Sparse Polynomial Interpolation With Arbitrary Orthogonal - PowerPoint PPT Presentation

Sparse Polynomial Interpolation With Arbitrary Orthogonal Polynomial Bases Erdal Imamoglu � Erich L. Kaltofen Zhengfeng Yang NC State University NC State University East China Normal University Duke University Raleigh, NC, USA Shanghai, China Raleigh-Durham, NC, USA

2 Outline 1. Chebyshev Polynomials 2. Problem Statement 3. Chebyshev Bases (With A Known Sparsity t ) 4. Deterministic Early Termination

3 Chebyshev Polynomials Let K be a field Chebyshev Polynomials of degree n Tchebyshev-1: T n ( x ) Chebyshev-2: U n ( x ) Chebyshev-3: V n ( x ) Chebyshev-4: W n ( x )

3 Chebyshev Polynomials Let K be a field Chebyshev Polynomials of degree n Tchebyshev-1: T n ( x ) Chebyshev-2: U n ( x ) Chebyshev-3: V n ( x ) Chebyshev-4: W n ( x ) T 0 ( x ) = 1 U 0 ( x ) = 1 V 0 ( x ) = 1 W 0 ( x ) = 1

3 Chebyshev Polynomials Let K be a field Chebyshev Polynomials of degree n Tchebyshev-1: T n ( x ) Chebyshev-2: U n ( x ) Chebyshev-3: V n ( x ) Chebyshev-4: W n ( x ) T 0 ( x ) = 1 T 1 ( x ) = x U 0 ( x ) = 1 U 1 ( x ) = 2 x V 0 ( x ) = 1 V 1 ( x ) = 2 x − 1 W 0 ( x ) = 1 W 1 ( x ) = 2 x + 1

3 Chebyshev Polynomials Let K be a field Chebyshev Polynomials of degree n Tchebyshev-1: T n ( x ) Chebyshev-2: U n ( x ) Chebyshev-3: V n ( x ) Chebyshev-4: W n ( x ) T 0 ( x ) = 1 T 1 ( x ) = x T n ( x ) = 2 xT n − 1 ( x ) − T n − 2 ( x ) U 0 ( x ) = 1 U 1 ( x ) = 2 x U n ( x ) = 2 xU n − 1 ( x ) − U n − 2 ( x ) V 0 ( x ) = 1 V 1 ( x ) = 2 x − 1 V n ( x ) = 2 xV n − 1 ( x ) − V n − 2 ( x ) W 0 ( x ) = 1 W 1 ( x ) = 2 x + 1 W n ( x ) = 2 xW n − 1 ( x ) − W n − 2 ( x ) for n ≥ 2

4 Problem Statement A black box for f ( x ) ∈ K [ x ] is given: β → � → a = f ( β )

4 Problem Statement A black box for f ( x ) ∈ K [ x ] is given: β → � → a = f ( β ) 1. If t (number of terms) is given, using 2 t evaluations write f ( x ) as t ∑ f ( x ) = c j T δ j ( x ) i. (Chebyshev-1 Basis) j = 1 t ∑ f ( x ) = c j U δ j ( x ) ii. (Chebyshev-2 Basis) j = 1 t ∑ iii. f ( x ) = c j V δ j ( x ) (Chebyshev-3 Basis) j = 1 t ∑ f ( x ) = c j W δ j ( x ) iv. (Chebyshev-4 Basis) j = 1 where c j � = 0 and 0 ≤ δ 1 < ··· < δ t

4 Problem Statement A black box for f ( x ) ∈ K [ x ] is given: β → � → a = f ( β ) 1. If t (number of terms) is given, using 2 t evaluations write f ( x ) as t ∑ f ( x ) = c j T δ j ( x ) i. (Chebyshev-1 Basis) j = 1 t ∑ f ( x ) = c j U δ j ( x ) ii. (Chebyshev-2 Basis) j = 1 t ∑ iii. f ( x ) = c j V δ j ( x ) (Chebyshev-3 Basis) j = 1 t ∑ f ( x ) = c j W δ j ( x ) iv. (Chebyshev-4 Basis) j = 1 where c j � = 0 and 0 ≤ δ 1 < ··· < δ t 2. If B ≥ t is given, interpolate f ( x ) with exactly t + B evaluations

