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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

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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

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants

Outline

1

Interpolation

2

Differentiation of barycentric rational interpolants

3

Linear barycentric rational finite differences

4

Integration of barycentric rational interpolants

Berrut Applications of LBR interpolation at equidistant nodes 2/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Introduction and notation

Interpolation

Berrut Applications of LBR interpolation at equidistant nodes 3/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

One-dimensional interpolation problem

Given: a ≤ x0 < x1 < . . . < xn ≤ b, n + 1 distinct nodes and f (x0), f (x1), . . . , f (xn), corresponding values. There exists a unique polynomial of degree ≤ n that interpolates the fi, i.e. pn[f ](xi) = fi, i = 0, 1, . . . , n. The Lagrange form of the polynomial interpolant is pn[f ](x) :=

n

  • j=0

fjℓj(x), ℓj(x) :=

  • k=j

(x − xk) (xj − xk).

Berrut Applications of LBR interpolation at equidistant nodes 4/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

One-dimensional interpolation problem

Given: a ≤ x0 < x1 < . . . < xn ≤ b, n + 1 distinct nodes and f (x0), f (x1), . . . , f (xn), corresponding values. There exists a unique polynomial of degree ≤ n that interpolates the fi, i.e. pn[f ](xi) = fi, i = 0, 1, . . . , n. The Lagrange form of the polynomial interpolant is pn[f ](x) :=

n

  • j=0

fjℓj(x), ℓj(x) :=

  • k=j

(x − xk) (xj − xk).

Berrut Applications of LBR interpolation at equidistant nodes 4/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

The first barycentric form

Denote the leading factors of the ℓj’s by νj :=

  • k=j

(xj − xk)−1, j = 0, 1, . . . , n, the so–called weights, which may be computed in advance. Rewrite the polynomial in its first barycentric form pn[f ](x) = L(x)

n

  • j=0

νj x − xj fj, where L(x) :=

n

  • k=0

(x − xk).

Berrut Applications of LBR interpolation at equidistant nodes 5/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

The first barycentric form

Denote the leading factors of the ℓj’s by νj :=

  • k=j

(xj − xk)−1, j = 0, 1, . . . , n, the so–called weights, which may be computed in advance. Rewrite the polynomial in its first barycentric form pn[f ](x) = L(x)

n

  • j=0

νj x − xj fj, where L(x) :=

n

  • k=0

(x − xk).

Berrut Applications of LBR interpolation at equidistant nodes 5/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Advantages

evaluation in O(n) operations, ease of adding new data (xn+1, fn+1), numerically best for evaluation.

Berrut Applications of LBR interpolation at equidistant nodes 6/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Advantages

evaluation in O(n) operations, ease of adding new data (xn+1, fn+1), numerically best for evaluation.

Berrut Applications of LBR interpolation at equidistant nodes 6/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Advantages

evaluation in O(n) operations, ease of adding new data (xn+1, fn+1), numerically best for evaluation.

Berrut Applications of LBR interpolation at equidistant nodes 6/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

The barycentric formula

The constant f ≡ 1 is represented exactly by its polynomial interpolant: 1 = L(x)

n

  • j=0

νj x − xj = pn[1](x). Dividing pn[f ] by 1 and cancelling L(x) gives the barycentric form of the polynomial interpolant pn[f ](x) =

n

  • j=0

νj x − xj fj

n

  • j=0

νj x − xj .

Berrut Applications of LBR interpolation at equidistant nodes 7/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

The barycentric formula

The constant f ≡ 1 is represented exactly by its polynomial interpolant: 1 = L(x)

n

  • j=0

νj x − xj = pn[1](x). Dividing pn[f ] by 1 and cancelling L(x) gives the barycentric form of the polynomial interpolant pn[f ](x) =

n

  • j=0

νj x − xj fj

n

  • j=0

νj x − xj .

Berrut Applications of LBR interpolation at equidistant nodes 7/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Advantages

Interpolation is guaranteed: lim

x→xk n

  • j=0
  • νj

x − xj fj

n

  • j=0
  • νj

x − xj = fk. Simplification of the weights: Cancellation of common factor leads to simplified weights. For equispaced nodes, ν∗

j = (−1)j

n j

  • .

