An extension of the FloaterâHormann family of barycentric rational ...
An extension of the Floater–Hormann family of barycentric rational interpolants Georges Klein University of Oxford
Fox Prize Meeting, Edinburgh, June 24, 2013
Introduction Extended FH Applications
Outline
1
Introduction: linear barycentric rational interpolation
2
Extended Floater–Hormann interpolation
3
Applications of extended Floater–Hormann interpolation
G. Klein
Extended Floater–Hormann interpolation
2/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
One-dimensional interpolation
Given: a ≤ x0 < x1 < . . . < xn ≤ b, f (x0 ), f (x1 ), . . . , f (xn ),
n + 1 distinct nodes and corresponding values.
We study functions g from a finite-dimensional linear subspace of (C [a, b], k · k∞ ) which interpolate f between the nodes, g (xi ) = f (xi ) = fi ,
i = 0, . . . , n.
Main interest: well-conditioned equispaced interpolation.
G. Klein
Extended Floater–Hormann interpolation
3/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
The polynomial in barycentric form There exists a unique polynomial pn of degree ≤ n, which interpolates the data and can be written in Lagrange form pn (x) =
n X
`i (x)fi
with
`i (x) =
i=0
Y x − xk , xi − xk
k6=i
in the first barycentric form (backward stable (Higham ’04)) pn (x) = L(x)
n X i=0
λi fi , x − xi
.Q Qn with λi = 1 k6=i (xi − xk ), and L(x) = k=0 (x − xk ), and in the second barycentric form (forward stable (Higham ’04)) Pn λi i=0 x−xi fi . pn (x) = Pn λ i
i=0 x−xi
G. Klein
Extended Floater–Hormann interpolation
4/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
From polynomial to linear rational interpolation Lemma Let {xi }ni=0 be a set of n + 1 distinct nodes, {fi }ni=0 corresponding real numbers and let {vi }ni=0 be any non-zero real numbers. Then (a) the barycentric rational function n X
R(x) =
vi fi x − xi
i=0 n X i=0
, vi x − xi
interpolates fk at xk : limx→xk R(x) = fk ; (b) conversely, every rational interpolant of the fi may be written in barycentric form for some weights vi . G. Klein
Extended Floater–Hormann interpolation
5/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Construction of Floater–Hormann interpolation for arbitrary nodes - Given n + 1 nodes, x0 , . . . , xn , and function values, f0 , . . . , fn , choose an integer d ∈ {0, 1, . . . , n}, “blending parameter”, - for i = 0, . . . , n − d, define pi (x), the polynomial of low degree ≤ d interpolating fi , fi+1 , . . . , fi+d . The d-th interpolant of the family is a “blend” of the pi (x), n−d X
rn (x) =
λi (x)pi (x)
i=0 n−d X
,
with
λi (x) =
(−1)i . (x − xi ) . . . (x − xi+d )
λi (x)
i=0
Notice that for d = n, rn simplifies to pn . G. Klein
Extended Floater–Hormann interpolation
6/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Linear barycentric rational form For its evaluation, we write rn in linear barycentric form n−d X
rn (x) =
n X
λi (x)pi (x)
i=0 n−d X
= λi (x)
i=0 n X i=0
i=0
wi fi x − xi . wi x − xi
For equispaced nodes, the weights wi do not depend on f , and oscillate in sign with absolute values 1, 1, . . . , 1, 1, 1 4, 1 4 8, 8,
1 2 , 1, 1, . . . , 1, 1, 3 4 , 1, 1, . . . , 1, 1, 7 8 , 1, 1, . . . , 1, 1,
··· G. Klein
d = 0, 1 2, 3 4, 7 8,
d = 1, 1 4, 4 1 8, 8,
d = 2, d = 3, d ≥ 4.
Extended Floater–Hormann interpolation
7/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Algebraic convergence and polynomial reproduction
Theorem (Floater–Hormann ’07) Let 1 ≤ d ≤ n and f ∈ C d+2 [a, b], h =
max (xi+1 − xi ), then
0≤i≤n−1
kf − rn k∞ ≤ Khd+1 , where K depends only on d, b − a and derivatives of f ; the analytic rational function rn has no real poles; rn reproduces polynomials of degree at least d.