5 Problem Statement: Previous results [Prony, 1795] → Interpolation in power basis [Bose, Chaudhuri, Hocquenghem, 1959] → Prony

5 Problem Statement: Previous results [Prony, 1795] → Interpolation in power basis [Bose, Chaudhuri, Hocquenghem, 1959] → Prony [Lakshman & Saunders, 1995] → Interpolation in Chebyshev-1 polynomials [Potts & Tasche, 2014] → Interpolation in Chebyshev-2 polynomials → Uses floating point arithmetic [Arnold & Kaltofen, 2015] → Interpolation in Chebyshev-1 polynomials → Reduction to power bases

5 Problem Statement: Previous results [Prony, 1795] → Interpolation in power basis [Bose, Chaudhuri, Hocquenghem, 1959] → Prony [Lakshman & Saunders, 1995] → Interpolation in Chebyshev-1 polynomials [Kaltofen & Lee, 2003] → Recovers unknown t from given a degree bound for f ( x ) [Potts & Tasche, 2014] → Interpolation in Chebyshev-2 polynomials → Uses floating point arithmetic [Arnold & Kaltofen, 2015] → Interpolation in Chebyshev-1 polynomials → Reduction to power bases

6 Chebyshev Bases (With A Known Sparsity t ) Reduction to power bases: Chebyshev-1: T n ( T m ( y )) = T mn ( y ) = T m ( T n ( y )) , ∀ m , n ∈ Z ≥ 0 y n + 1 � y + 1 � y n y = , ∀ n ≥ 0 T n 2 2

6 Chebyshev Bases (With A Known Sparsity t ) Reduction to power bases: Chebyshev-1: T n ( T m ( y )) = T mn ( y ) = T m ( T n ( y )) , ∀ m , n ∈ Z ≥ 0 y n + 1 � y + 1 � y n y = , ∀ n ≥ 0 T n (17 years) 2 2 � y + 1 � � � y − 1 1 = y n + 1 − y y n + 1 , ∀ n ≥ 0 Chebyshev-2: U n (17 Years) y 2

6 Chebyshev Bases (With A Known Sparsity t ) Reduction to power bases: Chebyshev-1: T n ( T m ( y )) = T mn ( y ) = T m ( T n ( y )) , ∀ m , n ∈ Z ≥ 0 y n + 1 � y + 1 � y n y = , ∀ n ≥ 0 T n (17 years) 2 2 � y + 1 � � � y − 1 1 = y n + 1 − y y n + 1 , ∀ n ≥ 0 Chebyshev-2: U n (17 Years) y 2 � y 2 + 1 � � � y + 1 1 y 2 = y 2 n + 1 + y 2 n + 1 , ∀ n ≥ 0 Chebyshev-3: V n y 2

6 Chebyshev Bases (With A Known Sparsity t ) Reduction to power bases: Chebyshev-1: T n ( T m ( y )) = T mn ( y ) = T m ( T n ( y )) , ∀ m , n ∈ Z ≥ 0 y n + 1 � y + 1 � y n y = , ∀ n ≥ 0 T n (17 years) 2 2 � y + 1 � � � y − 1 1 = y n + 1 − y y n + 1 , ∀ n ≥ 0 Chebyshev-2: U n (17 Years) y 2 � y 2 + 1 � � � y + 1 1 y 2 = y 2 n + 1 + y 2 n + 1 , ∀ n ≥ 0 Chebyshev-3: V n y 2 � y 2 + 1 � � � y − 1 1 y 2 = y 2 n + 1 − y 2 n + 1 , ∀ n ≥ 0 Chebyshev-4: W n y 2

7 Chebyshev Bases (With A Known Sparsity t ) t ∑ Chebyshev-2 Basis: Write f ( x ) as f ( x ) = c j U δ j ( x ) j = 1

7 Chebyshev Bases (With A Known Sparsity t ) t ∑ Chebyshev-2 Basis: Write f ( x ) as f ( x ) = c j U δ j ( x ) j = 1 Define � y + 1 � y − 1 t � � � � � � y − 1 y − 1 def y y ∑ g ( y ) = = f c j U δ j y 2 y 2 j = 1 t � � � � 1 y , 1 y δ j + 1 − ∑ = ∈ K c j y δ j + 1 y j = 1