Berrut Applications of LBR interpolation at equidistant nodes 8/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Advantages

Interpolation is guaranteed: lim

x→xk n

  • j=0
  • νj

x − xj fj

n

  • j=0
  • νj

x − xj = fk. Simplification of the weights: Cancellation of common factor leads to simplified weights. For equispaced nodes, ν∗

j = (−1)j

n j

  • .

Berrut Applications of LBR interpolation at equidistant nodes 8/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Form polynomial to rational interpolation

In the barycentric form of the polynomial interpolant pn[f ](x) =

n

  • j=0

νj x − xj fj

n

  • j=0

νj x − xj , the weights are defined in such a way that L(x)

n

  • j=0

νj x − xj = 1. Modification of these weights rational interpolant.

Berrut Applications of LBR interpolation at equidistant nodes 9/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Form polynomial to rational interpolation

In the barycentric form of the polynomial interpolant pn[f ](x) =

n

  • j=0

νj x − xj fj

n

  • j=0

νj x − xj , the weights are defined in such a way that L(x)

n

  • j=0

νj x − xj = 1. Modification of these weights rational interpolant.

Berrut Applications of LBR interpolation at equidistant nodes 9/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Lemma Let {xj}, j = 0, 1, . . . , n, be n + 1 distinct nodes, {fj} corresponding real numbers and let {vj} be any nonzero real

  • numbers. Then

(a) the rational function rn[f ](x) =

n

  • j=0

vj x − xj fj

n

  • j=0

vj x − xj , interpolates fk at xk: limx→xk rn[f ](x) = fk; (b) conversely, every rational interpolant of the fj may be written in barycentric form for some weights vj.

Berrut Applications of LBR interpolation at equidistant nodes 10/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Floater and Hormann interpolants

Weights suggested in B.(1988): (−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 Applications of LBR interpolation at equidistant nodes 11/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Floater and Hormann interpolants

Weights suggested in B.(1988): (−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 Applications of LBR interpolation at equidistant nodes 11/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Construction presented by Floater and Hormann

  • Choose an integer d ∈ {0, 1, . . . , n},
  • define pj(x), the polynomial of degree ≤ d interpolating

fj, fj+1, . . . , fj+d for j = 0, . . . , n − d. The d-th interpolant is given by rn[f ](x) =

n−d

  • j=0

λj(x)pj(x)

n−d

  • j=0

λj(x) , where λj(x) = (−1)j (x − xj) . . . (x − xj+d).

Berrut Applications of LBR interpolation at equidistant nodes 12/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Construction presented by Floater and Hormann

  • Choose an integer d ∈ {0, 1, . . . , n},
  • define pj(x), the polynomial of degree ≤ d interpolating

fj, fj+1, . . . , fj+d for j = 0, . . . , n − d. The d-th interpolant is given by rn[f ](x) =

n−d

  • j=0

λj(x)pj(x)

n−d

  • j=0

λj(x) , where λj(x) = (−1)j (x − xj) . . . (x − xj+d).

Berrut Applications of LBR interpolation at equidistant nodes 12/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Construction presented by Floater and Hormann

  • Choose an integer d ∈ {0, 1, . . . , n},
  • define pj(x), the polynomial of degree ≤ d interpolating

fj, fj+1, . . . , fj+d for j = 0, . . . , n − d. The d-th interpolant is given by rn[f ](x) =

n−d

  • j=0

λj(x)pj(x)

n−d

  • j=0

λj(x) , where λj(x) = (−1)j (x − xj) . . . (x − xj+d).

Berrut Applications of LBR interpolation at equidistant nodes 12/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Barycentric weights

Write rn[f ] in barycentric form rn[f ](x) =

n

  • j=0

vj x − xj fj

n

  • j=0

vj x − xj , with the weights vj =

  • i∈Jj

i+d

  • ℓ=i, ℓ=j

1 xj − xℓ .

Berrut Applications of LBR interpolation at equidistant nodes 13/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Barycentric weights

For equispaced nodes, the weights vj oscillate in sign with absolute values 1, 1, . . . , 1, 1, d = 0, (B.)

1 2, 1, 1, . . . , 1, 1, 1 2,

d = 1, (B.)

