Interpolation by Polynomials with Symmetries on the Imaginary - - PowerPoint PPT Presentation

Interpolation by Polynomials with Symmetries

• n

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) + βΨ(s)

Extensions

– p. 2

(Initial) Problem Formulation

Given:

F a family of rational functions F = { F : C → Cl×l}

Data points

x1 , . . . , xm ∈ C Y1 , . . . , Ym ∈ Cl×l

Find all minimal (McMillan) degree F ∈ F s.t.

Yj = F (xj) j = 1 , . . . , m

– p. 3

Interpolation - Convex Invertible Sets

F a family of rational functions Fa(s) Fb(s) in F

Yj=Fa(xj) j=1 , ... , m Yj=Fb(xj) j=1 , ... , m

• =

⇒ Yj = F (xj) j = 1 , . . . , m F ∈ conv(Fa , F −1

a

, Fb , F −1

b

)

Namely,

F =(1 − γ) (α Fa+(1 − α) Fb)+γ (β F −1

a

+(1 − β) F −1

b

)−1 α, β, γ ∈ [0, 1]

Polynomials: γ ≡ 0 (no linear fractional transformation)

– p. 4

Positive Functions

C+

(C+)

• pen (closed) right half plane

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 ∈ iR} ∃ lim

s → ∞ F (s)

⇐ ⇒ F = C(sI − A)−1B + D

KYP:

F ∈ GP ⇐ ⇒ ∃ H = H∗ nonsingular s.t. H 0

0 I

A B

C D

• +

A B

C D

∗ H 0

0 I

• ≥ 0

– p. 6

GP (cont.)

GP := { F (s) : (F + F ∗) ≥ 0 for almost all s ∈ iR} G#(s) := G∗(−s∗) F ∈ GP ⇐ ⇒ F (s) = G(s)P (s)G#(s) 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∗) Even = {F (s) : F #(s) = F (s) } Odd = {F (s) : F #(s) = −F (s) } F (s) = Feven(s) + Fodd(s) Feven ∈ Even Fodd ∈ Odd Even ⇒

• F (s)|s∈iR

∗ = F (s)|s∈iR Odd ⇒

• F (s)|s∈iR

∗ = −F (s)|s∈iR

– p. 8

Odd Functions

Odd = {F (s) : F #(s) = − F (s) } Odd ⊂ GP Odd P - Lossless, Foster, immittance of L-C circuits

– p. 9

Generalized Positive Even Functions

F ∈ GPE ⇐ ⇒      F (s) = F #(s) F (s)|s∈iR ≥ 0 for almost all s ∈ iR

Generalized Positive Even

GPE := GP Even F ∈ GP ⇐ ⇒ Feven ∈ GPE

– p. 10

GPE (cont.)

GPE ⇐ ⇒      F (s) = F #(s) F (s)|s∈iR ≥ almost for all s ∈ iR F ∈ GPE ⇐ ⇒ F (s) = G(s)G#(s) 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

Fa(s) Fb(s) in F F =α Fa+β Fb + (γ F −1

a

(s)+δ Fb)−1

α , β , γ , δ ≥ 0

F a Convex Invertible Cones of functions = ⇒ F ∈ F

Feedback loop - a CIC

Odd, Even, 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 Even

Foster= P Odd

GPE = GP Even GPE a Convex subCone of Even

– p. 13

Prototype Problem Formulation

Let F = { F : C → Cl×l} be

GPE

polynomials Given data points

x1 , . . . , xm ∈ C Y1 , . . . , Ym ∈ Cl×l

Find all minimal degree F ∈ F s.t.

Yj = F (xj) j = 1 , . . . , m

– p. 14

Two Interpolation Problems

Unstructured polynomial interpolation - Lagrange rational

GP

• Nevanlinna- Pick

– 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

x1, . . . , xm ∈ C

(distinct)

Y1 , . . . , Ym ∈ Cl×l

Find

P (s) =

q−1

• k=0

Cksk Ck ∈ Cl×l,

s.t.

Yj = P (xj) j = 1, . . . , m q

is minimal.

– p. 17

Unstructured Interpolation (cont.)

P (s) =

q−1

• k=0

Cksk Ck ∈ Cl×l,   

Il x1Il x2

1Il

... xm−1

1

Il

. . . . . . . . . . . . . . .

Il xnIl x2

nIl

... xm−1

n

Il

    

Co

. . .

Cn−1

  =  

Y1

. . .