G. Klein
Extended Floater–Hormann interpolation
8/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Lebesgue function and constant The Lebesgue constant associated with linear barycentric interpolation, X n X n wi |wi | , Λn,d = max Λn,d (x) = max a≤x≤b a≤x≤b |x − xi | x − xi i=0
i=0
is the condition number of the interpolation scheme. 4
15
3
10
2
5
1 −1
0
1
−1
0
1
Figure: Lebesgue function for Floater–Hormann interpolation with equispaced nodes in [−1, 1] with d = 2 and d = 5 and n = 40. G. Klein
Extended Floater–Hormann interpolation
9/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Lebesgue function with n = 40 and d = 0, . . . , 40
(loading movie...)
G. Klein
Extended Floater–Hormann interpolation
10/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Lebesgue constant Theorem (Bos–De Marchi–Hormann–K. ’12) Let 0 ≤ d ≤ n and the nodes xi , i = 0, . . . , n, be equispaced. Then n 2d−2 log − 1 ≤ Λn,d ≤ 2d−1 (2 + log n). d +1 d 108
8 d=3
7
107
6
106
5
105
d=2
n = 100
104
4 d=1
3
103 102
2 1
101
0 0
100 0
20
40
60
80
100 120 140 160 180 200
5
10
15
20
25
Figure: Logarithmic growth with n (left) and exponential with d (right). G. Klein
Extended Floater–Hormann interpolation
11/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Central question: How to choose d? Answer: It depends on the nodes and on the regularity of f . FH ’07: large d for very smooth functions, small d for less smooth functions; Numerical Recipes ’07: “It is wise to start with small values of d before trying larger values”; BMHK ’12: not too large, the Lebesgue constant grows exponentially with d; Cross validation (no extra knowledge on the data): remove a few well chosen data values, compute the interpolant for various values of d, compare the errors at the removed data points and keep the d that gives the smallest error (funqui); EFH: wider range of good choices for d. G. Klein
Extended Floater–Hormann interpolation
12/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Interpolation with FH and d = 15
(loading movie...)
G. Klein
Extended Floater–Hormann interpolation
13/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Interpolation of perturbed data with FH and d = 15
(loading movie...)
G. Klein
Extended Floater–Hormann interpolation
14/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Interpolation of f (x) = sin(20x) perturbed data 2
10
exact data FH funqui spline cheb
1
10
FH funqui spline cheb
0
10
−5
10
0
10
−10
10
−1
10
50
100
150
200
50
100
150
200
Figure: Errors with 8 ≤ n ≤ 200 using FH with d = 15, funqui, B-spline of degree 15, and polynomial interpolation in Chebyshev points.
G. Klein
Extended Floater–Hormann interpolation
15/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Differentiation of barycentric functions Proposition (Schneider–Werner ’86) Let rn be a rational function given in its barycentric form with non vanishing weights. Assume that x is not a pole of rn . Then for k ≥1 n X (k)
rn (x) = k!
j=0
wj rn [(x)k , xj ] x − xj n X j=0
(k) rn (xi )
= −k!
n X
wj x − xj
,
x not a node,
! . wi , wj rn [(xi )k , xj ]
i = 0, . . . , n.
j=0 j6=i
G. Klein
Extended Floater–Hormann interpolation
16/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Differentiation matrices Define the matrix D(1) (Baltensperger–Berrut–No¨el ’99): wj 1 , i 6= j, wi xi − xj n (1) X Dij = (1) − Di` , i = j, `=0 `6=i
and D(k) , k > 1, with the “hybrid formula” (Tee ’06), k wj (k−1) (k−1) D − D , i= 6 j, x − x w ii ij i j i n (k) X Dij = (k) − Di` , i = j. `=0 `6=i
G. Klein
Extended Floater–Hormann interpolation
17/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Evaluation of derivatives of rn at the nodes
With f = (f0 , . . . , fn )| , the product D(k) · f returns
| (k) (k) rn (x0 ), . . . , rn (xn ) ,
the vector of the k-th derivative of rn at the nodes.
