Applications of linear barycentric rational interpolation at - PowerPoint PPT Presentation

Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Applications of linear barycentric rational interpolation at equispaced nodes Jean-Paul Berrut (with Georges Klein and Michael Floater) University of Fribourg (Switzerland) jean-paul.berrut@unifr.ch math.unifr.ch/berrut SC2011, S. Margherita di Pula, Sardinia, October 2011 Berrut Applications of LBR interpolation at equidistant nodes 1/49

Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Outline Interpolation 1 Differentiation of barycentric rational interpolants 2 Linear barycentric rational finite differences 3 Integration of barycentric rational interpolants 4 Berrut Applications of LBR interpolation at equidistant nodes 2/49

Interpolation One-dimensional interpolation Differentiation of barycentric rational interpolants Barycentric Lagrange interpolation Linear barycentric rational finite differences Polynomial to rational interpolation Integration of barycentric rational interpolants Floater and Hormann interpolation Introduction and notation Interpolation Berrut Applications of LBR interpolation at equidistant nodes 3/49

Interpolation One-dimensional interpolation Differentiation of barycentric rational interpolants Barycentric Lagrange interpolation Linear barycentric rational finite differences Polynomial to rational interpolation Integration of barycentric rational interpolants Floater and Hormann interpolation One-dimensional interpolation problem Given: a ≤ x 0 < x 1 < . . . < x n ≤ b , n + 1 distinct nodes and f ( x 0 ) , f ( x 1 ) , . . . , f ( x n ) , corresponding values. There exists a unique polynomial of degree ≤ n that interpolates the f i , i.e. p n [ f ]( x i ) = f i , i = 0 , 1 , . . . , n . The Lagrange form of the polynomial interpolant is n � � ( x − x k ) p n [ f ]( x ) := f j ℓ j ( x ) , ℓ j ( x ) := ( x j − x k ) . j =0 k � = j Berrut Applications of LBR interpolation at equidistant nodes 4/49

Interpolation One-dimensional interpolation Differentiation of barycentric rational interpolants Barycentric Lagrange interpolation Linear barycentric rational finite differences Polynomial to rational interpolation Integration of barycentric rational interpolants Floater and Hormann interpolation One-dimensional interpolation problem Given: a ≤ x 0 < x 1 < . . . < x n ≤ b , n + 1 distinct nodes and f ( x 0 ) , f ( x 1 ) , . . . , f ( x n ) , corresponding values. There exists a unique polynomial of degree ≤ n that interpolates the f i , i.e. p n [ f ]( x i ) = f i , i = 0 , 1 , . . . , n . The Lagrange form of the polynomial interpolant is n � � ( x − x k ) p n [ f ]( x ) := f j ℓ j ( x ) , ℓ j ( x ) := ( x j − x k ) . j =0 k � = j Berrut Applications of LBR interpolation at equidistant nodes 4/49

Interpolation One-dimensional interpolation Differentiation of barycentric rational interpolants Barycentric Lagrange interpolation Linear barycentric rational finite differences Polynomial to rational interpolation Integration of barycentric rational interpolants Floater and Hormann interpolation The first barycentric form Denote the leading factors of the ℓ j ’s by � ( x j − x k ) − 1 , ν j := j = 0 , 1 , . . . , n , k � = j the so–called weights, which may be computed in advance. Rewrite the polynomial in its first barycentric form n � ν j p n [ f ]( x ) = L ( x ) f j , x − x j j =0 where n � L ( x ) := ( x − x k ) . k =0 Berrut Applications of LBR interpolation at equidistant nodes 5/49

Interpolation One-dimensional interpolation Differentiation of barycentric rational interpolants Barycentric Lagrange interpolation Linear barycentric rational finite differences Polynomial to rational interpolation Integration of barycentric rational interpolants Floater and Hormann interpolation The first barycentric form Denote the leading factors of the ℓ j ’s by � ( x j − x k ) − 1 , ν j := j = 0 , 1 , . . . , n , k � = j the so–called weights, which may be computed in advance. Rewrite the polynomial in its first barycentric form n � ν j p n [ f ]( x ) = L ( x ) f j , x − x j j =0 where n � L ( x ) := ( x − x k ) . k =0 Berrut Applications of LBR interpolation at equidistant nodes 5/49

