ( x j ) j =1 , ... , m = ⇒ Y j = F ( x j ) j = 1 , . . . , m Y j = F b ( x j ) j =1 , ... , m , F b , F − 1 F ∈ conv( F a , F − 1 ) a b Namely, + (1 − β ) F − 1 F = (1 − γ ) ( α F a + (1 − α ) F b )+ γ ( β F − 1 ) − 1 a b α, β, γ ∈ [0 , 1] Polynomials: γ ≡ 0 (no linear fractional transformation) – p. 4
Positive Functions ( C + ) open (closed) right half plane C + Positive functions P := { F ( s ) : ( F + F ∗ ) ≥ 0 s ∈ C + } Driving point immittance of R-L-C electrical networks Linear dissipative systems (a.k.a absolutely stable) Linear passive systems – p. 5
Generalized Positive Functions P := { F ( s ) : ( F + F ∗ ) ≥ 0 s ∈ C + } Generalized Positive functions GP := { F ( s ) : ( F + F ∗ ) ≥ 0 for almost all s ∈ i R } F = C ( sI − A ) − 1 B + D ∃ s → ∞ F ( s ) lim ⇐ ⇒ H = H ∗ nonsingular s.t. KYP: F ∈ GP ⇐ ⇒ ∃ � H 0 � � A B � A B � ∗ � H 0 � � + ≥ 0 0 I 0 I C D C D – p. 6
(cont.) GP GP := { F ( s ) : ( F + F ∗ ) ≥ 0 for almost all s ∈ i R } G # ( s ) := G ∗ ( − s ∗ ) F ( s ) = G ( s ) P ( s ) G # ( s ) F ∈ GP ⇐ ⇒ P ∈ P G and G − 1 analytic in C + C. Chamfy 1958 N.I. Akhiezer 1965 Ph. Delsartre, Y. Genin, Y. Kamp 1986 V. Derkach, S. Hassi, H. de-Snoo 1999 A. Dijksma, H. Langer, A. Luger, Yu. Shondin 2000 – p. 7
Even and Odd Functions F # ( s ) := F ∗ ( − s ∗ ) F # ( s ) = E ven = { F ( s ) : F ( s ) } F # ( s ) = O dd = { F ( s ) : − F ( s ) } F ( s ) = F even ( s ) + F odd ( s ) F even ∈ E ven F odd ∈ O dd � ∗ � E ven ⇒ F ( s ) | s ∈ i R = F ( s ) | s ∈ i R � ∗ � O dd ⇒ F ( s ) | s ∈ i R = − F ( s ) | s ∈ i R – p. 8
Odd Functions O dd = { F ( s ) : F # ( s ) = − F ( s ) } O dd ⊂ GP O dd � P - Lossless, Foster, immittance of L-C circuits – p. 9
Generalized Positive Even Functions F # ( s ) F ( s ) = F ∈ GPE ⇐ ⇒ F ( s ) | s ∈ i R ≥ 0 for almost all s ∈ i R GPE := GP � E ven Generalized Positive Even F ∈ GP ⇐ ⇒ F even ∈ GPE – p. 10
(cont.) GPE F # ( s ) F ( s ) = GPE ⇐ ⇒ F ( s ) | s ∈ i R ≥ 0 almost for all s ∈ i R ⇒ F ( s ) = G ( s ) G # ( s ) F ∈ GPE ⇐ G and G − 1 analytic in C + (Pseudo) spectral factorization P . Fuhrmann, L. Lerer, A. Ran, L. Roozemond, D.C. Youla ... – p. 11
Convex Invertible Cones of Functions F a family of rational functions F b ( s ) in F F a ( s ) F = α F a + β F b + ( γ F − 1 ( s )+ δ F b ) − 1 α , β , γ , δ ≥ 0 a F a Convex Invertible Cones of functions = ⇒ F ∈ F Feedback loop - a CIC O dd , E ven , GP are Convex Invertible Cones (CICs). – p. 12