Yn

  m ≥ q

– p. 18

The Lagrange Approach

Take

F (s) = ˜ F1(s) + ˜ F2(s) + ˜ F3(s) ˜ F1, ˜ F2, ˜ F3 ∈ F

s.t.

x1 x2 x3 ˜ F1(s) Y1 ˜ F2(s) Y2 ˜ F3(s) Y3.

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

F „ 1 2 3 « → „ 4 1 −4 « F (s)= ˜ F1(s)+ ˜ F2(s)+ ˜ F3(s)

with

˜ F1(s) = (4−s2)(9−s2)( ˜ α(1−s2)+ 1

6

) ˜ α≥0, ˜ F2(s) = (1−s2)(9−s2)( ˜ β(4−s2)− 1

60s2 )

˜ β≥0, ˜ F3(s) = (1−s2)(4−s2)( ˜ γ(9−s2)− 1

90s2 )

˜ γ≥0.

= ⇒

Accumulated into one parameter

F (s) = −s2 + 5 + β(1 − s2)(4 − s2)(9 − s2) β ≥

1 36

However true for

β ≥ 0 (min deg(F ) = 2)

– p. 20

Prototype Problem Formulation

Let F = { F : C → Cl×l} be

GPE

polynomials Given data points

x1 , . . . , xm ∈ C Y1 , . . . , Ym ∈ Cl×l

Find all minimal degree F ∈ F s.t.

Yj = F (xj) j = 1 , . . . , m

– p. 21

Recipe - Basic Idea

GPE = GP Even GPE a Convex subCone of Even P (s) polynomial in Even Ψ(s) polynomial in GPE      deg(Ψ) ≥ deg(P ) = ⇒ (P + βΨ) ∈ GPE for β > 0 sufficiently large

– p. 22

Data Reduction

Out of feasible points x1 , . . . , xm extract a maximal subset x1 , . . . , xn (m ≥ n) and the corresponding Y1, . . . , Yn s.t.

xj−xk=0 xj+x∗

k=0

n ≥ k > j ≥ 1

The choice x1 , . . . , xn is not unique, but

n is unique

All interpolating polynomials F (s) in GPE

2n ≥ deg(F )

– p. 23

Recipe

Construct the minimal degree interpolating P (s) in

Even, result: 2n − 1 ≥ deg(P )

Parameterize all Ψ ∈ GPE vanishing at x1 , . . . , xm result:

deg(Ψ) = 2n F (s) = P (s) + βΨ(s) β ∈ R

interpolating function in Even Find the minimal ˆ

β s.t. F ∈ GPE for all β ≥ ˆ β

– p. 24

Even Polynomial Interpolation

Given

x1, . . . , xm ∈ C Y1 , . . . , Ym ∈ Cl×l

Find

P (s) =

q−1

• k=0

Cksk Ck ∈ Cl×l

s.t.

P #(s) = P (s) P (xj) = Yj j=1, ... , m.

Feasible data,

xj−xk=0 = ⇒ Yj = Yk xj+x∗

k=0

= ⇒ Yj = Y ∗

k

m ≥ k > j ≥ 1.

– p. 25

Even Polynomial Interpolation (cont.)

Adaptation of an idea of Alpay, Bolotnikov, Loubaton 1996

P (s) =

q−1

• k=0

Cksk Ck ∈ Cl×l        

Il x1Il x2

1Il

... x2n−1

1

Il

. . . . . . . . . . . . . . .

Il xnIl x2

nIl

... x2n−1

n

Il Il −x∗

1Il (−x∗ 1)2Il ... (−x∗ 1)2n−1Il

. . . . . . . . . . . . . . .

Il −x∗

nIl (−x∗ n)2Il ... (−xn ∗)2n−1Il

             

Co

. . .

Cn−1 Cn

. . .

C2n−1

      =      

Y1

. . .

Yn Y ∗

1

. . .

Y ∗

n

      2n ≥ q (n dim. reduced set)

– p. 26

Neutral GPE Polynomials

Ψ(s) a minimal degree GPE

polynomial

Ψ(xj) = 0 j = 1 , . . . , m

data points

x1, . . . , xn ∈ C

the reduced set

(m ≥ n) = ⇒ Ψ(s) =

n

• j=1

(xj − s)M(x∗

j + s)

M ≥ 0 parameter F (s) = P (s) + βΨ(s)

– p. 27

ˆ β

F (s) = P (s) + βΨ(s) Ψ(s) =

n

• j=1

(xj − s)M(x∗

j + s)

M ≥ 0 parameter

Assume P (s) is not in GPE (else trivial) Take M > 0 parameter

F ∈ GPE ⇐ ⇒ β ≥ ˆ β ˆ β = − min

ω∈R ω=ixj

min

p=1 , ... , l λp

 

M −1

2n−1

P

k=0

ikCkωk

n

Q

j=1

|xj−iω|2

 

– p. 28

GPE Polynomial Interpolation

Find all minimal degree F (s) in GPE s.t.