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 Applications of LBR interpolation at equidistant nodes 14/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants One-dimensional interpolation Barycentric Lagrange interpolation Polynomial to rational interpolation Floater and Hormann interpolation

Theorem (Floater-Hormann (2007)) Let 0 ≤ d ≤ n and f ∈ C d+2[a, b], h := max

0≤i≤n−1(xi+1 − xi), then

the rational function rn[f ] has no poles in l R, if n − d is odd, then rn[f ] − f ≤ hd+1(b − a)f (d+2)

d+2

if d ≥ 1, rn[f ] − f ≤ h(1 + β)(b − a)f ′′

2

if d = 0; if n − d is even, then rn[f ] − f ≤ hd+1 (b − a)f (d+2)

d+2

+ f (d+1)

d+1

  • if d ≥ 1,

rn[f ] − f ≤ h(1 + β)

  • (b − a)f ′′

2

+ f ′

  • if d = 0.

β := max

1≤i≤n−2 min

|xi − xi+1| |xi − xi−1|, |xi+1 − xi| |xi+1 − xi+2|

  • Berrut

Applications of LBR interpolation at equidistant nodes 15/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Differentiation of barycentric rational interpolants

Berrut Applications of LBR interpolation at equidistant nodes 16/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Proposition (Schneider-Werner (1986)) Let rn[f ] be a rational function given in its barycentric form with non vanishing weights. Assume that x is not a pole of rn[f ]. Then for k ≥ 1 1 k!r (k)

n [f ](x)

=

n

  • j=0

vj x − xj rn[f ][(x)k, xj]

n

  • j=0

vj x − xj , x not a node, 1 k!r (k)

n [f ](xi)

= −

  • n
  • j=0

j=i

vjrn[f ][(xi)k, xj]

  • vi,

i = 0, . . . , n.

Berrut Applications of LBR interpolation at equidistant nodes 17/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Differentiation matrices

Define the matrices D(1) and D(2) (Baltensperger-B.-No¨ el (1999)): D(1)

ij

:=            vj vi 1 xi − xj , −

n

  • k=0

k=i

D(1)

ik ;

D(2)

ij

:=            2D(1)

ij

  • D(1)

ii

− 1 xi − xj

  • ,

i = j, −

n

  • k=0

k=i

D(2)

ik ,

i = j. If f := (f0, . . . , fn)T, then D(1) · f, respectively D(2) · f, returns the vector of the first, respectively second, derivative of rn[f ] at the nodes.

Berrut Applications of LBR interpolation at equidistant nodes 18/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Differentiation matrices

Define the matrices D(1) and D(2) (Baltensperger-B.-No¨ el (1999)): D(1)

ij

:=            vj vi 1 xi − xj , −

n

  • k=0

k=i

D(1)

ik ;

D(2)

ij

:=            2D(1)

ij

  • D(1)

ii

− 1 xi − xj

  • ,

i = j, −

n

  • k=0

k=i

D(2)

ik ,

i = j. If f := (f0, . . . , fn)T, then D(1) · f, respectively D(2) · f, returns the vector of the first, respectively second, derivative of rn[f ] at the nodes.

Berrut Applications of LBR interpolation at equidistant nodes 18/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Convergence rates for the derivatives

For x ∈ [a, b], we denote the error e(x) := f (x) − rn[f ](x). Theorem (B.-Floater-Klein) At the nodes, we have if d ≥ 0 and if f ∈ C d+2[a, b], then |e′(xj)| ≤ Chd, j = 0, 1, . . . , n; if d ≥ 1 and if f ∈ C d+3[a, b], then |e′′(xj)| ≤ Chd−1, j = 0, 1, . . . , n.

Berrut Applications of LBR interpolation at equidistant nodes 19/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Convergence rates for the derivatives

For x ∈ [a, b], we denote the error e(x) := f (x) − rn[f ](x). Theorem (B.-Floater-Klein) At the nodes, we have if d ≥ 0 and if f ∈ C d+2[a, b], then |e′(xj)| ≤ Chd, j = 0, 1, . . . , n; if d ≥ 1 and if f ∈ C d+3[a, b], then |e′′(xj)| ≤ Chd−1, j = 0, 1, . . . , n.