CICs of Functions (cont.) P a subCIC of GP P a maximal CIC of functions analytic in C + a non-empty intersection of CICs is a CIC. GPE = GP � E ven Foster = P � O dd GPE = GP � E ven GPE a Convex subCone of E ven – p. 13
Prototype Problem Formulation Let F = { F : C → C l × l } be polynomials GPE Given data points Y 1 , . . . , Y m ∈ C l × l x 1 , . . . , x m ∈ C Find all minimal degree F ∈ F s.t. Y j = F ( x j ) j = 1 , . . . , m – p. 14
Two Interpolation Problems Unstructured polynomial interpolation - Lagrange rational - Nevanlinna- Pick GP – p. 15
Nevanlinna - Pick Interpolation Generalized Schur - sample works J.A. Ball, 1983 J.A. Ball and J.W. Helton, 1985 D. Alpay, T.Ya. Azizov, A. Dijksma, H. Langer, G. Wanjala 2003 V. Bolotnikov, 2010 V.A. Derkach H. Dym, 2010 Generalized Nevanlinna - sample works D. Alpay, V. Bolotnikov, A. Dijksma, 1998 D. Alpay, A. Dijksma, H. Langer, S. Reich, D. Shoikhet, 2010 Neither polynomials nor GPE are singled out – p. 16
Unstructured Polynomial Interpolation Given Y 1 , . . . , Y m ∈ C l × l (distinct) x 1 , . . . , x m ∈ C Find q − 1 C k ∈ C l × l , C k s k � P ( s ) = k =0 s.t. Y j = P ( x j ) j = 1 , . . . , m is minimal. q – p. 17
Unstructured Interpolation (cont.) q − 1 C k ∈ C l × l , C k s k � P ( s ) = k =0 x m − 1 x 2 C o I l x 1 I l 1 I l ... I l Y 1 1 . . . . . . . = . . . . . . . . . . . . . . C n − 1 x m − 1 x 2 Y n I l x n I l n I l ... I l n m ≥ q – p. 18
The Lagrange Approach Take F ( s ) = ˜ F 1 ( s ) + ˜ F 2 ( s ) + ˜ F 1 , ˜ ˜ F 2 , ˜ F 3 ( s ) F 3 ∈ F s.t. x 1 x 2 x 3 ˜ F 1 ( s ) 0 0 Y 1 ˜ F 2 ( s ) 0 0 Y 2 ˜ F 3 ( s ) 0 0 Y 3 . Apparently due to E. Waring, 1779 Simple, elegant, but never the minimal degree – p. 19
The Lagrange Approach Example Taking F = GPE , find all minimal degree „ 1 4 « „ « F ( s )= ˜ F 1 ( s )+ ˜ F 2 ( s )+ ˜ → 1 F 3 ( s ) F 2 − 4 3 ˜ F 1 ( s ) = (4 − s 2 )(9 − s 2 )( α (1 − s 2 )+ 1 ˜ ) α ≥ 0 , ˜ 6 60 s 2 ) with F 2 ( s ) = (1 − s 2 )(9 − s 2 )( ˜ ˜ ˜ = ⇒ β (4 − s 2 ) − 1 β ≥ 0 , 90 s 2 ) ˜ F 3 ( s ) = (1 − s 2 )(4 − s 2 )( ˜ γ (9 − s 2 ) − 1 γ ≥ 0 . ˜ Accumulated into one parameter F ( s ) = − s 2 + 5 + β (1 − s 2 )(4 − s 2 )(9 − s 2 ) 1 β ≥ 36 However true for β ≥ 0 (min deg( F ) = 2) – p. 20
Prototype Problem Formulation Let F = { F : C → C l × l } be polynomials GPE Given data points Y 1 , . . . , Y m ∈ C l × l x 1 , . . . , x m ∈ C Find all minimal degree F ∈ F s.t. Y j = F ( x j ) j = 1 , . . . , m – p. 21
Recipe - Basic Idea GPE = GP � E ven GPE a Convex subCone of E ven P ( s ) polynomial in E ven deg(Ψ) ≥ deg( P ) = ⇒ Ψ( s ) polynomial in GPE ( P + β Ψ) ∈ GPE for β > 0 sufficiently large – p. 22