Interpolation One-dimensional interpolation Differentiation of barycentric rational interpolants Barycentric Lagrange interpolation Linear barycentric rational finite differences Polynomial to rational interpolation Integration of barycentric rational interpolants Floater and Hormann interpolation Advantages evaluation in O ( n ) operations, ease of adding new data ( x n +1 , f n +1 ), numerically best for evaluation. Berrut Applications of LBR interpolation at equidistant nodes 6/49

Interpolation One-dimensional interpolation Differentiation of barycentric rational interpolants Barycentric Lagrange interpolation Linear barycentric rational finite differences Polynomial to rational interpolation Integration of barycentric rational interpolants Floater and Hormann interpolation Advantages evaluation in O ( n ) operations, ease of adding new data ( x n +1 , f n +1 ), numerically best for evaluation. Berrut Applications of LBR interpolation at equidistant nodes 6/49

Interpolation One-dimensional interpolation Differentiation of barycentric rational interpolants Barycentric Lagrange interpolation Linear barycentric rational finite differences Polynomial to rational interpolation Integration of barycentric rational interpolants Floater and Hormann interpolation Advantages evaluation in O ( n ) operations, ease of adding new data ( x n +1 , f n +1 ), numerically best for evaluation. Berrut Applications of LBR interpolation at equidistant nodes 6/49

Interpolation One-dimensional interpolation Differentiation of barycentric rational interpolants Barycentric Lagrange interpolation Linear barycentric rational finite differences Polynomial to rational interpolation Integration of barycentric rational interpolants Floater and Hormann interpolation The barycentric formula The constant f ≡ 1 is represented exactly by its polynomial interpolant: n � ν j 1 = L ( x ) = p n [1]( x ) . x − x j j =0 Dividing p n [ f ] by 1 and cancelling L ( x ) gives the barycentric form of the polynomial interpolant n � ν j f j x − x j j =0 p n [ f ]( x ) = . n � ν j x − x j j =0 Berrut Applications of LBR interpolation at equidistant nodes 7/49

Interpolation One-dimensional interpolation Differentiation of barycentric rational interpolants Barycentric Lagrange interpolation Linear barycentric rational finite differences Polynomial to rational interpolation Integration of barycentric rational interpolants Floater and Hormann interpolation The barycentric formula The constant f ≡ 1 is represented exactly by its polynomial interpolant: n � ν j 1 = L ( x ) = p n [1]( x ) . x − x j j =0 Dividing p n [ f ] by 1 and cancelling L ( x ) gives the barycentric form of the polynomial interpolant n � ν j f j x − x j j =0 p n [ f ]( x ) = . n � ν j x − x j j =0 Berrut Applications of LBR interpolation at equidistant nodes 7/49

Interpolation One-dimensional interpolation Differentiation of barycentric rational interpolants Barycentric Lagrange interpolation Linear barycentric rational finite differences Polynomial to rational interpolation Integration of barycentric rational interpolants Floater and Hormann interpolation Advantages Interpolation is guaranteed : n � � ν j f j x − x j j =0 lim = f k . n � x → x k � ν j x − x j j =0 Simplification of the weights: Cancellation of common factor leads to simplified weights. For equispaced nodes, � n � ν ∗ j = ( − 1) j . j Berrut Applications of LBR interpolation at equidistant nodes 8/49

Interpolation One-dimensional interpolation Differentiation of barycentric rational interpolants Barycentric Lagrange interpolation Linear barycentric rational finite differences Polynomial to rational interpolation Integration of barycentric rational interpolants Floater and Hormann interpolation Advantages Interpolation is guaranteed : n � � ν j f j x − x j j =0 lim = f k . n � x → x k � ν j x − x j j =0 Simplification of the weights : Cancellation of common factor leads to simplified weights. For equispaced nodes, � n � ν ∗ j = ( − 1) j . j Berrut Applications of LBR interpolation at equidistant nodes 8/49

Interpolation One-dimensional interpolation Differentiation of barycentric rational interpolants Barycentric Lagrange interpolation Linear barycentric rational finite differences Polynomial to rational interpolation Integration of barycentric rational interpolants Floater and Hormann interpolation Form polynomial to rational interpolation In the barycentric form of the polynomial interpolant n � ν j f j x − x j j =0 p n [ f ]( x ) = , n � ν j x − x j j =0 the weights are defined in such a way that n � ν j L ( x ) = 1 . x − x j j =0 Modification of these weights � rational interpolant. Berrut Applications of LBR interpolation at equidistant nodes 9/49