Berrut Applications of LBR interpolation at equidistant nodes 19/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Theorem (B.-Floater-Klein) (continued) With the intermediate points, we have if d ≥ 1 and if f ∈ C d+3[a, b], then e′ ≤ Chd if d ≥ 2, e′ ≤ C(β + 1)h if d = 1; if d ≥ 2 and if f ∈ C d+4[a, b], then e′′ ≤ C(β + 1)hd−1 if d ≥ 3, e′′ ≤ C(β2 + β + 1)h if d = 2. Mesh ratio β := max

  • max

1≤i≤n−1

|xi − xi+1| |xi − xi−1|, max

0≤i≤n−2

|xi+1 − xi| |xi+1 − xi+2|

  • .

Berrut Applications of LBR interpolation at equidistant nodes 20/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Remarks

In the important cases k = 1, 2 the convergence rate of the k-th derivative is O(hd+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 Applications of LBR interpolation at equidistant nodes 21/49

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SLIDE 34

Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Remarks

In the important cases k = 1, 2 the convergence rate of the k-th derivative is O(hd+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 Applications of LBR interpolation at equidistant nodes 21/49

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SLIDE 35

Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Remarks

In the important cases k = 1, 2 the convergence rate of the k-th derivative is O(hd+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 Applications of LBR interpolation at equidistant nodes 21/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Runge’s function

Table: Error in the interpolation and the derivatives of the rational interpolant of 1/(1 + x2) in [−5, 5] for d = 3.

Interpolation First derivative Second derivative n error

  • rder

error

  • rder

error

  • rder

10 6.9e−02 3.9e−01 1.5e+00 20 2.8e−03 4.6 3.1e−02 3.7 2.6e−01 2.5 40 4.3e−06 9.4 7.8e−05 8.6 1.5e−03 7.4 80 5.1e−08 6.4 1.2e−06 6.0 6.1e−05 4.6 160 3.0e−09 4.1 1.0e−07 3.6 9.4e−06 2.7 320 1.8e−10 4.0 1.2e−08 3.1 1.2e−06 2.9 640 1.1e−11 4.0 1.5e−09 3.0 3.0e−07 2.0

Berrut Applications of LBR interpolation at equidistant nodes 22/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Runge’s function

Table: Error in the interpolation and the derivatives of the rational interpolant of 1/(1 + x2) in [−5, 5] for d = 3.

Interpolation First derivative Second derivative n error

  • rder

error

  • rder

error

  • rder

10 6.9e−02 3.9e−01 1.5e+00 20 2.8e−03 4.6 3.1e−02 3.7 2.6e−01 2.5 40 4.3e−06 9.4 7.8e−05 8.6 1.5e−03 7.4 80 5.1e−08 6.4 1.2e−06 6.0 6.1e−05 4.6 160 3.0e−09 4.1 1.0e−07 3.6 9.4e−06 2.7 320 1.8e−10 4.0 1.2e−08 3.1 1.2e−06 2.9 640 1.1e−11 4.0 1.5e−09 3.0 3.0e−07 2.0

Berrut Applications of LBR interpolation at equidistant nodes 22/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Runge’s function

Table: Error in the interpolation and the derivatives of the rational interpolant of 1/(1 + x2) in [−5, 5] for d = 3.

Interpolation First derivative Second derivative n error

  • rder

error

  • rder

error

  • rder

10 6.9e−02 3.9e−01 1.5e+00 20 2.8e−03 4.6 3.1e−02 3.7 2.6e−01 2.5 40 4.3e−06 9.4 7.8e−05 8.6 1.5e−03 7.4 80 5.1e−08 6.4 1.2e−06 6.0 6.1e−05 4.6 160 3.0e−09 4.1 1.0e−07 3.6 9.4e−06 2.7 320 1.8e−10 4.0 1.2e−08 3.1 1.2e−06 2.9 640 1.1e−11 4.0 1.5e−09 3.0 3.0e−07 2.0

Berrut Applications of LBR interpolation at equidistant nodes 22/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Runge’s function

Table: Error in the interpolation and the derivatives of the rational interpolant of 1/(1 + x2) in [−5, 5] for d = 3.