F @ 1 2 3 1 A → @ 18 75 50 1 A

P (s) = −3s4 + 34s2 − 13 6 × 6 Vandermonde matrix

minimal deg. interpolating in Even (−P ∈ GPE)

F (s) = P (s) +β Ψ(s) = −3s4+34s2−13 +β (1−s2)(4−s2)(9−s2)

β ∈ R ⇒ F ∈ Even F ∈ GPE ⇐ ⇒ β ≥ ˆ β = 1

2

– p. 29

Problem Formulation

Given:

F a family of

structured polynomials

F = { F : C → Cl×l}

Data points

x1 , . . . , xm ∈ C Y1 , . . . , Ym ∈ Cl×l

Find all minimal (McMillan) degree F ∈ F s.t.

Yj = F (xj) j = 1 , . . . , m

– p. 30

Partial Structure Polynomial Interpolation

Given

x1, . . . , xm ∈ C Y1 , . . . , Ym ∈ Cl×l

Find

P (s) =

q−1

• k=0

Cksk Ck ∈ Cl×l,

s.t.

P #(s) = AP (s)B A,B∈Cl×l non−singular P (xj) = Yj j=1, ... , m.

Feasible data,

xj−xk=0 = ⇒ Yj = Yk xj+x∗

k=0

= ⇒ Yj = (AYkB)∗

m ≥ k > j ≥ 1.

– p. 31

Partial Structure Interpolation (cont.)

P (s) =

q−1

• k=0

Cksk Ck ∈ Cl×l        

Il x1Il x2

1Il

... x2n−1

1

Il

. . . . . . . . . . . . . . .

Il xnIl x2

nIl

... x2n−1

n

Il Il −x∗

1Il (−x∗ 1)2Il ... (−x∗ 1)2n−1Il

. . . . . . . . . . . . . . .

Il −x∗

nIl (−x∗ n)2Il ... (−xn ∗)2n−1Il

             

Co

. . .

Cn−1 Cn

. . .

C2n−1

      =       

Y1

. . .

Yn (AY1B)∗

. . .

(AYnB)∗

       2n ≥ q (n dim. reduced set)

– p. 32

Structured Polynomials Interpolation

P (s) =

q−1

• k=0

Cksk Ck ∈ Cl×l,

s.t.

P #(s) = AP (s)(A∗)−1 A∈Cl×l non−singular P (xj) = Yj j=1, ... , m. Ψ(s)=

n

Q

j=1

(xj−s)M(x∗

j +s)

AM=(AM)∗ M parameter

F (s) = P (s) + βΨ(s) interpolating polynomial

Compute ˆ

β s.t. F ∈ F ⇐ ⇒ β ≥ ˆ β

– p. 33

Relaxing Convexity

F (s) = P (s) + βΨ(s) interpolating polynomial Ψ ∈ F

a Convex subCone of all P (s)

deg(Ψ) ≥ deg(P ) = ⇒

Compute ˆ

β s.t. F ∈ F ⇐ ⇒ β ≥ ˆ β

Taking β > 0 sufficiently large, all we need is “absorbing"

– p. 34

Non-Singular Hermitian Matrices

Non-Convex:

−1 0

2

• +

2

0 −1

• = ( 1 0

0 1 )

Ap and AΨ l × l Hermitian matrices. AΨ nonsingular:

π eigenvalues in C+ l−π eigenvalues in C−

β > ApA−1

Ψ =

⇒ (Ap + βAΨ) has

π eigenvalues in C+ l−π eigenvalues in C−

– p. 35

J-Spectral Factorizable Polynomials

F =

• G(s)
• −Iν

Il−ν

• G#(s) : ν ∈ [0, l]

G(s) polynomial

• ν = 0 =

⇒ F = GPE ν = l = ⇒ F = −GPE

Fix ν ∈ [1, l − 1] (F is not convex) Given a feasible, reduced data set

x1 , . . . , xn ∈ C Y1 , . . . , Yn ∈ Cl×l

Find (all) low degree interpolating F ∈ F

F (xj) = Yj j = 1, . . . , n

– p. 36

J-Spectral Factorizable (cont.)

