Convergence rates of derivatives of a family of barycentric rational interpolants

Transcription

1 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 michaelf@ifi.uio.no georges.klein@unifr.ch Avignon, June 29, 2010 Berrut-Floater-Klein Convergence rates of r (k) (x) 1/28

2 Outline 1 Interpolation 2 3 Berrut-Floater-Klein Convergence rates of r (k) (x) 2/28

3 Introduction and notation Interpolation Berrut-Floater-Klein Convergence rates of r (k) (x) 3/28

4 One-dimensional interpolation problem 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. There exists a unique polynomial of degree n that interpolates the data, i.e. p n [f ](x i ) = f i, i = 0,1,...,n. The Lagrange form of the polynomial interpolant is given by p n [f ](x) := n f j l j (x), l j (x) := k j (x x k ) (x j x k ). Berrut-Floater-Klein Convergence rates of r (k) (x) 4/28

5 One-dimensional interpolation problem 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. There exists a unique polynomial of degree n that interpolates the data, i.e. p n [f ](x i ) = f i, i = 0,1,...,n. The Lagrange form of the polynomial interpolant is given by p n [f ](x) := n f j l j (x), l j (x) := k j (x x k ) (x j x k ). Berrut-Floater-Klein Convergence rates of r (k) (x) 4/28

6 The first barycentric form Denote the leading factors of the l j s by λ j := k j(x j x k ) 1, j = 0,1,...,n, which may be computed in advance. Rewrite the polynomial in its first barycentric form p n [f ](x) = L(x) n λ j x x j f j, where n L(x) := (x x k ). k=0 Berrut-Floater-Klein Convergence rates of r (k) (x) 5/28

7 The first barycentric form Denote the leading factors of the l j s by λ j := k j(x j x k ) 1, j = 0,1,...,n, which may be computed in advance. Rewrite the polynomial in its first barycentric form p n [f ](x) = L(x) n λ j x x j f j, where n L(x) := (x x k ). k=0 Berrut-Floater-Klein Convergence rates of r (k) (x) 5/28

8 Advantages evaluation in O(n) operations, ease of adding new data (x n+1,f n+1 ), numerically best for the evaluation. Berrut-Floater-Klein Convergence rates of r (k) (x) 6/28

9 Advantages evaluation in O(n) operations, ease of adding new data (x n+1,f n+1 ), numerically best for the evaluation. Berrut-Floater-Klein Convergence rates of r (k) (x) 6/28

10 Advantages evaluation in O(n) operations, ease of adding new data (x n+1,f n+1 ), numerically best for the evaluation. Berrut-Floater-Klein Convergence rates of r (k) (x) 6/28

11 The barycentric formula The constant f 1 is represented exactly by its polynomial interpolant: n λ j 1 = L(x) = p n [1](x). x x j Dividing p n [f ] by 1 and cancelling L(x) gives the barycentric form of the polynomial interpolant n λ j f j x x j p n [f ](x) =. n λ j x x j Berrut-Floater-Klein Convergence rates of r (k) (x) 7/28

12 The barycentric formula The constant f 1 is represented exactly by its polynomial interpolant: n λ j 1 = L(x) = p n [1](x). x x j Dividing p n [f ] by 1 and cancelling L(x) gives the barycentric form of the polynomial interpolant n λ j f j x x j p n [f ](x) =. n λ j x x j Berrut-Floater-Klein Convergence rates of r (k) (x) 7/28

13 Advantages Interpolation is guaranteed: lim p n [f ](x) = lim x x k x x k λ k x x k f k λ k x x k = f k. Simplification of the weights: Cancellation of common factor leads to simplified weights. For equidistant nodes, ( ) n λ j = ( 1) j. j Berrut-Floater-Klein Convergence rates of r (k) (x) 8/28

14 Advantages Interpolation is guaranteed: lim p n [f ](x) = lim x x k x x k λ k x x k f k λ k x x k = f k. Simplification of the weights: Cancellation of common factor leads to simplified weights. For equidistant nodes, ( ) n λ j = ( 1) j. j Berrut-Floater-Klein Convergence rates of r (k) (x) 8/28

15 Form polynomial to rational interpolation In the barycentric form of the polynomial interpolant p n [f ](x) = n n λ j x x j f j λ j x x j, the weights are defined in such a way that L(x) n λ j x x j = 1. Modification of these weights rational interpolant. Berrut-Floater-Klein Convergence rates of r (k) (x) 9/28

16 Form polynomial to rational interpolation In the barycentric form of the polynomial interpolant p n [f ](x) = n n λ j x x j f j λ j x x j, the weights are defined in such a way that L(x) n λ j x x j = 1. Modification of these weights rational interpolant. Berrut-Floater-Klein Convergence rates of r (k) (x) 9/28