Interpolation First derivative Second derivative n error

  • rder

error

  • rder

error

  • rder

10 6.9e−02 3.9e−01 1.5e+00 20 2.8e−03 4.6 3.1e−02 3.7 2.6e−01 2.5 40 4.3e−06 9.4 7.8e−05 8.6 1.5e−03 7.4 80 5.1e−08 6.4 1.2e−06 6.0 6.1e−05 4.6 160 3.0e−09 4.1 1.0e−07 3.6 9.4e−06 2.7 320 1.8e−10 4.0 1.2e−08 3.1 1.2e−06 2.9 640 1.1e−11 4.0 1.5e−09 3.0 3.0e−07 2.0

Berrut Applications of LBR interpolation at equidistant nodes 22/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Differentiation matrices Convergence rates Example

Comparison with cubic spline

10

1

10

2

10

3

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10 n error FH d=3, cubic spline (spline toolbox) FH interpolant Cubic spline 1st derivative FH 1st derivative spline 2nd derivative FH 2nd derivative spline Berrut Applications of LBR interpolation at equidistant nodes 23/49

slide-41
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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Higher order derivatives Linear barycentric rational FD RFD weights Examples

Higher order derivatives and application to rational finite differences

Berrut Applications of LBR interpolation at equidistant nodes 24/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Higher order derivatives Linear barycentric rational FD RFD weights Examples

Quasi-equispaced nodes

Let us now investigate the convergence rate of the k-th derivative, k = 1, . . . , d + 1, of rn[f ] at equispaced or quasi-equispaced nodes. By quasi-equispaced nodes (Elling 2007) we shall mean here points whose minimal spacing hmin satisfies hmin ≥ ch, where c is a constant.

Berrut Applications of LBR interpolation at equidistant nodes 25/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Higher order derivatives Linear barycentric rational FD RFD weights Examples

Convergence rates for higher order derivatives

Theorem Suppose n, d, d ≤ n, and k, k ≤ d + 1, are positive integers and f ∈ C d+1+k[a, b]. If the nodes xj, j = 0, . . . , n, are equispaced or quasi-equispaced, then |e(k)(xj)| ≤ Chd+1−k, 0 ≤ j ≤ n, where C only depends on d, k and derivatives of f .

Berrut Applications of LBR interpolation at equidistant nodes 26/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Higher order derivatives Linear barycentric rational FD RFD weights Examples

Rational finite differences (RFD)

Let us introduce rational finite difference (RFD) formulas for the approximation, at a node xi, of the k-th derivative of a C d+1+k function, dkf dxk

  • x=xi

≈ dk dxk rn[f ]

  • x=xi

=

n

  • j=0

D(k)

ij fj,

where D(k)

ij

is the k-th derivative of the j-th Lagrange fundamental rational function at the node xi.

Berrut Applications of LBR interpolation at equidistant nodes 27/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Higher order derivatives Linear barycentric rational FD RFD weights Examples

Rational finite differences (RFD)

Let us introduce rational finite difference (RFD) formulas for the approximation, at a node xi, of the k-th derivative of a C d+1+k function, dkf dxk

  • x=xi

≈ dk dxk rn[f ]

  • x=xi

=

n

  • j=0

D(k)

ij fj,

where D(k)

ij

is the k-th derivative of the j-th Lagrange fundamental rational function at the node xi.

Berrut Applications of LBR interpolation at equidistant nodes 27/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Higher order derivatives Linear barycentric rational FD RFD weights Examples

Rational finite differences (RFD)

Let us introduce rational finite difference (RFD) formulas for the approximation, at a node xi, of the k-th derivative of a C d+1+k function, dkf dxk

  • x=xi

≈ dk dxk rn[f ]

  • x=xi

=

n

  • j=0

D(k)

ij fj,

where D(k)

ij

is the k-th derivative of the j-th Lagrange fundamental rational function at the node xi.

Berrut Applications of LBR interpolation at equidistant nodes 27/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Higher order derivatives Linear barycentric rational FD RFD weights Examples

Rational finite differences (RFD)

In order to establish formulas for the RFD weights D(k)

ij , we use the

differentiation matrix D(1) defined earlier for the first order derivative and the “hybrid formula” (Tee 2006), D(k)

ij

:=            k xi − xj vj vi D(k−1)

ii

− D(k−1)

ij

  • ,

i = j, −

n

  • ℓ=0

ℓ=i

D(k)

iℓ ,

i = j, for higher order derivatives.

Berrut Applications of LBR interpolation at equidistant nodes 28/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Higher order derivatives Linear barycentric rational FD RFD weights Examples

Weights for the first centered RFD formulas

Table: Weights for d = 4 for the approximation of the 2-nd and 4-th

  • rder derivatives at x = 0 on an equispaced grid.

