Issues in Multivariate Polynomial Interpolation Carl de Boor - PDF document

Issues in Multivariate Polynomial Interpolation Carl de Boor 27sep11 pages.cs.wisc.edu/ ∼ deboor/multiint/multiint.html 46

Univariate polynomial interpolation Given an infinite sequence z := ( z 1 , z 2 , . . . ) ∈ FN , F = R or C , the corresponding sequence k ∏ ω k,z ( x ) := ( x − z i ) , k = 0 , 1 , 2 , . . . , i =1 is a basis for the space Π( F 1 ) of univariate polynomials. 45

Univariate polynomial interpolation Given an infinite sequence z := ( z 1 , z 2 , . . . ) ∈ FN , F = R or C , the corresponding sequence k ∏ ω k,z ( x ) := ( x − z i ) , k = 0 , 1 , 2 , . . . , i =1 is a basis for the space Π( F 1 ) of univariate polynomials. Hence, every p ∈ Π( F 1 ) has a unique expansion as the corresponding Newton series ∞ ∑ p =: ω j − 1 ,z ∆( z ≤ j ) p, j =1 with ∆( z ≤ j ) := ∆( z 1 , . . . , z j ) the linear functionals on Π( F 1 ) defined thereby, and this par- ticular notation to be justified in a moment. 44

Univariate polynomial interpolation Given an infinite sequence z := ( z 1 , z 2 , . . . ) ∈ FN , F = R or C , the corresponding sequence k ∏ ω k,z ( x ) := ( x − z i ) , k = 0 , 1 , 2 , . . . , i =1 is a basis for the space Π( F 1 ) of univariate polynomials. Hence, every p ∈ Π( F 1 ) has a unique expansion as the corresponding Newton series ∞ ∑ p =: ω j − 1 ,z ∆( z ≤ j ) p, j =1 with ∆( z ≤ j ) := ∆( z 1 , . . . , z j ) the linear functionals on Π( F 1 ) defined thereby, and this par- ticular notation to be justified in a moment. By Taylor’s theorem, ∞ ∑ ( x − a ) j − 1 D j − 1 p ( a ) p ( x ) = ( j − 1)! , j =1 therefore (with z j = a , all j ) ) p = D j − 1 p ( a ) ∆( a, . . . , a ( j − 1)! , � �� � j terms while p ( z 1 ) = ∆( z 1 ) p p ( z 2 ) = ∆( z 1 ) p + ( x − z 1 )∆( z 1 , z 2 ) p + ( x − z 1 )( x − z 2 ) q ( x ) , hence, if x = z 2 ̸ = z 1 , then p ( z 2 ) − p ( z 1 ) = ∆( z 1 .z 2 ) p, z 2 − z 1 a divided difference . 43

Univariate polynomial interpolation Given an infinite sequence z := ( z 1 , z 2 , . . . ) ∈ FN , F = R or C , the corresponding sequence k ∏ ω k,z ( x ) := ( x − z i ) , k = 0 , 1 , 2 , . . . , i =1 is a basis for the space Π( F 1 ) of univariate polynomials. Hence, every p ∈ Π( F 1 ) has a unique expansion as the corresponding Newton series ∞ ∑ p =: ω j − 1 ,z ∆( z ≤ j ) p, j =1 with ∆( z ≤ j ) := ∆( z 1 , . . . , z j ) the linear functionals on Π( F 1 ) defined thereby, and this par- ticular notation to be justified in a moment. ∑ n ∑ Then, for any n , p − ω j − 1 ,z ∆( z ≤ j ) p = ω j − 1 ,z ∆( z ≤ j ) p . j =1 j>n � �� � � �� � =: p n =: ω n,z q n 42

Univariate polynomial interpolation Given an infinite sequence z := ( z 1 , z 2 , . . . ) ∈ FN , F = R or C , the corresponding sequence k ∏ ω k,z ( x ) := ( x − z i ) , k = 0 , 1 , 2 , . . . , i =1 is a basis for the space Π( F 1 ) of univariate polynomials. Hence, every p ∈ Π( F 1 ) has a unique expansion as the corresponding Newton series ∞ ∑ p =: ω j − 1 ,z ∆( z ≤ j ) p, j =1 with ∆( z ≤ j ) := ∆( z 1 , . . . , z j ) the linear functionals on Π( F 1 ) defined thereby, and this par- ticular notation to be justified in a moment. ∑ n ∑ Then, for any n , p − ω j − 1 ,z ∆( z ≤ j ) p = ω j − 1 ,z ∆( z ≤ j ) p . j =1 j>n � �� � � �� � =: p n =: ω n,z q n • p n is the remainder of the division of p by ω n,z , hence the unique element of Π <n agreeing with p at z ≤ n = ( z 1 , . . . , z n ) (counting multiplicities), and depends continuously on z ≤ n . 41

Univariate polynomial interpolation Given an infinite sequence z := ( z 1 , z 2 , . . . ) ∈ FN , F = R or C , the corresponding sequence k ∏ ω k,z ( x ) := ( x − z i ) , k = 0 , 1 , 2 , . . . , i =1 is a basis for the space Π( F 1 ) of univariate polynomials. n ∑ ∑ Then, for any n , p − ω j − 1 ,z ∆( z ≤ j ) p = ω j − 1 ,z ∆( z ≤ j ) p . j =1 j>n � �� � � �� � =: p n =: ω n,z q n • p n is the remainder of the division of p by ω n,z , hence the unique element of Π <n agreeing with p at z ≤ n = ( z 1 , . . . , z n ) (counting multiplicities), and depends continuously on z ≤ n . • Hermite interpolation is the limit of Lagrange interpolation as data sites appro- priately coalesce. 40

