The Condition of Linear Barycentric Rational Interpolation from Equispaced Samples

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1 The Condition of Linear Barycentric Rational Interpolation from Equispaced Samples Georges Klein University of Fribourg (Switzerland) homeweb.unifr.ch/kleing/pub Vancouver, ICIAM, July 21, 2011 G. Klein Condition of LBRI from equispaced samples 1/23

2 Outline 1 Barycentric rational interpolation 2 3 G. Klein Condition of LBRI from equispaced samples 2/23

3 Introduction and notation One-dimensional interpolation Floater Hormann interpolation Barycentric rational interpolation G. Klein Condition of LBRI from equispaced samples 3/23

4 One-dimensional interpolation One-dimensional interpolation Floater Hormann interpolation Given: a x 0 < x 1 <... < x n b, f(x 0 ),f(x 1 ),...,f(x n ), n+1 distinct nodes and corresponding values. Aim: find a function g from a finite-dimensional linear subspace of (C[a,b], ) such that g interpolates f at the nodes, g(x j ) = f(x j ) = f j, j = 0,...,n. G. Klein Condition of LBRI from equispaced samples 4/23

5 One-dimensional interpolation Floater Hormann interpolation Construction presented by Floater and Hormann - Choose an integer d {0,1,...,n}, - for j = 0,...,n d, define p j (x), the polynomial of degree d interpolating f j,f j+1,...,f j+d. The d-th interpolant of the family is given by r n [f](x) = n d λ j (x)p j (x) j=0 n d λ j (x) j=0, where λ j (x) = ( 1) j (x x j )...(x x j+d ). G. Klein Condition of LBRI from equispaced samples 5/23

6 One-dimensional interpolation Floater Hormann interpolation Construction presented by Floater and Hormann - Choose an integer d {0,1,...,n}, - for j = 0,...,n d, define p j (x), the polynomial of degree d interpolating f j,f j+1,...,f j+d. The d-th interpolant of the family is given by r n [f](x) = n d λ j (x)p j (x) j=0 n d λ j (x) j=0, where λ j (x) = ( 1) j (x x j )...(x x j+d ). G. Klein Condition of LBRI from equispaced samples 5/23

7 One-dimensional interpolation Floater Hormann interpolation Construction presented by Floater and Hormann - Choose an integer d {0,1,...,n}, - for j = 0,...,n d, define p j (x), the polynomial of degree d interpolating f j,f j+1,...,f j+d. The d-th interpolant of the family is given by r n [f](x) = n d λ j (x)p j (x) j=0 n d λ j (x) j=0, where λ j (x) = ( 1) j (x x j )...(x x j+d ). G. Klein Condition of LBRI from equispaced samples 5/23

8 One-dimensional interpolation Floater Hormann interpolation Lemma (Berrut Baltensperger Mittelmann (2005)) Let {x j } n j=0 be a set of n+1 distinct nodes, {f j} n j=0 corresponding real numbers and let {v j } n j=0 be any non-zero real numbers. Then (a) the barycentric rational function R(x) = n j=0 n j=0 v j x x j f j v j x x j, interpolates f k at x k : lim x xk R(x) = f k ; (b) conversely, every rational interpolant of the f j may be written in barycentric form for some weights v j. G. Klein Condition of LBRI from equispaced samples 6/23

9 Barycentric weights One-dimensional interpolation Floater Hormann interpolation Write r n [f] in barycentric form r n [f](x) = n j=0 n j=0 w j x x j f j w j x x j For equispaced nodes, the weights w j oscillate in sign with absolute values 1,1,...,1,1, d = 0, (Berrut) 1 2,1,1,...,1,1, 1 2, d = 1, (Berrut) 1 4, 3 4,1,1,...,1,1, 3 4, 1 4, d = 2, (Floater Hormann) 1 8, 4 8, 7 8,1,1,...,1,1, 7 8, 4 8, 1 8, d = 3. (Floater Hormann). G. Klein Condition of LBRI from equispaced samples 7/23

10 One-dimensional interpolation Floater Hormann interpolation Theorem (Floater Hormann (2007)) Let 0 d n and f C d+2 [a,b], h = max 0 i n 1 (x i+1 x i ), then the rational function r n [f] has no poles in lr, if d 1, r n [f] f = max a x b r n[f](x) f(x) Ch d+1, if d = 0, r n [f] f Cβh, where β is a mesh ratio and C is a generic constant, independent of n. G. Klein Condition of LBRI from equispaced samples 8/23