−4 −3 −2 −1 1 2 3 4

2nd derivative (order 3)

− 1

12 4 3

− 5

2 4 3

− 1

12 1 63

− 5

28 11 7

− 355

126 11 7

− 5

28 1 63

− 1

128 5 72

− 11

32 15 8

− 1835

576 15 8

− 11

32 5 72

− 1

128

4th derivative (order 1)

1 −4 6 −4 1 − 109

441 365 147

− 1133

147 4826 441

− 1133

147 365 147

− 109

441 1763 12288

− 2845

2304 17017 3072

− 3415

256 327787 18432

− 3415

256 17017 3072

− 2845

2304 1763 12288 Berrut Applications of LBR interpolation at equidistant nodes 29/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Higher order derivatives Linear barycentric rational FD RFD weights Examples

Weights for the first one-sided RFD formulas

Table: Weights for d = 4 for the approximation of the 2-nd and 4-th

  • rder derivatives at x = 0 on an equispaced grid.

1 2 3 4 5 6 7 8

2nd derivative (order 3)

35 12

− 26

3 19 2

− 14

3 11 12 15 4

− 77

6 107 6

−13

61 12

− 5

6 319 90

− 25

2 77 4

− 161

9

11 − 41

10 25 36 379 105

− 529

42 8129 420

− 809

42 211 14

− 1903

210 293 84

− 127

210 42143 11760

− 1055

84 3245 168

− 1615

84 337 21

− 1727

140 429 56

− 1775

588 179 336

4th derivative (order 1)

1 −4 6 −4 1 3 −14 26 −24 11 −2

1774 1125

− 83

10 2827 150

− 5383

225 451 25

− 5741

750 637 450 9701 4410

− 3127

294 33253 1470

− 26069

882 2719 98

− 27577

1470 6901 882

− 2113

1470 326620243 172872000

− 785833

82320 17221193 823200

− 6868019

246960 2892553 102900

− 16757309

686000 40726213 2469600

− 3976513

576240 2097749 1646400

Berrut Applications of LBR interpolation at equidistant nodes 30/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Higher order derivatives Linear barycentric rational FD RFD weights Examples

Weights for the first centered RFD formulas

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

  • 20
  • 18
  • 16
  • 14
  • 12
  • 10
  • 8
  • 6
  • 4
  • 2

2 4 6 8 10 12 16 18 20 n=4 n=10 n=16 n=22 n=28 n=34 n=40

Figure: Absolute values of the weights for d = 3 for the approximation of the first order derivative at x = 0 on an equispaced grid.

Berrut Applications of LBR interpolation at equidistant nodes 31/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Higher order derivatives Linear barycentric rational FD RFD weights Examples

Weights for the first one-sided RFD formulas

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 n=3 n=5 n=7 n=9 n=11 n=13 n=15 n=17 n=19

Figure: Absolute values of the weights for d = 3 for the approximation of the first order derivative at x = 0 on an equispaced grid.

Berrut Applications of LBR interpolation at equidistant nodes 32/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Higher order derivatives Linear barycentric rational FD RFD weights Examples

Relative errors in centered FD, resp. RFD for d = 4

10

1

10

2

10

−10

10

−8

10

−6

10

−4

10

−2

10 n relative error FD, k=2 RFD d=4, k=2 FD, k=4 RFD d=4, k=4

Figure: Relative errors in the approximation at x = 0 of the second and fourth order derivatives of 1/(1 + 25x2) sampled in [−5, 5].

Berrut Applications of LBR interpolation at equidistant nodes 33/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Higher order derivatives Linear barycentric rational FD RFD weights Examples

Errors in one-sided FD, resp. RFD for d = 4

10

1

10

2

10

3

10

−9

10

−6

10

−3

10 10

3

10

6

10

9

10

12

n error FD, k=2 RFD d=4, k=2 FD, k=4 RFD d=4, k=4

Figure: Errors in the approximation at x = −5 of the second and fourth

  • rder derivatives of 1/(1 + x2) sampled in [−5, 5].

