Interpolation problems on cycloidal spaces J. M. Carnicer, E. Mainar - PowerPoint PPT Presentation

Interpolation problems on cycloidal spaces J. M. Carnicer, E. Mainar and J. M. Pe˜ na Departamento de Matem´ atica Aplicada, Universidad de Zaragoza Multivariate Approximation and Interpolation with Applications 25–30 September 2013 J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 1 / 32

Interpolation on polynomial spaces � 1 , x , . . . , x n � P n := Hermite interpolation problems on P n For x 0 , . . . , x n , not necessarily distinct, find p ∈ P n such that λ i p = λ i f , i = 0 , . . . , n , where λ i f := f ( r i − 1) ( x i ) , r i = # { j ≤ i | x j = x i } . The Hermite interpolation problem in P n has always a unique solution P ( f ; x 0 , . . . , x n ) for any set of nodes x 0 , . . . , x n . J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 2 / 32

Newton basis Newton basis functions Given x 0 , . . . , x n − 1 , not necessarily distinct, the Newton basis function ω n ( x ) := ( x − x 0 ) · · · ( x − x n − 1 ) . For n = 0, define ω 0 ( x ) = 1. The Newton basis function ω n ( x ) is a function in P n vanishing on x 0 , . . . , x n − 1 and whose coefficient in x n with respect to the basis (1 , x , . . . , x n ) is 1. This function can be regarded as the interpolation error of the function x n at x 0 , . . . , x n − 1 ω n ( x ) = x n − P ( · ) n ; x 0 , . . . , x n − 1 � � ( x ) The set of functions ( ω k ( x )) k =0 ,..., n form a basis of P n . J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 3 / 32

Divided differences and Newton interpolation formula Divided difference [ x 0 , . . . , x n ] f is the coefficient in x n with respect to the basis (1 , x , . . . , x n ) of the interpolant P ( f ; x 0 , . . . , x n ) Newton interpolation formula n � P ( f ; x 0 , . . . , x n )( x ) = [ x 0 , . . . , x k ] f ω k ( x ) k =0 J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 4 / 32

Aitken-Neville formula and recurrence relations Neville formula for the polynomial interpolant P ( f ; x 0 , . . . , x n ) = x n − x P ( f ; x 0 , . . . , x n − 1 ) + x − x 0 P ( f ; x 1 , . . . , x n ) . x n − x 0 x n − x 0 Recurrence relations for divided differences [ x 0 , . . . , x n ] f = [ x 1 , . . . , x n ] f − [ x 0 , . . . , x n − 1 ] f . x n − x 0 Divided differences at a single point [ x 0 , . . . , x n ] f = f ( n ) ( x 0 ) , x 0 = · · · = x n . n ! f ( k ) ( x 0 ) Follows from Taylor formula: P ( f ; x 0 , . . . , x 0 ) = � n ( x − x 0 ) k . k =0 k ! J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 5 / 32

Interpolation on Chebyshev spaces Given u 0 , . . . , u n a set of linearly independent functions, U n = � u 0 , . . . , u n � Hermite interpolation problems on U n : For x 0 , . . . , x n , not necessarily distinct, find u ∈ U n such that λ i u = λ i f , i = 0 , . . . , n , where λ i f := f ( r i − 1) ( x i ), r i = # { j ≤ i | x j = x i } . Hermite interpolation problems do not have a solution for any set of nodes. A condition is required. J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 6 / 32

Extended collocation matrices If we express the interpolant u ( x ) = � n k =0 c k u k ( x ) in terms of the given basis, the interpolation conditions lead to a linear system     λ 0 f c 0 M ∗ � u 0 , . . . , u n . . � . .  =     . . x 0 , . . . , x n    c n λ n f whose coefficient matrix M ∗ � u 0 , . . . , u n � := ( u ( m i ) ( t i )) 0 ≤ i ≤ n ; 0 ≤ j ≤ n , m i := # { k < i | x k = x i } j x 0 , . . . , x n is called the extended collocation matrix . Theorem The Hermite problem has a unique solution if and only if the corresponding extended collocation matrix has nonzero determinant. J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 7 / 32

Extended Chebyshev systems Definition A system of functions is extended Chebyshev (ET) on [ a , b ] if all extended collocation matrices have positive determinants, det M ∗ � u 0 , . . . , u n � > 0 , for all x 0 ≤ · · · ≤ x n in [ a , b ] . x 0 , . . . , x n An extended Chebyshev (ET) space is a space generated by an extended Chebyshev basis. If U n is ET, then the Hermite interpolation problem at an arbitrary extended sequence of nodes has always a unique solution. J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 8 / 32