G. Klein
Extended Floater–Hormann interpolation
18/42
Introduction Extended FH Applications
Construction Convergence and stability Examples
Extended Floater–Hormann interpolation
G. Klein
Extended Floater–Hormann interpolation
19/42
Introduction Extended FH Applications
Construction Convergence and stability Examples
Extended Floater–Hormann interpolation For given d, Λn (x) typically has at most d large local maxima at the ends the interval if the nodes are equispaced. 15 10 5 −1
0
1
Idea: construct 2d new data values e f−d , . . . , e f−1 and e fn+1 , . . . , e fn+d out of the given data f0 , . . . , fn . We choose positive integers e n n and de ≤ e n, and compute (k) (k) e ren (x0 ) and ren (xn ), k = 1, . . . , d. G. Klein
Extended Floater–Hormann interpolation
20/42
Introduction Extended FH Applications
Construction Convergence and stability Examples
Extended Floater–Hormann interpolation We set e d X (xi − x0 )k (k) f + r (x ) , −d ≤ i ≤ −1, 0 0 e n k! k=1 e 0 ≤ i ≤ n, fi := fi , e d X (xi − xn )k (k) , n + 1 ≤ i ≤ n + d. f + r (x ) n n e n k! k=1
The extension of the Floater–Hormann family then is n+d X
e rn (x) :=
i=−d n+d X
wi e fi x − xi
i=−d G. Klein
. wi x − xi
Extended Floater–Hormann interpolation
21/42
Introduction Extended FH Applications
Construction Convergence and stability Examples
Convergence of extended Floater–Hormann interpolation Notation: e D := min{d, d}. Theorem Suppose n, d, e n < n and de ≤ e n are positive integers and assume e d+2 that f ∈ C [a − dh, b + dh] ∩ C 2d+1 ([a, a + e nh] ∪ [b − e nh, b]) is sampled at n + 1 equispaced nodes in [a, b]. Then kf − e rn k∞ ≤ KhD+1 . e rn is analytic, has no real poles and reproduces polynomials of degree ≤ D.
G. Klein
Extended Floater–Hormann interpolation
22/42
Introduction Extended FH Applications
Construction Convergence and stability Examples
A generalised Hermite formula (Platte’s method) Let L be a general linear approximation, n X L(x) = bi (x)fi . i=0
Suppose f is analytic inside a simple, closed, rectifiable curve C contained in a closed simply connected region around the nodes, then the error satisfies Z f (s) 1 f (x) − L(x) = R(x, s) ds, 2πi C s − x where R(x, s) = 1 − (s − x)
n X bi (x) s − xi i=0
is the relative error in the approximation of 1/(s − x) by L. G. Klein
Extended Floater–Hormann interpolation
23/42
Introduction Extended FH Applications
Construction Convergence and stability Examples
Interpolation of analytic functions −1 0.6
original FH
extended FH
−2 −3
0.4
−4 0.2
−5 −6
0 −7 −8
−0.2
−9 −0.4 −0.6 −1.5
−10 spline 15 −1
−11
funqui −0.5
0
0.5
1
1.5
−12
Figure: Comparison of the relative error of the prototype function 1/(s − x) with FH with d = 15, EFH with d = 15, de = e n = 15, B-spline of degree 15 and funqui for n = 100 on a log10 scale. G. Klein
Extended Floater–Hormann interpolation
24/42
Introduction Extended FH Applications
Construction Convergence and stability Examples
Lebesgue constant for extended FH interpolation
Theorem en associated For positive integers n and d, the Lebesgue constant Λ with extended Floater–Hormann interpolation is bounded as en ≤ 2 + log(n + 2d). Λ
Logarithmic growth with n and d.
G. Klein
Extended Floater–Hormann interpolation
25/42
Introduction Extended FH Applications
Construction Convergence and stability Examples
Lebesgue function for extended FH interpolation
(loading movie...)
G. Klein
Extended Floater–Hormann interpolation
26/42
Introduction Extended FH Applications
Construction Convergence and stability Examples
Lebesgue functions EFH
2nd kind
1st kind
3.5
3.5
3.5
3
3
3
2.5
2.5
2.5
2
2
2
1.5
1.5
1.5
1 −1
0
1
1 −1
0
1
1 −1
0
1
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
Extended Floater–Hormann interpolation
27/42
Introduction Extended FH Applications
Construction Convergence and stability Examples
Perturbed data with EFH and d = 15, de = e n = 15
(loading movie...)
G. Klein
Extended Floater–Hormann interpolation
28/42
Introduction Extended FH Applications
Construction Convergence and stability Examples
Interpolation of f (x) = sin(20x) perturbed data 2
10
exact data EFH FH funqui spline cheb
1
10
0
10
EFH FH funqui spline cheb
0
10
−5
10
−10
10
−1
10
50
100
150
200
50
100
150
200
Figure: Errors with 8 ≤ n ≤ 200 using EFH with d = 15, de = e n = 15, FH with d = 15, funqui, B-spline of degree 15, and polynomial interpolation in Chebyshev points.