Data Reduction Out of feasible points x 1 , . . . , x m extract a maximal subset x 1 , . . . , x n ( m ≥ n ) and the corresponding Y 1 , . . . , Y n s.t. x j − x k � =0 n ≥ k > j ≥ 1 x j + x ∗ k � =0 The choice x 1 , . . . , x n is not unique, but n is unique All interpolating polynomials F ( s ) in GPE 2 n ≥ deg( F ) – p. 23
Recipe Construct the minimal degree interpolating P ( s ) in E ven , result: 2 n − 1 ≥ deg( P ) Parameterize all Ψ ∈ GPE vanishing at x 1 , . . . , x m result: deg(Ψ) = 2 n F ( s ) = P ( s ) + β Ψ( s ) β ∈ R interpolating function in E ven Find the minimal ˆ β s.t. F ∈ GPE for all β ≥ ˆ β – p. 24
Even Polynomial Interpolation Given Y 1 , . . . , Y m ∈ C l × l x 1 , . . . , x m ∈ C Find q − 1 C k ∈ C l × l C k s k � P ( s ) = k =0 s.t. P # ( s ) = P ( s ) P ( x j ) = j =1 , ... , m. Y j Feasible data, x j − x k =0 = ⇒ Y j = Y k m ≥ k > j ≥ 1 . x j + x ∗ Y j = Y ∗ k =0 = ⇒ k – p. 25
Even Polynomial Interpolation (cont.) Adaptation of an idea of Alpay, Bolotnikov, Loubaton 1996 q − 1 C k s k C k ∈ C l × l � P ( s ) = k =0 x 2 n − 1 x 2 I l x 1 I l 1 I l ... I l Y 1 C o 1 . . . . . . . . . . . . . . . . . . . . . x 2 x 2 n − 1 Y n C n − 1 I l x n I l n I l ... I l n = Y ∗ I l − x ∗ 1 I l ( − x ∗ 1 ) 2 I l ... ( − x ∗ 1 ) 2 n − 1 I l C n 1 . . . . . . . . . . . . . . . . . . . . . Y ∗ C 2 n − 1 I l − x ∗ n I l ( − x ∗ ∗ ) 2 n − 1 I l n ) 2 I l ... ( − x n n 2 n ≥ q ( n dim. reduced set) – p. 26
Neutral GPE Polynomials Ψ( s ) a minimal degree GPE polynomial data points Ψ( x j ) = 0 j = 1 , . . . , m the reduced set x 1 , . . . , x n ∈ C ( m ≥ n ) = ⇒ n ( x j − s ) M ( x ∗ � M ≥ 0 parameter Ψ( s ) = j + s ) j =1 F ( s ) = P ( s ) + β Ψ( s ) – p. 27
ˆ β F ( s ) = P ( s ) + β Ψ( s ) n ( x j − s ) M ( x ∗ � M ≥ 0 parameter Ψ( s ) = j + s ) j =1 Assume P ( s ) is not in GPE (else trivial) ⇒ β ≥ ˆ Take M > 0 parameter F ∈ GPE ⇐ β 2 n − 1 M − 1 i k C k ω k P ˆ β = − min min k =0 p =1 , ... , l λ p n ω ∈ R Q | x j − iω | 2 ω � = ix j j =1 – p. 28
Polynomial Interpolation GPE Find all minimal degree F ( s ) in GPE s.t. 1 18 0 1 0 1 A → F 2 75 @ @ A 3 50 P ( s ) = − 3 s 4 + 34 s 2 − 13 6 × 6 Vandermonde matrix minimal deg. interpolating in E ven ( − P ∈ GPE ) F ( s ) = P ( s ) + β Ψ( s ) = − 3 s 4 +34 s 2 − 13 + β (1 − s 2 )(4 − s 2 )(9 − s 2 ) ⇒ β ≥ ˆ β = 1 β ∈ R ⇒ F ∈ E ven F ∈ GPE ⇐ 2 – p. 29
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