Berrut Applications of LBR interpolation at equidistant nodes 34/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Integration of barycentric rational interpolants

Berrut Applications of LBR interpolation at equidistant nodes 35/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Quadrature from equispaced samples

Problem: Given a real integrable function f sampled at n + 1 points, approximate I := b

a

f (x)dx by a linear quadrature rule n

k=0 wkfk, where f0, . . . , fn are the

given data. Two main situations: We can choose the points Gauss quadrature, Clenshaw-Curtis, ... f is sampled at n + 1 equispaced points Newton-Cotes: unstable as n → ∞.

Berrut Applications of LBR interpolation at equidistant nodes 36/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Quadrature from equispaced samples

Problem: Given a real integrable function f sampled at n + 1 points, approximate I := b

a

f (x)dx by a linear quadrature rule n

k=0 wkfk, where f0, . . . , fn are the

given data. Two main situations: We can choose the points Gauss quadrature, Clenshaw-Curtis, ... f is sampled at n + 1 equispaced points Newton-Cotes: unstable as n → ∞.

Berrut Applications of LBR interpolation at equidistant nodes 36/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Quadrature from equispaced samples

Problem: Given a real integrable function f sampled at n + 1 points, approximate I := b

a

f (x)dx by a linear quadrature rule n

k=0 wkfk, where f0, . . . , fn are the

given data. Two main situations: We can choose the points Gauss quadrature, Clenshaw-Curtis, ... f is sampled at n + 1 equispaced points Newton-Cotes: unstable as n → ∞.

Berrut Applications of LBR interpolation at equidistant nodes 36/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Integration of rational interpolants

Every linear interpolation formula trivially leads to a linear quadrature rule. For a barycentric rational interpolant, we have: I = b

a

f (x)dx ≈ b

a

rn[f ](x)dx = b

a

n

k=0 vk x−xk fk

n

j=0 vj x−xj

dx =

n

  • k=0

wkfk =: Qn, where wk := b

a vk x−xk

n

j=0 vj x−xj

dx.

Berrut Applications of LBR interpolation at equidistant nodes 37/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Integration of rational interpolants

Every linear interpolation formula trivially leads to a linear quadrature rule. For a barycentric rational interpolant, we have: I = b

a

f (x)dx ≈ b

a

rn[f ](x)dx = b

a

n

k=0 vk x−xk fk

n

j=0 vj x−xj

dx =

n

  • k=0

wkfk =: Qn, where wk := b

a vk x−xk

n

j=0 vj x−xj

dx.

Berrut Applications of LBR interpolation at equidistant nodes 37/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Integration of rational interpolants

Every linear interpolation formula trivially leads to a linear quadrature rule. For a barycentric rational interpolant, we have: I = b

a

f (x)dx ≈ b

a

rn[f ](x)dx = b

a

n

k=0 vk x−xk fk

n

j=0 vj x−xj

dx =

n

  • k=0

wkfk =: Qn, where wk := b

a vk x−xk

n

j=0 vj x−xj

dx.

Berrut Applications of LBR interpolation at equidistant nodes 37/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Our suggestions

For true rational interpolants whose denominator degree exceeds 4, there is no straightforward way to establish a linear rational quadrature rule. We are describing two ideas on how to do this, a direct and an indirect one, avoiding expensive partial fraction decompositions and algebraic methods.

Berrut Applications of LBR interpolation at equidistant nodes 38/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Our suggestions

For true rational interpolants whose denominator degree exceeds 4, there is no straightforward way to establish a linear rational quadrature rule. We are describing two ideas on how to do this, a direct and an indirect one, avoiding expensive partial fraction decompositions and algebraic methods.

Berrut Applications of LBR interpolation at equidistant nodes 38/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Direct rational quadrature (DRQ)

Direct rational quadrature rules are based on the numerical stability of the rational interpolant and on well-behaved quadrature rules such as Gauss-Legendre or Clenshaw-Curtis. Let wD

k , k = 0, . . . , n, be some approximations of the weights wk

in Qn; then the direct rational quadrature rule reads I = b

a

f (x)dx ≈

n

  • k=0

wD

k fk,

instead of Qn.

Berrut Applications of LBR interpolation at equidistant nodes 39/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Direct rational quadrature (DRQ)

Direct rational quadrature rules are based on the numerical stability of the rational interpolant and on well-behaved quadrature rules such as Gauss-Legendre or Clenshaw-Curtis. Let wD

k , k = 0, . . . , n, be some approximations of the weights wk

in Qn; then the direct rational quadrature rule reads I = b

a

f (x)dx ≈

n

  • k=0

wD

k fk,

instead of Qn.