11 Lebesgue function and constant Lower and upper bound Condition of Floater Hormann interpolation in equispaced nodes G. Klein Condition of LBRI from equispaced samples 9/23

12 Lebesgue function and constant Lebesgue function and constant Lower and upper bound The Lebesgue constant n w j x x j Λ n = max Λ j=0 n(x) = max, a x b a x b n w j x x j is the condition number of the interpolation method. It is the maximum of the Lebesgue function Λ n (x) in [a,b]. j=0 G. Klein Condition of LBRI from equispaced samples 10/23

13 Lebesgue function and constant Lower and upper bound Lebesgue function for Floater Hormann interpolation Figure: Lebesgue function for Floater Hormann interpolation in equispaced nodes in [ 1,1] with d = 2,3,4,5 and n = 40 G. Klein Condition of LBRI from equispaced samples 11/23

14 Lebesgue function and constant Lower and upper bound Lebesgue constant for Floater Hormann interpolation Theorem (Bos De Marchi Hormann K. (2011)) Let 0 d n and the nodes x j, j = 0,...,n, be equispaced. Then 2 d 2 ( n ) d +1 ln d 1 Λ n 2 d 1 (2+ln(n)). Logarithmic growth with n but exponential growth with d. G. Klein Condition of LBRI from equispaced samples 12/23

15 Lebesgue function and constant Lower and upper bound Lebesgue constant for Floater Hormann interpolation Theorem (Bos De Marchi Hormann K. (2011)) Let 0 d n and the nodes x j, j = 0,...,n, be equispaced. Then 2 d 2 ( n ) d +1 ln d 1 Λ n 2 d 1 (2+ln(n)). Logarithmic growth with n but exponential growth with d. G. Klein Condition of LBRI from equispaced samples 12/23

16 Construction of EFH Examples Extended Floater Hormann interpolation G. Klein Condition of LBRI from equispaced samples 13/23

17 Construction of EFH Examples It turns out that for given d, Λ n (x) has at most d high oscillations at the ends and is much smaller in the remaining part of the interval if the nodes are equispaced. Idea: we add 2d new data values f d,..., f 1 and f n+1,..., f n+d, corresponding to additional nodes x d,...,x 1 and x n+1,...,x n+d. The global data set is then interpolated and evaluated only in the interval [a, b]. To be precise, we choose positive integers ñ n and d ñ, and compute r (k) ñ [f](x 0) and r (k) ñ [f](x n), k = 1,..., d, where rñ[f](x 0 ) is the rational interpolant of the values f 0,...,fñ, evaluated in x 0, and rñ[f](x n ) is the rational interpolant of the values f n ñ,...,f n, evaluated in x n, both with parameter d. G. Klein Condition of LBRI from equispaced samples 14/23

18 Construction of EFH Examples It turns out that for given d, Λ n (x) has at most d high oscillations at the ends and is much smaller in the remaining part of the interval if the nodes are equispaced. Idea: we add 2d new data values f d,..., f 1 and f n+1,..., f n+d, corresponding to additional nodes x d,...,x 1 and x n+1,...,x n+d. The global data set is then interpolated and evaluated only in the interval [a, b]. To be precise, we choose positive integers ñ n and d ñ, and compute r (k) ñ [f](x 0) and r (k) ñ [f](x n), k = 1,..., d, where rñ[f](x 0 ) is the rational interpolant of the values f 0,...,fñ, evaluated in x 0, and rñ[f](x n ) is the rational interpolant of the values f n ñ,...,f n, evaluated in x n, both with parameter d. G. Klein Condition of LBRI from equispaced samples 14/23

19 Construction of EFH Examples It turns out that for given d, Λ n (x) has at most d high oscillations at the ends and is much smaller in the remaining part of the interval if the nodes are equispaced. Idea: we add 2d new data values f d,..., f 1 and f n+1,..., f n+d, corresponding to additional nodes x d,...,x 1 and x n+1,...,x n+d. The global data set is then interpolated and evaluated only in the interval [a, b]. To be precise, we choose positive integers ñ n and d ñ, and compute r (k) ñ [f](x 0) and r (k) ñ [f](x n), k = 1,..., d, where rñ[f](x 0 ) is the rational interpolant of the values f 0,...,fñ, evaluated in x 0, and rñ[f](x n ) is the rational interpolant of the values f n ñ,...,f n, evaluated in x n, both with parameter d. G. Klein Condition of LBRI from equispaced samples 14/23