In order to derive Newton and Aitken-Neville formulae, it is required that interpolation problems at k ≤ n nodes x i , . . . , x i + k have a unique solution in the space U k = � u 0 , . . . , u k � . Definition A system of functions is extended complete Chebyshev (ECT) on [ a , b ] if all systems ( u 0 , . . . , u k ), k = 0 , . . . , n , are extended Chebyshev. M¨ uhlbach derived Newton formulae and Aitken-Neville formulae for ECT spaces on [ a , b ]. Remark An ET space on [ a , b ] is ECT on sufficiently small subintervals. The hypothesis that we have a ECT basis might lead to strong restrictions on the domain. For our purposes we need to discuss the validity of the formulae under weaker hypotheses. J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 9 / 32

Critical length J. M. Carnicer, E. Mainar, J. M. Pe˜ na; Critical Length for Design Purposes and Extended Chebyshev Spaces, Const. Approx. 20 , 55–71. Definition Let U n be a space of differentiable functions which is invariant under translations. The critical length of U is the number ℓ n ∈ (0 , + ∞ ] such that U is ET on any interval I if and only if I does not contain a compact interval of length ℓ n . J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 10 / 32

Critical length Proposition Let U n be an ( n + 1)-dimensional space of differentiable functions which is invariant under translations and reflections. Let ( u 0 , . . . , u n ) be a basis such that W ( u 0 , . . . , u n )(0) is lower triangular with positive diagonal entries. Then U n is an ET space on each interval of length less than or equal to α if and only if w k , n ( x ) := det W ( u k , . . . , u n )( x ) > 0 , ∀ k > n / 2 , t ∈ (0 , α ] . If the space is invariant under reflections, the critical length can be identified as the first positive zero of the functions w k , n , k > n / 2, that is ℓ n := min k > n / 2 min { α ; w k , n ( α ) = 0 , α > 0 } . J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 11 / 32

Cycloidal spaces General Cycloidal spaces C n C 1 := � cos x , sin x � cos x , sin x , 1 , x , . . . , x n − 2 � � C n := , n ≥ 2 An alternative basis to (cos x , sin x , 1 , x , x 2 , . . . , x n − 2 ) for C n is given by ϕ 0 ( x ) := cos x , � x � x 1 ( x − t ) i − 1 cos tdt , i = 1 , . . . , n . ϕ i ( x ) := ϕ i − 1 ( y ) dy = ( i − 1)! 0 0 Clearly, ϕ k ∈ C k and k (0) = · · · = ϕ ( k − 1) ϕ ( k ) ϕ k (0) = ϕ ′ (0) = 0 , k (0) = 1 . k J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 12 / 32

Fundamental functions ϕ 0 ( x ) = cos x = 1 − x 2 2! + · · · , ϕ 1 ( x ) = sin x = x − x 3 3! + · · · , ϕ 2 ( x ) = 1 − cos x = x 2 2! − x 4 4! + · · · , ϕ 3 ( x ) = x − sin x = x 3 3! − x 5 5! + · · · , ϕ 4 ( x ) = cos x − 1 + x 2 2! = x 4 4! − x 6 6! + · · · , ϕ 5 ( x ) = sin x − x + x 3 3! = x 5 5! − x 7 7! + · · · , J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 13 / 32

The Hermite interpolation problem on cycloidal spaces Hermite interpolation problems on C n For x 0 , . . . , x n , not necessarily distinct, find c ∈ C n such that λ i f := f ( r i − 1) ( x i ) , λ i c = λ i f , i = 0 , . . . , n , r i = # { j ≤ i | x j = x i } . Hermite interpolation problems on cycloidal spaces do not have solution on any sequence of nodes. Notation If the solution exists and is unique, we denote it by C ( f ; x 0 , . . . , x n ). J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 14 / 32

Existence of the cycloidal interpolant They are spaces invariant under translations and reflections and therefore they have a critical length ℓ n . Proposition The cycloidal space C n is ET on [ a , b ] if b − a < ℓ n Therefore a sufficient condition on x 0 ≤ · · · ≤ x n for the existence of solution of the Hermite interpolation problem is that x n − x 0 < ℓ n . J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 15 / 32

A Taylor formula The Taylor problem has solution because x 0 = · · · = x n and x n − x 0 = 0 < ℓ n . Taylor formula n − 2 n f ( k ) ( x 0 )( x − x 0 ) k C ( f ; x [ n +1] � � f ( k ) ( x 0 ) ϕ k ( x − x 0 ) . )( x ) = + 0 k ! k =0 k = n − 1 J. M. Carnicer (Universidad de Zaragoza) Interpolation on cycloidal spaces MAIA 2013 16 / 32