G. Klein
Extended Floater–Hormann interpolation
29/42
Introduction Extended FH Applications
Construction Convergence and stability Examples
Interpolation of f (x) = sin(20x)
EFH FH spline funqui
−5
10
−10
10
5
10
15
20
25
30
Figure: Errors with 1 ≤ d ≤ 30 and n = 200 using EFH with de = e n = 15, FH and B-spline of degree d.
G. Klein
Extended Floater–Hormann interpolation
30/42
Introduction Extended FH Applications
Differentiation Integration
Applications
G. Klein
Extended Floater–Hormann interpolation
31/42
Introduction Extended FH Applications
Differentiation Integration
Approximation of derivatives via finite differences
Extended rational finite difference methods (ERFD) for the approximation at the nodes xi , 0 ≤ i ≤ n, of the k-th derivative of a sufficiently smooth function, (k)
f (k) (xi ) ≈ e rn (xi ) =
n+d X
(k) (k) Dij e fj =: e fi .
j=−d (k)
The weights Dij are the elements from the (d + 1)st to the (n + d + 1)st row of the (n + 2d + 1) × (n + 2d + 1) differentiation matrix D (k) .
G. Klein
Extended Floater–Hormann interpolation
32/42
Introduction Extended FH Applications
Differentiation Integration
Convergence of ERFD
Theorem e are positive Suppose n, d, e n < n, de ≤ e n and k ≤ D = min{d, d} integers and assume that e f ∈ C d+1+k [a − dh, b + dh] ∩ C 2d+1 ([a, a + e nh] ∪ [b − e nh, b]) is sampled at n + 1 equispaced nodes a = x0 < x1 < . . . < xn = b. Then (k)
|e rn (xj ) − f (k) (xj )| ≤ KhD+1−k ,
G. Klein
−d ≤ j ≤ n + d.
Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Differentiation Integration
Approximation of derivatives at intermediate points For the approximation of the k-th derivative of a function f at intermediate points x ∈ [a, b], there are several possibilities, among them: interpolation of the derivatives at the nodes: en(k) (x) := R
n+d X i=−d
wi (k) e rn (xi ) x − xi
n+d X i=−d
wi . x − xi
approximate e rn in Chebfun and differentiate the resulting chebfun. Usually not much loss in accuracy with such cheaper methods.
G. Klein
Extended Floater–Hormann interpolation
34/42
Introduction Extended FH Applications
Differentiation Integration
2nd derivative with FH and d = 15
(loading movie...)
G. Klein
Extended Floater–Hormann interpolation
35/42
Introduction Extended FH Applications
Differentiation Integration
2nd derivative with EFH and d = 15, de = e n = 15
(loading movie...)
G. Klein
Extended Floater–Hormann interpolation
36/42
Introduction Extended FH Applications
Differentiation Integration
2nd derivative of f (x) = 1/(1 + 100x 2 ) 5
10
EFH FH funqui spline cheb
0
10
−5
10
50
100
150
200
250
300
Figure: Errors with 20 ≤ n ≤ 300 using EFH with d = 15, de = e n = 15, FH with d = 15, funqui, B-spline of degree 15, and polynomial interpolation in Chebyshev points.
G. Klein
Extended Floater–Hormann interpolation
37/42
Introduction Extended FH Applications
Differentiation Integration
Integration of rational interpolants Linear interpolation formulas leads to a linear quadrature rules. For linear barycentric rational interpolant, e rn in particular, we have: Z
b
Z
b
f (x) dx ≈ a
a
Z e rn (x) dx = =
b
Pn+d
a n+d X
wk e k=−d x−xk fk wj j=−d x−xj
Pn+d
dx
ωk e fk ,
k=−d
where Z ωk := a
b
wk x−xk Pn+d wj j=−d x−xj
G. Klein
dx.