Univariate polynomial interpolation Given an infinite sequence z := ( z 1 , z 2 , . . . ) ∈ FN , F = R or C , the corresponding sequence k ∏ ω k,z ( x ) := ( x − z i ) , k = 0 , 1 , 2 , . . . , i =1 is a basis for the space Π( F 1 ) of univariate polynomials. n ∑ ∑ Then, for any n , p − ω j − 1 ,z ∆( z ≤ j ) p = ω j − 1 ,z ∆( z ≤ j ) p . j =1 j>n � �� � � �� � =: p n =: ω n,z q n • p n is the remainder of the division of p by ω n,z , hence the unique element of Π <n agreeing with p at z ≤ n = ( z 1 , . . . , z n ) (counting multiplicities), and depends continuously on z ≤ n . Note: • ∆( z ≤ n ) depends on the multiset { z 1 , . . . , z n } (and not on the ordering). • ∑ ω j − 1 ,z ω n,z ∆( z ≤ j ) p = q n = ∆( z ≤ n , · ) p = ⇒ ∆( z ≤ n +2 ) = ∆( z n +1 , z n +2 )∆( z ≤ n , · ) j>n Hence, by induction, • ∆( z ≤ n , · ) = ∆( z n , · )∆( z n − 1 , · ) · · · ∆( z 2 , · )∆( z 1 , · ), showing why ∆( z ≤ n +1 ) is called an n th divided difference . 39

Univariate polynomial interpolation Given an infinite sequence z := ( z 1 , z 2 , . . . ) ∈ FN , F = R or C , the corresponding sequence k ∏ ω k,z ( x ) := ( x − z i ) , k = 0 , 1 , 2 , . . . , i =1 is a basis for the space Π( F 1 ) of univariate polynomials. n ∑ ∑ Then, for any n , p − ω j − 1 ,z ∆( z ≤ j ) p = ω j − 1 ,z ∆( z ≤ j ) p . j =1 j>n � �� � � �� � =: p n =: ω n,z q n • p n is the remainder of the division of p by ω n,z , hence the unique element of Π <n agreeing with p at z ≤ n = ( z 1 , . . . , z n ) (counting multiplicities), and depends continuously on z ≤ n . Note: • ∆( z ≤ n ) depends on the multiset { z 1 , . . . , z n } (and not on the ordering). • ∑ ω j − 1 ,z ω n,z ∆( z ≤ j ) p = q n = ∆( z ≤ n , · ) p = ⇒ ∆( z ≤ n +2 ) = ∆( z n +1 , z n +2 )∆( z ≤ n , · ) j>n Hence, by induction, • ∆( z ≤ n , · ) = ∆( z n , · )∆( z n − 1 , · ) · · · ∆( z 2 , · )∆( z 1 , · ), showing why ∆( z ≤ n +1 ) is called an n th divided difference . ∫ 1 • (error ` 0 Dp ( x + s ( y − x )) d s (even if x = y ), a la Genocchi-Hermite): Since ∆( x, y ) p = get ∫ 1 ∫ s 1 ∫ s n − 1 ( D n p ) ( z 1 + s 1 ( z 2 − z 1 ) + · · · + s n ( x − z n )) d s n · · · d s 1 · · · q n ( x ) = 0 0 0 ∫ M ( z 1 , . . . , z n , x | · ) D n p =: 38

Polynomial interpolation 37

Multivariate polynomial interpolation dim Π ≤ 1 ( R 2 ) = 3 < 4 < 6 = dim Π ≤ 2 ( R 2 ) 36

Multivariate polynomial interpolation Now, even Π ≤ 2 ( R 2 ) won’t do; need at least Π ≤ 3 ( R 2 ). 35

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in F d , how to choose a space F of polynomials so that, for every choice of data values ∈ F T , a = ( a ( τ ) : τ ∈ T) there is exactly one element f ∈ F that matches this information, i.e., satisfies f ( τ ) = a ( τ ) , τ ∈ T . I.e., want F to be correct for T. unisolvent, poised, properly posed, regular, . . . F is R or C . 34

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in F d , how to choose a space F of polynomials so that, for every choice of data values ∈ F T , a = ( a ( τ ) : τ ∈ T) there is exactly one element f ∈ F that matches this information, i.e., satisfies f ( τ ) = a ( τ ) , τ ∈ T . I.e., want F to be correct for T. unisolvent, poised, properly posed, regular, . . . F is R or C . Equivalently, want the linear map δ T : g �→ g | T := ( g ( τ ) : τ ∈ T) ∈ F T (of restriction to T) to be onto and 1-1 when restricted to F . Then ∑ P T := (( δ T ) | F ) − 1 δ T : g �→ ℓ τ g ( τ ) τ ∈ T is the linear projector that associates each g with its unique interpolant P T g at T in F and ∑ τ ∈ T ℓ τ g ( τ ) is its Lagrange form. 33

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in F d , how to choose a polynomial space F correct for T. 32

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in F d , how to choose a polynomial space F correct for T. Want a map T �→ Π T with the following properties: • Π T is a pol. space correct for T, with P T g the interpolant from Π T at T to g ; 31

Multivariate polynomial interpolation Basic Problem: Given a finite set T of data sites τ in F d , how to choose a polynomial space F correct for T. Want a map T �→ Π T with the following properties: • Π T is a pol. space correct for T, with P T g the interpolant from Π T at T to g ; • T �→ Π T is monotone, i.e., T ⊂ Σ = ⇒ Π T ⊂ Π Σ ; 30