17 Lemma Let {(x j,f j )}, j = 0,1,...,n be n + 1 pairs of distinct real numbers and let {u j } be any nonzero real numbers. Then (a) the rational function r n [f ](x) = n n u j x x j f j u j x x j, interpolates f k at x k : lim x xk r n [f ](x) = f k ; (b) conversely, every rational interpolant of the f j may be written in barycentric form for some weights u j. Berrut-Floater-Klein Convergence rates of r (k) (x) 10/28

18 Floater and Hormann interpolants Weights suggested by Berrut: ( 1) j ; 1/2,1,1,...,1,1,1/2 with oscillating sign. Floater and Hormann in 2007: new choice for the weights family of barycentric rational interpolants. Berrut-Floater-Klein Convergence rates of r (k) (x) 11/28

19 Floater and Hormann interpolants Weights suggested by Berrut: ( 1) j ; 1/2,1,1,...,1,1,1/2 with oscillating sign. Floater and Hormann in 2007: new choice for the weights family of barycentric rational interpolants. Berrut-Floater-Klein Convergence rates of r (k) (x) 11/28

20 Construction presented by Floater and Hormann - Choose an integer d {0,1,...,n}, - define p j (x), the polynomial of degree d interpolating f j,f j+1,...,f j+d, for j = 0,...,n d. The d-th interpolant is given by r n [f ](x) = n d λ j (x)p j (x) n d λ j (x), where λ j (x) = ( 1) j (x x j )... (x x j+d ). Berrut-Floater-Klein Convergence rates of r (k) (x) 12/28

21 Construction presented by Floater and Hormann - Choose an integer d {0,1,...,n}, - define p j (x), the polynomial of degree d interpolating f j,f j+1,...,f j+d, for j = 0,...,n d. The d-th interpolant is given by r n [f ](x) = n d λ j (x)p j (x) n d λ j (x), where λ j (x) = ( 1) j (x x j )... (x x j+d ). Berrut-Floater-Klein Convergence rates of r (k) (x) 12/28

22 Construction presented by Floater and Hormann - Choose an integer d {0,1,...,n}, - define p j (x), the polynomial of degree d interpolating f j,f j+1,...,f j+d, for j = 0,...,n d. The d-th interpolant is given by r n [f ](x) = n d λ j (x)p j (x) n d λ j (x), where λ j (x) = ( 1) j (x x j )... (x x j+d ). Berrut-Floater-Klein Convergence rates of r (k) (x) 12/28

23 Barycentric weights Write r n [f ] in barycentric form r n [f ](x) = n n w j x x j f j w j x x j For equidistant nodes, the weights 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). Berrut-Floater-Klein Convergence rates of r (k) (x) 13/28

24 Theorem Let 0 d n and f C d+2 [a,b], then the rational function r n [f ] has no poles in lr, if n d is odd, then r n [f ] f h d+1 (b a) f (d+2) d+2 if d 1, r n [f ] f h(1 + β)(b a) f 2 if d = 0; if n d is even, then r n [f ] f h d+1( (b a) f (d+2) d+2 + f (d+1) ) d+1 if d 1, r n [f ] f h(1 + β) ( (b a) f 2 + f ) if d = 0. β := { max min xi x i+1 1 i n 2 x i x i 1, x i+1 x i } x i+1 x i+2 Berrut-Floater-Klein Convergence rates of r (k) (x) 14/28

25 Differentiation of barycentric rational interpolants Berrut-Floater-Klein Convergence rates of r (k) (x) 15/28

26 Proposition (Schneider and Werner (1986)) Let r n [f ] be a rational function given in its barycentric form with non vanishing weights. Assume that x is not a pole of r n [f ]. Then for k 1 1 k! r(k) n [f ](x) = 1 k! r(k) n n w j x x j r n [f ][(x) k,x j ] n w j, x not a node, x x j ( n / [f ](x i ) = w j r n [f ][(x i ) k,x j ]) w i, i = 0,...,n. j i Berrut-Floater-Klein Convergence rates of r (k) (x) 16/28

27 Differentiation matrices Define the matrices D (1) and D (2) : w j 1, w i x i x j D (1) ij := n D (1) ik ; D (2) ij := k=0 k i ( 2D (1) ij D (1) ii 1 ), i j, x i x j n D (2) ik, i = j. k=0 k i If f := (f 0,...,f n ) T, then D (1) f, respectively D (2) f, returns the vector of the first, respectively second, derivative of r n [f ] at the nodes. Berrut-Floater-Klein Convergence rates of r (k) (x) 17/28