Extended Floater–Hormann interpolation
38/42
Introduction Extended FH Applications
Differentiation Integration
Convergence of EFH integration
Theorem Suppose n, d, e n < n and de ≤ e n are positive integers and assume e d+2 that f ∈ C [a − dh, b + dh] ∩ C 2d+1 ([a, a + e nh] ∪ [b − e nh, b]) is sampled at n + 1 equispaced nodes in [a, b]. Let the integral of e rn be approximated by a method of order at least O(hd+2 ). Then for any x ∈ [a, b], Z x Z x e f (y ) dy − rn (y ) dy ≤ KhD+2 log(n). a
a
G. Klein
Extended Floater–Hormann interpolation
39/42
Introduction Extended FH Applications
Differentiation Integration
Antiderivative of f (x) = 1/(1 + 16 cos(3x)2 ) 0
10
EFH FH funqui spline cheb
−10
10
50
100
150
200
250
300
Figure: Errors with 20 ≤ n ≤ 300 using EFH with d = 15, de = e n = 15, FH with d = 15, funqui, B-spline of degree 15, and polynomial interpolation in Chebyshev points.
G. Klein
Extended Floater–Hormann interpolation
40/42
Introduction Extended FH Applications
Summary
Floater–Hormann interpolation Stability and choice of d Extended Floater–Hormann interpolation Enhanced stability, convergence, parameter choice Applications: differentiation, integration
G. Klein
Extended Floater–Hormann interpolation
41/42
Introduction Extended FH Applications
Thank you for your attention!
G. Klein
Extended Floater–Hormann interpolation
42/42
Fox Prize Meeting, Edinburgh, June 24, 2013
Introduction Extended FH Applications
Outline
1
Introduction: linear barycentric rational interpolation
2
Extended Floater–Hormann interpolation
3
Applications of extended Floater–Hormann interpolation
G. Klein
Extended Floater–Hormann interpolation
2/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
One-dimensional interpolation
Given: a ≤ x0 < x1 < . . . < xn ≤ b, f (x0 ), f (x1 ), . . . , f (xn ),
n + 1 distinct nodes and corresponding values.
We study functions g from a finite-dimensional linear subspace of (C [a, b], k · k∞ ) which interpolate f between the nodes, g (xi ) = f (xi ) = fi ,
i = 0, . . . , n.
Main interest: well-conditioned equispaced interpolation.
G. Klein
Extended Floater–Hormann interpolation
3/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
The polynomial in barycentric form There exists a unique polynomial pn of degree ≤ n, which interpolates the data and can be written in Lagrange form pn (x) =
n X
`i (x)fi
with
`i (x) =
i=0
Y x − xk , xi − xk
k6=i
in the first barycentric form (backward stable (Higham ’04)) pn (x) = L(x)
n X i=0
λi fi , x − xi
.Q Qn with λi = 1 k6=i (xi − xk ), and L(x) = k=0 (x − xk ), and in the second barycentric form (forward stable (Higham ’04)) Pn λi i=0 x−xi fi . pn (x) = Pn λ i
i=0 x−xi
G. Klein
Extended Floater–Hormann interpolation
4/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
From polynomial to linear rational interpolation Lemma Let {xi }ni=0 be a set of n + 1 distinct nodes, {fi }ni=0 corresponding real numbers and let {vi }ni=0 be any non-zero real numbers. Then (a) the barycentric rational function n X
R(x) =
vi fi x − xi
i=0 n X i=0
, vi x − xi
interpolates fk at xk : limx→xk R(x) = fk ; (b) conversely, every rational interpolant of the fi may be written in barycentric form for some weights vi . G. Klein
Extended Floater–Hormann interpolation
5/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Construction of Floater–Hormann interpolation for arbitrary nodes - Given n + 1 nodes, x0 , . . . , xn , and function values, f0 , . . . , fn , choose an integer d ∈ {0, 1, . . . , n}, “blending parameter”, - for i = 0, . . . , n − d, define pi (x), the polynomial of low degree ≤ d interpolating fi , fi+1 , . . . , fi+d . The d-th interpolant of the family is a “blend” of the pi (x), n−d X
rn (x) =
λi (x)pi (x)
i=0 n−d X
,
with
λi (x) =
(−1)i . (x − xi ) . . . (x − xi+d )
λi (x)
i=0
Notice that for d = n, rn simplifies to pn . G. Klein
Extended Floater–Hormann interpolation
6/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Linear barycentric rational form For its evaluation, we write rn in linear barycentric form n−d X
rn (x) =
n X
λi (x)pi (x)
i=0 n−d X
= λi (x)
i=0 n X i=0
i=0
wi fi x − xi . wi x − xi
For equispaced nodes, the weights wi do not depend on f , and oscillate in sign with absolute values 1, 1, . . . , 1, 1, 1 4, 1 4 8, 8,
1 2 , 1, 1, . . . , 1, 1, 3 4 , 1, 1, . . . , 1, 1, 7 8 , 1, 1, . . . , 1, 1,
··· G. Klein
d = 0, 1 2, 3 4, 7 8,
d = 1, 1 4, 4 1 8, 8,
d = 2, d = 3, d ≥ 4.
Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Algebraic convergence and polynomial reproduction
Theorem (Floater–Hormann ’07) Let 1 ≤ d ≤ n and f ∈ C d+2 [a, b], h =
max (xi+1 − xi ), then
0≤i≤n−1
kf − rn k∞ ≤ Khd+1 , where K depends only on d, b − a and derivatives of f ; the analytic rational function rn has no real poles; rn reproduces polynomials of degree at least d.
G. Klein
Extended Floater–Hormann interpolation
8/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Lebesgue function and constant The Lebesgue constant associated with linear barycentric interpolation, X n X n wi |wi | , Λn,d = max Λn,d (x) = max a≤x≤b a≤x≤b |x − xi | x − xi i=0
i=0
is the condition number of the interpolation scheme. 4
15
3
10
2
5
1 −1
0
1
−1
0
1
Figure: Lebesgue function for Floater–Hormann interpolation with equispaced nodes in [−1, 1] with d = 2 and d = 5 and n = 40. G. Klein
Extended Floater–Hormann interpolation
9/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Lebesgue function with n = 40 and d = 0, . . . , 40
(loading movie...)
G. Klein
Extended Floater–Hormann interpolation
10/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Lebesgue constant Theorem (Bos–De Marchi–Hormann–K. ’12) Let 0 ≤ d ≤ n and the nodes xi , i = 0, . . . , n, be equispaced. Then n 2d−2 log − 1 ≤ Λn,d ≤ 2d−1 (2 + log n). d +1 d 108
8 d=3
7
107
6
106
5
105
d=2
n = 100
104
4 d=1
3
103 102
2 1
101
0 0
100 0
20
40
60
80
100 120 140 160 180 200
5
10
15
20
25
Figure: Logarithmic growth with n (left) and exponential with d (right). G. Klein
Extended Floater–Hormann interpolation
11/42
Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Central question: How to choose d? Answer: It depends on the nodes and on the regularity of f . FH ’07: large d for very smooth functions, small d for less smooth functions; Numerical Recipes ’07: “It is wise to start with small values of d before trying larger values”; BMHK ’12: not too large, the Lebesgue constant grows exponentially with d; Cross validation (no extra knowledge on the data): remove a few well chosen data values, compute the interpolant for various values of d, compare the errors at the removed data points and keep the d that gives the smallest error (funqui); EFH: wider range of good choices for d. G. Klein
Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Interpolation with FH and d = 15
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Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Interpolation of perturbed data with FH and d = 15
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Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Interpolation of f (x) = sin(20x) perturbed data 2
10
exact data FH funqui spline cheb
1
10
FH funqui spline cheb
0
10
−5
10
0
10
−10
10
−1
10
50
100
150
200
50
100
150
200
Figure: Errors with 8 ≤ n ≤ 200 using FH with d = 15, funqui, B-spline of degree 15, and polynomial interpolation in Chebyshev points.
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Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Differentiation of barycentric functions Proposition (Schneider–Werner ’86) Let rn be a rational function given in its barycentric form with non vanishing weights. Assume that x is not a pole of rn . Then for k ≥1 n X (k)
rn (x) = k!
j=0
wj rn [(x)k , xj ] x − xj n X j=0
(k) rn (xi )
= −k!
n X
wj x − xj
,
x not a node,
! . wi , wj rn [(xi )k , xj ]
i = 0, . . . , n.
j=0 j6=i
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Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Differentiation matrices Define the matrix D(1) (Baltensperger–Berrut–No¨el ’99): wj 1 , i 6= j, wi xi − xj n (1) X Dij = (1) − Di` , i = j, `=0 `6=i
and D(k) , k > 1, with the “hybrid formula” (Tee ’06), k wj (k−1) (k−1) D − D , i= 6 j, x − x w ii ij i j i n (k) X Dij = (k) − Di` , i = j. `=0 `6=i
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Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Floater–Hormann interpolation Convergence and stability Differentiation
Evaluation of derivatives of rn at the nodes
With f = (f0 , . . . , fn )| , the product D(k) · f returns
| (k) (k) rn (x0 ), . . . , rn (xn ) ,
the vector of the k-th derivative of rn at the nodes.