20 Construction of EFH Examples We set f 0 + r (k) ñ [f](x 0) (x j x 0 ) k, d j 1, k! k=1 fj := f j, 0 j n, f n + d d r (k) k=1 ñ [f](x n) (x j x n ) k, n+1 j n+d. k! Our extension of the Floater Hormann family then is r n [f](x) := n+d j= d n+d j= d w j x x j fj w j x x j. G. Klein Condition of LBRI from equispaced samples 15/23

21 Construction of EFH Examples We set f 0 + r (k) ñ [f](x 0) (x j x 0 ) k, d j 1, k! k=1 fj := f j, 0 j n, f n + d d r (k) k=1 ñ [f](x n) (x j x n ) k, n+1 j n+d. k! Our extension of the Floater Hormann family then is r n [f](x) := n+d j= d n+d j= d w j x x j fj w j x x j. G. Klein Condition of LBRI from equispaced samples 15/23

22 Construction of EFH Examples Convergence of extended Floater Hormann interpolation Notation: D := min{d, d}. Theorem (K. (2011)) Suppose n, d, ñ < n and d ñ are positive integers and assume that f C d+2 [a dh,b +dh] C 2 d+1 ([a,a+ñh] [b ñh,b]) is sampled at n+1 equispaced nodes in [a,b]. Then r n [f] f = max a x b r n[f](x) f(x) Ch D+1. r n has no poles in lr. G. Klein Condition of LBRI from equispaced samples 16/23

23 Construction of EFH Examples Lebesgue constant for extended FH interpolation Theorem (K. (2011)) For positive integers n and d, the Lebesgue constant Λ n associated with extended Floater Hormann interpolation is bounded as Λ n 2+ln(n+2d). Logarithmic growth with n and d. G. Klein Condition of LBRI from equispaced samples 17/23

24 Construction of EFH Examples Lebesgue constant for extended FH interpolation Theorem (K. (2011)) For positive integers n and d, the Lebesgue constant Λ n associated with extended Floater Hormann interpolation is bounded as Λ n 2+ln(n+2d). Logarithmic growth with n and d. G. Klein Condition of LBRI from equispaced samples 17/23

25 Lebesgue functions Construction of EFH Examples Figure: Lebesgue function with n = 50 for extended FH interpolation in equispaced nodes with d = 3 (left), for polynomial interpolation in Chebyshev points of the second kind (center) and first kind (right) G. Klein Condition of LBRI from equispaced samples 18/23

26 Lebesgue constants and bound Construction of EFH Examples FH EFH bound for EFH Figure: Lebesgue constants associated with FH and EFH interpolation with d = 8 and 8 n 1000, together with the upper bound on the EFH Lebesgue constant G. Klein Condition of LBRI from equispaced samples 19/23

27 Lebesgue constants Construction of EFH Examples 10 6 FH EFH Figure: Lebesgue constants associated with FH and EFH interpolation in equispaced nodes with n = 200 and 1 d 25 G. Klein Condition of LBRI from equispaced samples 20/23

28 Construction of EFH Examples Interpolation of Runge s example 1/(1+x 2 ) in [ 5,5] 10 2 spline FH EFH Figure: Error behaviour of spline, FH and EFH interpolation of 1/(1+x 2 ) with d = 4 and 20 n 1000 G. Klein Condition of LBRI from equispaced samples 21/23

29 Construction of EFH Examples Perturbed Runge s example 1/(1+x 2 ) in [ 5,5] 10 4 spline FH EFH Figure: Error behaviour of spline, FH and EFH interpolation of 1/(1+x 2 ) with sign alternating perturbation of the data, n = 1000 and 1 d 50 G. Klein Condition of LBRI from equispaced samples 22/23

30 Thank you for your attention! G. Klein Condition of LBRI from equispaced samples 23/23

Convergence rates of derivatives of a family of barycentric rational interpolants

Convergence rates of derivatives of a family of barycentric rational interpolants J.-P. Berrut, M. S. Floater and G. Klein University of Fribourg (Switzerland) CMA / IFI, University of Oslo jean-paul.berrut@unifr.ch