Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Nontraditional Notions of Polynomial Ordering with Computational Applications Steve Hussung Indiana University Bloomington Midwestern Workshop on Asymptotic Analysis Purdue University Fort Wayne October 4-6, 2019 1/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Table of contents Intro 1 Potential Theoretic Background 2 Convex Bodies, Orderings 3 WAMS, Discrete Sequences 4 2/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Introduction and Motivation Why consider nontraditional notions of degree? Suppose we approximate some function f , analytic in the hypercube H d := [ − 1 , 1] d . Then consider degree( p ) ≤ n � f − p � H d inf In [T17], it is shown that for analytic functions with an analytic continuation to a ”particular” set around the hypercube. √ � d + ǫ ) , O ( ρ − n / for traditional degree O ( ρ − n + ǫ ) , for two selected nontraditional degrees for any ǫ > 0, where ρ > 0 depends on the “particular set”. 3/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Notation • We begin with a compact set K ⊂ C d for positive d ∈ Z . • Then for positive k ∈ Z , define the polynomial spaces P Σ ( k ) as all polynomials of standard degree less than or equal to k . • Let M k be the dimension of P Σ ( k ). • For positive s ∈ Z , let α ( s ) be an enumeration of the multi-exponents, so that e s = z α ( s ) for s = 1 , . . . , M k forms a basis for P Σ ( k ). 4/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Vandermonde Matrix and Determinant Then for a given k and sequence ( z i ) s i =1 ⊂ K , we can form the s by s Vandermonde matrix VDM. [VDM( z 1 , z 2 , . . . , z s )] i , j = z α ( j ) i And its determinant V ( z 1 , . . . , z s ) = | det(VDM( z 1 , . . . , z s )) | We will focus on maximizing V ( z 1 , . . . , z s ) over sets of s points in K . 5/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Vandermonde Matrix and Determinant Note: In one dimension, the determinant of the Vandermonde is just the product of the distances between each pair of points. s s � � V ( z 1 , . . . , z s ) = | z i − z j | i =1 j > i 6/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Fekete Points • For any s , there exist set(s) of points ( ζ i ) s i =1 ⊂ K that maximize V . We call these Fekete Points . s • We define the measure µ s on K via µ s ( z ) = 1 � δ ζ i ( z ). s i =1 • These measures converge weak-*: µ s ⇀ µ K , where µ K is the potential theoretic equilibrium measure 7/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Fekete Points: Example If K is the unit complex disk, { z : | z | ≤ 1 } , then the s th order Fekete points are the roots of unity of order s . 8/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Fekete Points: Asymptotics k � Let L k = i ( M i − M i − 1 ). i =1 1 / L K . k →∞ V ( ζ ( k ) 1 , . . . , ζ ( k ) and we want to define τ ( K ) = lim M k ) In one dimension, it straightforward to prove that V ( ζ 1 , . . . , ζ M k ) 1 / L k is decreasing in k , and has a limit as k → ∞ which we call the transfinite diameter [R95] In multiple dimensions, this is much more difficult , but was done in [Z75]. � M k � z ( k ) We say that an array i =1 for each k is Asymptotically Fekete i if 1 / L k → τ ( K ) V ( z ( k ) 1 , . . . , z ( k ) M k ) as k → ∞ 9/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences A Preview of Orderings Definition Let ≺ denote the grevlex ordering on Z d , [M19], where α ≺ β if • | α | < | β | ; or • | α | = | β | , and there exists k ∈ { 1 , . . . , d } such that α j = β j for all j < k , and α k < β k Ex: For d = 2, this gives 1 , x , y , x 2 , xy , y 2 , x 3 , x 2 y , . . . We enumerate the monomials using this ordering, so e 1 ( z ) , e 2 ( z ) , . . . are a polynomial basis. 10/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Monomial Classes and Tchebyshev Constants We then, following [Z75] in [BBCL92], define the following monomial class, Definition We define the s th monomial class, s − 1 � M ( s ) := p : p ( z ) = e s ( z ) + c j e j ( z ) : c j ∈ C j =1 Definition We define the discrete Tchebyshev constant T s := inf {� p � K : p ∈ M ( s ) } 1 / deg( e s ) And in preparation, we define the set D = { x 1 , . . . , x d ∈ R + : � x i = 1 } . 