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Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Construction Convergence and stability Examples
Extended Floater–Hormann interpolation
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Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Construction Convergence and stability Examples
Extended Floater–Hormann interpolation For given d, Λn (x) typically has at most d large local maxima at the ends the interval if the nodes are equispaced. 15 10 5 −1
0
1
Idea: construct 2d new data values e f−d , . . . , e f−1 and e fn+1 , . . . , e fn+d out of the given data f0 , . . . , fn . We choose positive integers e n n and de ≤ e n, and compute (k) (k) e ren (x0 ) and ren (xn ), k = 1, . . . , d. G. Klein
Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Construction Convergence and stability Examples
Extended Floater–Hormann interpolation We set e d X (xi − x0 )k (k) f + r (x ) , −d ≤ i ≤ −1, 0 0 e n k! k=1 e 0 ≤ i ≤ n, fi := fi , e d X (xi − xn )k (k) , n + 1 ≤ i ≤ n + d. f + r (x ) n n e n k! k=1
The extension of the Floater–Hormann family then is n+d X
e rn (x) :=
i=−d n+d X
wi e fi x − xi
i=−d G. Klein
. wi x − xi
Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Construction Convergence and stability Examples
Convergence of extended Floater–Hormann interpolation Notation: e D := min{d, d}. Theorem Suppose n, d, e n < n and de ≤ e n are positive integers and assume e d+2 that f ∈ C [a − dh, b + dh] ∩ C 2d+1 ([a, a + e nh] ∪ [b − e nh, b]) is sampled at n + 1 equispaced nodes in [a, b]. Then kf − e rn k∞ ≤ KhD+1 . e rn is analytic, has no real poles and reproduces polynomials of degree ≤ D.
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Introduction Extended FH Applications
Construction Convergence and stability Examples
A generalised Hermite formula (Platte’s method) Let L be a general linear approximation, n X L(x) = bi (x)fi . i=0
Suppose f is analytic inside a simple, closed, rectifiable curve C contained in a closed simply connected region around the nodes, then the error satisfies Z f (s) 1 f (x) − L(x) = R(x, s) ds, 2πi C s − x where R(x, s) = 1 − (s − x)
n X bi (x) s − xi i=0
is the relative error in the approximation of 1/(s − x) by L. G. Klein
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Introduction Extended FH Applications
Construction Convergence and stability Examples
Interpolation of analytic functions −1 0.6
original FH
extended FH
−2 −3
0.4
−4 0.2
−5 −6
0 −7 −8
−0.2
−9 −0.4 −0.6 −1.5
−10 spline 15 −1
−11
funqui −0.5
0
0.5
1
1.5
−12
Figure: Comparison of the relative error of the prototype function 1/(s − x) with FH with d = 15, EFH with d = 15, de = e n = 15, B-spline of degree 15 and funqui for n = 100 on a log10 scale. G. Klein
Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Construction Convergence and stability Examples
Lebesgue constant for extended FH interpolation
Theorem en associated For positive integers n and d, the Lebesgue constant Λ with extended Floater–Hormann interpolation is bounded as en ≤ 2 + log(n + 2d). Λ
Logarithmic growth with n and d.
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Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Construction Convergence and stability Examples
Lebesgue function for extended FH interpolation
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Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Construction Convergence and stability Examples
Lebesgue functions EFH
2nd kind
1st kind
3.5
3.5
3.5
3
3
3
2.5
2.5
2.5
2
2
2
1.5
1.5
1.5
1 −1
0
1
1 −1
0
1
1 −1
0
1
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).
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Introduction Extended FH Applications
Construction Convergence and stability Examples
Perturbed data with EFH and d = 15, de = e n = 15
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Introduction Extended FH Applications
Construction Convergence and stability Examples
Interpolation of f (x) = sin(20x) perturbed data 2
10
exact data EFH FH funqui spline cheb
1
10
0
10
EFH FH funqui spline cheb
0
10
−5
10
−10
10
−1
10
50
100
150
200
50
100
150
200
Figure: Errors with 8 ≤ n ≤ 200 using EFH with d = 15, de = e n = 15, FH with d = 15, funqui, B-spline of degree 15, and polynomial interpolation in Chebyshev points.