28 Differentiation matrices Define the matrices D (1) and D (2) : w j 1, w i x i x j D (1) ij := n D (1) ik ; D (2) ij := k=0 k i ( 2D (1) ij D (1) ii 1 ), i j, x i x j n D (2) ik, i = j. k=0 k i If f := (f 0,...,f n ) T, then D (1) f, respectively D (2) f, returns the vector of the first, respectively second, derivative of r n [f ] at the nodes. Berrut-Floater-Klein Convergence rates of r (k) (x) 17/28

29 Convergence rates for the derivatives For x [a,b], we call the error e(x) := f (x) r n [f ](x). Theorem At the nodes, we have if d 0 and if f C d+2 [a,b], then e (x j ) Ch d, j = 0,1,...,n; if d 1 and if f C d+3 [a,b], then e (x j ) Ch d 1, j = 0,1,...,n. Berrut-Floater-Klein Convergence rates of r (k) (x) 18/28

30 Convergence rates for the derivatives For x [a,b], we call the error e(x) := f (x) r n [f ](x). Theorem At the nodes, we have if d 0 and if f C d+2 [a,b], then e (x j ) Ch d, j = 0,1,...,n; if d 1 and if f C d+3 [a,b], then e (x j ) Ch d 1, j = 0,1,...,n. Berrut-Floater-Klein Convergence rates of r (k) (x) 18/28

31 Theorem (continued) At intermediate points, we have if d 1 and if f C d+3 [a,b], then e Ch d if d 2, e C(β + 1)h if d = 1; if d 2 and if f C d+4 [a,b], then Mesh ratio { β := max e C(β + 1)h d 1 if d 3, e C(β 2 + β + 1)h if d = 2. max 1 i n 1 x i x i+1 x i x i 1, max 0 i n 2 x i+1 x i }. x i+1 x i+2 Berrut-Floater-Klein Convergence rates of r (k) (x) 19/28

32 Remarks In the important cases k = 1,2 the convergence rate of the k-th derivative is O(h d+1 k ) as h 0: In short: Loss of one order per differentiation. Stricter conditions on the differentiability of f compared to the interpolant. Berrut-Floater-Klein Convergence rates of r (k) (x) 20/28

33 Remarks In the important cases k = 1,2 the convergence rate of the k-th derivative is O(h d+1 k ) as h 0: In short: Loss of one order per differentiation. Stricter conditions on the differentiability of f compared to the interpolant. Berrut-Floater-Klein Convergence rates of r (k) (x) 20/28

34 Remarks In the important cases k = 1,2 the convergence rate of the k-th derivative is O(h d+1 k ) as h 0: In short: Loss of one order per differentiation. Stricter conditions on the differentiability of f compared to the interpolant. Berrut-Floater-Klein Convergence rates of r (k) (x) 20/28

35 Proof for the first derivative at the nodes The Newton error formula f (x) p i (x) = (x x i ) (x x i+d )[x i,x i+1,...,x i+d,x]f leads to e(x) = where after cancellation and n d i=0 λ i(x)(f (x) p i (x)) n d i=0 λ = A(x) i(x) B(x), n d A(x) := ( 1) i [x i,...,x i+d,x]f i=0 n d B(x) := λ i (x). i=0 Berrut-Floater-Klein Convergence rates of r (k) (x) 21/28

36 Proof for the first derivative at the nodes The Newton error formula f (x) p i (x) = (x x i ) (x x i+d )[x i,x i+1,...,x i+d,x]f leads to e(x) = where after cancellation and n d i=0 λ i(x)(f (x) p i (x)) n d i=0 λ = A(x) i(x) B(x), n d A(x) := ( 1) i [x i,...,x i+d,x]f i=0 n d B(x) := λ i (x). i=0 Berrut-Floater-Klein Convergence rates of r (k) (x) 21/28

37 Proof for the first derivative at the nodes The Newton error formula f (x) p i (x) = (x x i ) (x x i+d )[x i,x i+1,...,x i+d,x]f leads to e(x) = where after cancellation and n d i=0 λ i(x)(f (x) p i (x)) n d i=0 λ = A(x) i(x) B(x), n d A(x) := ( 1) i [x i,...,x i+d,x]f i=0 n d B(x) := λ i (x). i=0 Berrut-Floater-Klein Convergence rates of r (k) (x) 21/28