7 Chebyshev Bases (With A Known Sparsity t ) t ∑ Chebyshev-2 Basis: Write f ( x ) as f ( x ) = c j U δ j ( x ) j = 1 Define � y + 1 � y − 1 t � � � � � � y − 1 y − 1 def y y ∑ g ( y ) = = f c j U δ j y 2 y 2 j = 1 t � � � � 1 y , 1 y δ j + 1 − ∑ = ∈ K c j y δ j + 1 y j = 1 � ω i + 1 � 1 � � � � ω i − 1 ω i a i = g ( ω i ) = = − g = − a − i for ω ∈ K f ω i ω i 2 Free evaluation: a 0 = 0 Prony’s algorithm [Prony, 1795] can reconstruct g ( y )

8 Chebyshev Bases (With A Known Sparsity t ) Chebyshev-3 and Chebyshev-4 Bases can be done in a similar way � y 2 + 1 � � � y + 1 1 y 2 = y 2 n + 1 + Chebyshev-3: V n y 2 n + 1 y 2 � y 2 + 1 � � � y − 1 1 y 2 = y 2 n + 1 − Chebyshev-4: W n y 2 n + 1 y 2

8 Chebyshev Bases (With A Known Sparsity t ) Chebyshev-3 and Chebyshev-4 Bases can be done in a similar way y n + 1 � y + 1 � y n y = Chebyshev-1: T n 2 2 � y + 1 � � � y − 1 1 y = y n + 1 − Chebyshev-2: U n y n + 1 2 y � y 2 + 1 � � � y + 1 1 y 2 = y 2 n + 1 + Chebyshev-3: V n y 2 n + 1 y 2 � y 2 + 1 � � � y − 1 1 y 2 = y 2 n + 1 − Chebyshev-4: W n y 2 n + 1 y 2

8 Chebyshev Bases (With A Known Sparsity t ) Chebyshev-3 and Chebyshev-4 Bases can be done in a similar way y n + 1 � y + 1 � y n y = Chebyshev-1: T n 2 2 � y + 1 � � � y − 1 1 y = y n + 1 − Chebyshev-2: U n y n + 1 2 y � y 2 + 1 � y + 1 � � � � y + 1 y 2 y = T 2 n + 1 Chebyshev-3: V n → Chebyshev-1 y 2 2 � y 2 + 1 � y + 1 � � y 2 y = U 2 n Chebyshev-4: W n → Chebyshev-2 2 2

9 Chebyshev Bases (With A Known Sparsity t ) t ∑ Chebyshev-1 Basis: Write f ( x ) as f ( x ) = c j T δ j ( x ) j = 1 Define � y + 1 � y + 1 t � � def y y ∑ g ( y ) = f = c j T δ j 2 2 j = 1 t � � � � c j y δ j + 1 y , 1 ∑ = ∈ K y δ j 2 y j = 1 � ω i + 1 � 1 � � ω i a i = g ( ω i ) = f = g = a − i for ω ∈ K ω i 2 Prony’s algorithm [Prony, 1795] can reconstruct g ( y ) [Arnold & Kaltofen, 2015] uses 2 t + 1 evaluations if δ 1 � = 0

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11 Chebyshev Bases (With A Known Sparsity t ) [Arnold & Kaltofen, 2015] uses 2 t + 1 evaluations if δ 1 � = 0

11 Chebyshev Bases (With A Known Sparsity t ) [Arnold & Kaltofen, 2015] uses 2 t + 1 evaluations if δ 1 � = 0 Let ω ∈ K ω δ i 1 + ω δ i 2 + 1 1 � �� � 1 1 ω δ i 1 ω δ i 2 � � ω δ i 1 , ω δ i 2 , � = ω δ i 1 , ⇐ ⇒ � ≥ 3 � � ω δ i 2 2 2 �

11 Chebyshev Bases (With A Known Sparsity t ) [Arnold & Kaltofen, 2015] uses 2 t + 1 evaluations if δ 1 � = 0 Let ω ∈ K ω δ i 1 + ω δ i 2 + 1 1 � �� � 1 1 ω δ i 1 ω δ i 2 � � ω δ i 1 , ω δ i 2 , � = ω δ i 1 , ⇐ ⇒ � ≥ 3 � � ω δ i 2 2 2 � � 1 � �� ω δ t ,..., 1 � � ω δ 1 , ω δ 1 ,..., ω δ t � = 2 t or = 2 t − 1 with δ 1 = 0 Let � � �