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Introduction Extended FH Applications
Construction Convergence and stability Examples
Interpolation of f (x) = sin(20x)
EFH FH spline funqui
−5
10
−10
10
5
10
15
20
25
30
Figure: Errors with 1 ≤ d ≤ 30 and n = 200 using EFH with de = e n = 15, FH and B-spline of degree d.
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Introduction Extended FH Applications
Differentiation Integration
Applications
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Introduction Extended FH Applications
Differentiation Integration
Approximation of derivatives via finite differences
Extended rational finite difference methods (ERFD) for the approximation at the nodes xi , 0 ≤ i ≤ n, of the k-th derivative of a sufficiently smooth function, (k)
f (k) (xi ) ≈ e rn (xi ) =
n+d X
(k) (k) Dij e fj =: e fi .
j=−d (k)
The weights Dij are the elements from the (d + 1)st to the (n + d + 1)st row of the (n + 2d + 1) × (n + 2d + 1) differentiation matrix D (k) .
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Introduction Extended FH Applications
Differentiation Integration
Convergence of ERFD
Theorem e are positive Suppose n, d, e n < n, de ≤ e n and k ≤ D = min{d, d} integers and assume that e f ∈ C d+1+k [a − dh, b + dh] ∩ C 2d+1 ([a, a + e nh] ∪ [b − e nh, b]) is sampled at n + 1 equispaced nodes a = x0 < x1 < . . . < xn = b. Then (k)
|e rn (xj ) − f (k) (xj )| ≤ KhD+1−k ,
G. Klein
−d ≤ j ≤ n + d.
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Introduction Extended FH Applications
Differentiation Integration
Approximation of derivatives at intermediate points For the approximation of the k-th derivative of a function f at intermediate points x ∈ [a, b], there are several possibilities, among them: interpolation of the derivatives at the nodes: en(k) (x) := R
n+d X i=−d
wi (k) e rn (xi ) x − xi
n+d X i=−d
wi . x − xi
approximate e rn in Chebfun and differentiate the resulting chebfun. Usually not much loss in accuracy with such cheaper methods.
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Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Differentiation Integration
2nd derivative with FH and d = 15
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Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Differentiation Integration
2nd derivative with EFH and d = 15, de = e n = 15
(loading movie...)
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Introduction Extended FH Applications
Differentiation Integration
2nd derivative of f (x) = 1/(1 + 100x 2 ) 5
10
EFH FH funqui spline cheb
0
10
−5
10
50
100
150
200
250
300
Figure: Errors with 20 ≤ n ≤ 300 using EFH with d = 15, de = e n = 15, FH with d = 15, funqui, B-spline of degree 15, and polynomial interpolation in Chebyshev points.
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Introduction Extended FH Applications
Differentiation Integration
Integration of rational interpolants Linear interpolation formulas leads to a linear quadrature rules. For linear barycentric rational interpolant, e rn in particular, we have: Z
b
Z
b
f (x) dx ≈ a
a
Z e rn (x) dx = =
b
Pn+d
a n+d X
wk e k=−d x−xk fk wj j=−d x−xj
Pn+d
dx
ωk e fk ,
k=−d
where Z ωk := a
b
wk x−xk Pn+d wj j=−d x−xj
G. Klein
dx.
Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Differentiation Integration
Convergence of EFH integration
Theorem Suppose n, d, e n < n and de ≤ e n are positive integers and assume e d+2 that f ∈ C [a − dh, b + dh] ∩ C 2d+1 ([a, a + e nh] ∪ [b − e nh, b]) is sampled at n + 1 equispaced nodes in [a, b]. Let the integral of e rn be approximated by a method of order at least O(hd+2 ). Then for any x ∈ [a, b], Z x Z x e f (y ) dy − rn (y ) dy ≤ KhD+2 log(n). a
a
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Introduction Extended FH Applications
Differentiation Integration
Antiderivative of f (x) = 1/(1 + 16 cos(3x)2 ) 0
10
EFH FH funqui spline cheb
−10
10
50
100
150
200
250
300
Figure: Errors with 20 ≤ n ≤ 300 using EFH with d = 15, de = e n = 15, FH with d = 15, funqui, B-spline of degree 15, and polynomial interpolation in Chebyshev points.
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Extended Floater–Hormann interpolation
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Introduction Extended FH Applications
Summary
Floater–Hormann interpolation Stability and choice of d Extended Floater–Hormann interpolation Enhanced stability, convergence, parameter choice Applications: differentiation, integration
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Introduction Extended FH Applications
Thank you for your attention!
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