F (s) = P (s) + βΨ(s) interpolating polynomial P ∈ Even interpolating polynomial 2n − 1 ≥ deg(P ) Ψ(s) =

n

• j=1

(xj − s)R

• −Iν

Il−ν

• R∗(x∗

j + s)

R parameter ˆ β = max

ω∈R

• 2n−1

P

k=0

R−1Ck(R∗)−1(iω)k

n

Q

j=1

|xj−iω|2

• (If xj = iωj

ωj ∈ R is an interpolation point, a whole

neighborhood of ωj is excluded from the above max

ω∈R )

β > ˆ β = ⇒ F ∈ F

– p. 37

GP - Revisited

GP := { F (s) : (F + F ∗) ≥ 0 for almost all s ∈ iR} P is a subCIC of GP F (s) is in GP ⇐ ⇒ F 1

s

• is in

GP F (s) is in P ⇐ ⇒ F 1

s

• is in

P

– p. 38

GP - Revisited (cont.)

G#(s) := G∗(−s∗) F ∈ GP ⇐ ⇒ F (s) = G(s)P (s)G#(s) P ∈ P G and G−1 analytic in C+ F (s) is in GP ⇐ ⇒ G(s)F (s)G#(s) is in GP F ∈ GP a polynomial of degree 2n ⇐ ⇒ (sn)F 1

s

• (sn)# a polynomial in

GP

– p. 39

CRC of GP or GE Polynomials

F (s) =

2n

• k=0

Cksk is

in GP or GE.

R(F ) is the reverse of F :

R(F (s)) := snF( 1

s)(sn)#

= (−1)n

2n

P

k=0

Cks2n−k = (−1)n(Cos2n+C1s2n−1+...+C2n−1s+C2n)

For n prescribed, GP polynomials a Convex Reversible Cone (CRC)

GPE polynomials - a subCRC

– p. 40

Interpolation - Convex Invertible Sets

F a family of rational functions Fa(s) Fb(s) in F

Yj=Fa(xj) j=1 , ... , m Yj=Fb(xj) j=1 , ... , m

• =

⇒ Yj = F (xj) j = 1 , . . . , m F ∈ conv(Fa , F −1

a

, Fb , F −1

b

)

Namely,

F =(1 − γ) (α Fa+(1 − α) Fb)+γ (β F −1

a

+(1 − β) F −1

b

)−1 α, β, γ ∈ [0, 1]

Polynomials:

γ ≡ 0 (no linear fractional transformation)

– p. 41

CRCs of Polynomials (cont.)

F a family of polynomials Fa(s) Fb(s) in F

Yj=Fa(xj) j=1 , ... , m Yj=Fb(xj) j=1 , ... , m

• =

⇒ Yj = F (xj) j = 1 , . . . , m F = (1 − γ) (α Fa+(1 − α) Fb)+γ R (β R(Fa)+(1 − β) R(Fb)) α, β, γ ∈ [0, 1] F (s) =

2n

• k=0

Cksk F can be GP or GPE

– p. 42

CICs - References

• D. Alpay, I. Lewkowicz, “Convex Cones of Generalized

Positive Rational Functions and the Nevanlinna-Pick Interpolation," to appear in Lin. Alg & Appl. Available at http://arxiv.org/abs/1010.0546.

• N. Cohen, I. Lewkowicz, “Convex Invertible Cones and

the Lyapunov Equation", Lin. Alg. & Appl., Vol. 250,

• pp. 105-131, 1997.
• N. Cohen, I. Lewkowicz, “Convex Invertible Cones of

State Space Systems", Mathematics of Control Signals and

Systems, Vol. 10, pp. 265-285, 1997.

– p. 43

CICs - References (cont.)

• N. Cohen, I. Lewkowicz, “Convex Invertible Cones and

Positive Real Analytic Functions", Lin. Alg. & Appl., Vol. 425,

• pp. 797-813, 2007.
• N. Cohen, I. Lewkowicz, “The Lyapunov Order for Real

Matrices", Lin. Alg. & Appl., Vol. 430, pp. 1489-1866, 2009.

• I. Lewkowicz, L. Rodman, E. Yarkony, “Convex Invertible

Sets and Matrix Sign Function", Lin. Alg. & Appl., Vol. 396,

• pp. 329-352, 2005

– p. 44