Unusual convergence behaviour of certain rational interpolants - - PowerPoint PPT Presentation

Unusual convergence behaviour of certain rational interpolants

Joris Van Deun

Outline

Chebyshev polynomials and rational functions Barycentric interpolation Conformal maps Strange convergence

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Outline

Chebyshev polynomials and rational functions Barycentric interpolation Conformal maps Strange convergence

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Chebyshev polynomials

Using the Joukowski map x = 1 2

• z + 1

z

• = J(z),

the Chebyshev polynomial of the first kind Tn can be defined as Tn(x) = 1 2

• zn + 1

zn

• .

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Chebyshev rational functions

◮ Take n poles {α1, α2, . . . , αn}

◮ outside [−1, 1], ◮ real or complex conjugate, ◮ possibly infinite.

◮ Put βk = J−1(αk) for k = 1, . . . , n, such that |βk| < 1. ◮ Define the (finite) Blaschke product

Bn(z) = z − β1 1 − ¯ β1z · · · · · z − βn 1 − ¯ βnz .

◮ If all poles αk at infinity, then all βk = 0 and Bn(z) = zn. 5/30

Chebyshev rational functions

Define Tn(x) = 1 2

• Bn(z) +

1 Bn(z)

• ,

then Tn(x) is a [n/n] rational function with poles {α1, . . . , αn} that attract the zeros.

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Outline

Chebyshev polynomials and rational functions Barycentric interpolation Conformal maps Strange convergence

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Polynomial Lagrange interpolation

◮ Let n + 1 distinct interpolation points xj, j = 0, . . . , n, be given,

together with function values fj = f (xj).

◮ The polynomial pn of degree n that interpolates f at the points

xj is given by pn(x) =

n

• j=0

fjlj(x) where the Lagrange polynomials lj(x) are given by lj(x) = n

k=0,k=j(x − xk)

n

k=0,k=j(xj − xk) . 8/30

Barycentric formula

The above Lagrange formula for pn can be rewritten in barycentric form as pn(x) =

n

• j=0

wj x − xj fj

n

• j=0

wj x − xj , where the barycentric weights wj are given by wj = 1

• k=j(xj − xk) .

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Rational interpolation

If this relation between weights wj and nodes xk is changed, then

◮ the interpolation property still holds, but ◮ the interpolant becomes a rational function of degree at most n

in numerator and denominator. For a given set of poles and interpolation points, we can easily determine the corresponding weights.

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Interpolation in Chebyshev zeros

◮ Polynomial interpolant in the zeros of Tn+1(x)

xj = cos θ(p)

j

= cos (2j + 1)π 2n + 2 , wj = (−1)j sin θ(p)

j

= (−1)j 1 − x2

j . ◮ Rational interpolant in the zeros of Tn+1(x) with poles

{α1, . . . , αn} xj = cos θ(r)

j ,

wj = (−1)j sin θ(r)

j

Hn+1(xj) , with Hn(x) = 1 n

n

• k=1
• α2

k − 1

αk − x .

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Example

Take f (x) = p(x) sinh p(x) with p(x) = π ω(x2 − c2) and ω = 0.01 and c = 0.6. Poles are at ±

• c2 ± kiω,

k = 1, 2, . . .

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Outline

Chebyshev polynomials and rational functions Barycentric interpolation Conformal maps Strange convergence

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Convergence rate

◮ Assume that f is analytic in the interior of and on an ellipse Cρ

with foci at ±1 and sum of major and minox axes equal to 2ρ. Then |f (x) − pn(x)| = O(ρ−n) where pn is the polynomial interpolating f at the zeros of Tn+1.

◮ If f has singularities close to [−1, 1], then ρ ≈ 1 and

convergence is slow.

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Transformed Chebyshev points

◮ Let g : D1 → D2 be a conformal map such that

g([−1, 1]) = [−1, 1].