38 By definition and from the fact that e(x j ) = 0, we have Defining the functions where we see that e e(x) (x j ) = lim. x xj x x j B j (x) := i+d ( 1) i 1, x x k i I j C j (x) := i I \I j ( 1) i k=i k j i+d k=i 1 x x k, I = {0,1,...,n d} and I j = {i I : j d i j}, (x x j )B(x) = B j (x) + (x x j )C j (x). Berrut-Floater-Klein Convergence rates of r (k) (x) 22/28

39 By definition and from the fact that e(x j ) = 0, we have Defining the functions where we see that e e(x) (x j ) = lim. x xj x x j B j (x) := i+d ( 1) i 1, x x k i I j C j (x) := i I \I j ( 1) i k=i k j i+d k=i 1 x x k, I = {0,1,...,n d} and I j = {i I : j d i j}, (x x j )B(x) = B j (x) + (x x j )C j (x). Berrut-Floater-Klein Convergence rates of r (k) (x) 22/28

40 By definition and from the fact that e(x j ) = 0, we have Defining the functions where we see that e e(x) (x j ) = lim. x xj x x j B j (x) := i+d ( 1) i 1, x x k i I j C j (x) := i I \I j ( 1) i k=i k j i+d k=i 1 x x k, I = {0,1,...,n d} and I j = {i I : j d i j}, (x x j )B(x) = B j (x) + (x x j )C j (x). Berrut-Floater-Klein Convergence rates of r (k) (x) 22/28

41 We have now e(x) x x j = A(x) B j (x) + (x x j )C j (x), and taking the limit of both sides as x x j gives e (x j ) = A(x j) B j (x j ). Berrut-Floater-Klein Convergence rates of r (k) (x) 23/28

42 We have now e(x) x x j = A(x) B j (x) + (x x j )C j (x), and taking the limit of both sides as x x j gives e (x j ) = A(x j) B j (x j ). Berrut-Floater-Klein Convergence rates of r (k) (x) 23/28

43 With x = x j in B j (x), the products alternate in sign as ( 1) i does, so that all the terms in the sum have the same sign and B j (x j ) = i+d x j x k 1. i I j k=i k j Therefore, by choosing any i I j, we deduce that i+d 1 B j (x j ) k=i k j x j x k Ch d, i I j. On the other hand Floater and Hormann had already shown that A(x) C, x [a,b], whence the bound follows. Berrut-Floater-Klein Convergence rates of r (k) (x) 24/28

44 With x = x j in B j (x), the products alternate in sign as ( 1) i does, so that all the terms in the sum have the same sign and B j (x j ) = i+d x j x k 1. i I j k=i k j Therefore, by choosing any i I j, we deduce that i+d 1 B j (x j ) k=i k j x j x k Ch d, i I j. On the other hand Floater and Hormann had already shown that A(x) C, x [a,b], whence the bound follows. Berrut-Floater-Klein Convergence rates of r (k) (x) 24/28

45 With x = x j in B j (x), the products alternate in sign as ( 1) i does, so that all the terms in the sum have the same sign and B j (x j ) = i+d x j x k 1. i I j k=i k j Therefore, by choosing any i I j, we deduce that i+d 1 B j (x j ) k=i k j x j x k Ch d, i I j. On the other hand Floater and Hormann had already shown that A(x) C, x [a,b], whence the bound follows. Berrut-Floater-Klein Convergence rates of r (k) (x) 24/28

46 Numerical Berrut-Floater-Klein Convergence rates of r (k) (x) 25/28

47 Runge s function Table: Error in the interpolation and the derivatives of the rational interpolant of 1/(1 + x 2 ) for d = 3 Interpolation First derivative Second derivative n error order error order error order e e e e e e e e e e e e e e e e e e e e e Berrut-Floater-Klein Convergence rates of r (k) (x) 26/28

48 Runge s function Table: Error in the interpolation and the derivatives of the rational interpolant of 1/(1 + x 2 ) for d = 3 Interpolation First derivative Second derivative n error order error order error order e e e e e e e e e e e e e e e e e e e e e Berrut-Floater-Klein Convergence rates of r (k) (x) 26/28

49 Runge s function Table: Error in the interpolation and the derivatives of the rational interpolant of 1/(1 + x 2 ) for d = 3 Interpolation First derivative Second derivative n error order error order error order e e e e e e e e e e e e e e e e e e e e e Berrut-Floater-Klein Convergence rates of r (k) (x) 26/28

50 Comparison with cubic spline 10 0 FH d=3, cubic spline (spline toolbox) log(error) FH interpolant Cubic spline 1st derivative FH 1st derivative spline 2nd derivative FH 2nd derivative spline log(n) Berrut-Floater-Klein Convergence rates of r (k) (x) 27/28

51 Thank you for your attention! Berrut-Floater-Klein Convergence rates of r (k) (x) 28/28