11/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Zaharjuta Conclusion Definition For θ ∈ int( C ), the directional Chebyshev constant is the function T ( K , θ ) := lim sup T s α ( s ) s →∞ , deg( es ) → θ And this gives us the formula for the transfinite diameter 1 / ( M k − M k − 1 ) � log( τ ( K )) = lim k →∞ log T s deg( e s )= k 1 � = ln T ( K , θ ) d θ meas( D ) D 12/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Zaharjuta Conclusion Lastly, from [Z75] and [BBCL92], we note the connection between the maximum Vandermonde matrix determinant, V ( ζ 1 , . . . , ζ s ), and these averages. For k ≥ 1, � M k � M k � � T deg( s ) T deg( s ) � � ≤ V ( ζ 1 , . . . , ζ M k ) ≤ M k ! s s s =1 s =1 13/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Leja Points Fekete sequences are very difficult to find, so for given k we define a Leja Sequence ( ℓ i ) s i =1 ⊂ K as a sequence given by the following procedure. • Choose ℓ 1 ∈ K at a maximum of z α (1) . • Assuming ℓ 1 , . . . , ℓ s − 1 have been chosen, • For each subsequent point ℓ s , choose a maximum of ℓ → V ( ℓ 1 , . . . , ℓ s − 1 , ℓ ). We know that Leja points are asymptotically Fekete, in both the one and several variable cases. 14/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Leja Points: Example If K is the unit complex disk, { z : | z | ≤ 1 } , then the 2 n th order Leja points are the roots of unity of order 2 n . 15/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Leja Points in C d : Proof of Asymptotically Fekete From [BBCL92], • Let L s = V ( ℓ 1 , . . . , ℓ s ). Then L s ≤ V ( ζ 1 , . . . , ζ s ) is clear. • V ( ℓ 1 , . . . , ℓ s − 1 , ℓ ) = p s ( ℓ ). V ( ℓ 1 , . . . , ℓ s − 1 ) • p s ( ℓ ) is monic, with e s ( ℓ ) being the monic term. • Further, L s / L s − 1 = � p s � K ≥ T deg( e s ) . s M k L M k L M k − 1 . . . L 1 T deg( e s ) • So L M k = � 1 = � p s � K ≥ . s L M k − 1 L M k − 2 s =1 • Taking L k th roots, we achieve L s ≥ V s / M k !. Thus, in C d , Leja point sequences are asymptotically Fekete. 16/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Polynomial Spaces associated with Convex Bodies So far, we have used P Σ ( k ) to denote our polynomial space. We let � d � � Σ := x 1 , . . . , x d : x i ≥ 0 , x i ≤ 1 i =1 + be a convex body containing 1 More generally, let C ∈ R d n Σ for some positive n . 17/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences C Polynomial Spaces Then we define the polynomial space P C ( k ) as � c J z J P C ( k ) := p ( z ) = J ∈ C ∩ Z d + (Which encompasses our use of P Σ ( k ) thus far.) With these new polynomials comes the question of C-degree, which we answer deg C ( p ) = p ∈P C ( k ) ( k ) min 18/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Grevlex Ordering Definition Let ≺ denote the grevlex ordering on Z d , [M19], where α ≺ β if • | α | < | β | ; or • | α | = | β | , and there exists k ∈ { 1 , . . . , d } such that α j = β j for all j < k , and α k < β k Ex: For d = 2, this gives 1 , x , y , x 2 , xy , y 2 , x 3 , x 2 y , . . . We note that this definition pays no attention to the gradiation given by the convex body C . 19/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Grevlex Ordering Definition Let ≺ denote the grevlex ordering on Z d , [M19], where α ≺ β if • | α | < | β | ; or • | α | = | β | , and there exists k ∈ { 1 , . . . , d } such that α j = β j for all j < k , and α k < β k 20/47
Intro Potential Theoretic Background Convex Bodies, Orderings WAMS, Discrete Sequences Nested and Additive We now deal with the question of ordering within these monomial classes. Two properties are desireable for an ordering < . Definition (Additivity) For any α , β , δ such that α < β , we have α + δ < β + δ Definition (Nested) For k 1 < k 2 , both in Z + , let α ∈ P C ( k 1 ) , β ∈ P C ( k 2 ) \ P C ( k 1 ). Then α < β We often say that an order “respects the C-degree”, if it is nested. 21/47
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