◮ Let f : D2 → C be such that f ◦ g : D1 → C is analytic inside

and on an ellipse C˜

ρ. ◮ Put xj = g(yj) where yj are the zeros of Tn+1. ◮ Put

rn(x) = n

j=0 wj x−xj fj

n

j=0 wj x−xj

, where fj = f (xj) and wj = (−1)j 1 − x2

j . 15/30

New convergence rate

Theorem (Baltensperger, Berrut, No¨ el)

With the definitions given before, it holds that |f (x) − rn(x)| = O(˜ ρ−n) for every x ∈ [−1, 1]. Construction of appropriate conformal map g usually based on knowledge about singularities of f .

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“Blaschke” map

◮ Let xj denote the zeros of Tn+1 and yj those of Tn+1, then

yj = (J ◦ B

1 n+1

n+1 ◦ J−1)(xj) = g−1(xj)

where Bn+1(x) is the Blaschke product and J(x) the Joukowski map.

◮ Map g(x) is implicitly defined. ◮ Requires n + 1 poles, why not take m < n + 1 “poles” in

definition of g?

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Example

−2 −1 1 2 −0.25 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 0.25 −2 −1 1 2 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5

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Outline

Chebyshev polynomials and rational functions Barycentric interpolation Conformal maps Strange convergence

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Example

◮ Suppose we interpolate

f (x) = x2 x2 + a2 which has simples poles at ±ia.

◮ This suggests taking m = 2 and α1 = ia, α2 = −ia in the

definition of g(x), which yields g(x) = ax √ a2 + 1 − x2 .

◮ It is easily computed that

(f ◦ g)(x) = x2 a2 + 1

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Example with a = 0.01

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Example — different map

◮ It seems that we have exact interpolation when n = 2k for

k = 2, 3, 4, . . .

◮ Now take m = 3 and α1 = ia, α2 = −ia and α3 = ∞ in the

definition of g(x).

◮ Explicit expression for g(x) no longer available, but it can be

computed that ˜ ρ = 1.15.

◮ It seems that we have exact interpolation when n = 3k for

k = 2, 3, 4, . . .

◮ What happens when n = mk ? 22/30

Example — different map

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Explanation

◮ For n = mk the interpolation points are the zeros of

Tn(x) = 1 2

• Bk

m(z) +

1 Bk

m(z)

• = 1

2

• Bmk(z) +

1 Bmk(z)

• .

◮ The barycentric interpolant rn−1(x) turns out to be

rn−1(x) = 1 Hm(x)

n−1

• j=0

Tn(x) T ′

n(xj)(x − xj)Hm(xj)f (xj)

• nly when n = mk.

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Estimated poles

◮ Suppose we again interpolate

f (x) = x2 x2 + a2 .

◮ Now take the map

g(x) = ˜ ax √ ˜ a2 + 1 − x2 corresponding to Bm(z) = B2(z) with α1 = i˜ a and α2 = −i˜ a.

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Example, a = 0.001 and ˜ a = 0.00116

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Convergence speed

◮ Example:

|f (x) − rn−1(x)| ≤ max{a2, ˜ a2} |a2 − ˜ a2| · 1 |Tn(ia)| = O(Bn(ib)), for n = 2k.

◮ General convergence result for meromorphic f with poles aj:

|f (x) − rn−1(x)| = O  

∞

• j=1

Bn(bj)   , for n = mk (result obtained using Mittag-Leffler expansions).

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Final example

Take f (x) = 1 2 erf x √ 2ǫ + 3 2

• e−x.

Entire, but for ǫ → 0 almost discontinuous. Take poles of [2/2] Pad´ e approximant to erf(x): a1,2 = ±1.732i and multiply by √ 2ǫ.

−1 −0.5 0.5 1 0.5 1 1.5 2 2.5 3

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α1,2 = ±1.732i √ 2ǫ; ǫ = 0.0001

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α1,2 = ±1.732i √ 2ǫ; α3 = ∞

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