Berrut Applications of LBR interpolation at equidistant nodes 39/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Convergence and degree of precision of DRQ in general

Error in interpolation: O(hp), error in the quadrature approximating b

a rn[f ](x)dx: O(hq).

If q ≥ p, then

  • b

a

f (x)dx −

n

  • k=0

wD

k fk

  • ≤

b

a

|f (x) − rn[f ](x)|dx +

  • b

a

rn[f ](x)dx −

n

  • k=0

wD

k fk

  • ≤ Chp.

Similar arguments hold for the degree of precision.

Berrut Applications of LBR interpolation at equidistant nodes 40/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Convergence and degree of precision of DRQ in general

Error in interpolation: O(hp), error in the quadrature approximating b

a rn[f ](x)dx: O(hq).

If q ≥ p, then

  • b

a

f (x)dx −

n

  • k=0

wD

k fk

  • ≤

b

a

|f (x) − rn[f ](x)|dx +

  • b

a

rn[f ](x)dx −

n

  • k=0

wD

k fk

  • ≤ Chp.

Similar arguments hold for the degree of precision.

Berrut Applications of LBR interpolation at equidistant nodes 40/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Convergence and degree of precision of DRQ in general

Error in interpolation: O(hp), error in the quadrature approximating b

a rn[f ](x)dx: O(hq).

If q ≥ p, then

  • b

a

f (x)dx −

n

  • k=0

wD

k fk

  • ≤

b

a

|f (x) − rn[f ](x)|dx +

  • b

a

rn[f ](x)dx −

n

  • k=0

wD

k fk

  • ≤ Chp.

Similar arguments hold for the degree of precision.

Berrut Applications of LBR interpolation at equidistant nodes 40/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Convergence rates of DRQ in a particular case

Theorem Suppose n and d, d ≤ n/2 − 1, are nonnegative integers, f ∈ C d+3[a, b] and rn[f ] belongs to the family of interpolants presented by Floater and Hormann, interpolating f at equispaced

  • nodes. Let the quadrature weights wk in Qn be approximated by a

quadrature rule converging at least at the rate of O(hd+2). Then

  • b

a

f (x)dx −

n

  • k=0

wD

k fk

  • ≤ Chd+2,

where C is a constant depending only on d, on derivatives of f and

  • n the interval length b − a.

Berrut Applications of LBR interpolation at equidistant nodes 41/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Indirect rational quadrature (IRQ)

Indirect quadrature means that we approximate a primitive in the interval [a, b] by a linear rational interpolant. For x ∈ [a, b], we write the problem rn[u](x) ≈ x

a

f (y)dy as an ODE r ′

n[u](x) ≈ f (x),

rn[u](a) = 0 and collocate at the interpolation points.

Berrut Applications of LBR interpolation at equidistant nodes 42/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Indirect rational quadrature (IRQ)

Indirect quadrature means that we approximate a primitive in the interval [a, b] by a linear rational interpolant. For x ∈ [a, b], we write the problem rn[u](x) ≈ x

a

f (y)dy as an ODE r ′

n[u](x) ≈ f (x),

rn[u](a) = 0 and collocate at the interpolation points.

Berrut Applications of LBR interpolation at equidistant nodes 42/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Indirect rational quadrature (IRQ)

To this end we make use of the formula for the first derivative of a rational interpolant explained earlier, giving the vector u′ of the first derivative of rn[u] at the interpolation points u′ = Du, where Dij := D(1)

ij

=            vj vi 1 xi − xj , i = j, −

n

  • k=0

k=i

D(1)

ik ,

i = j. Remark: The matrix D is centro-skew symmetric.

Berrut Applications of LBR interpolation at equidistant nodes 43/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Indirect rational quadrature (IRQ)

To this end we make use of the formula for the first derivative of a rational interpolant explained earlier, giving the vector u′ of the first derivative of rn[u] at the interpolation points u′ = Du, where Dij := D(1)

ij

=            vj vi 1 xi − xj , i = j, −

n

  • k=0

k=i

D(1)

ik ,

i = j. Remark: The matrix D is centro-skew symmetric.

Berrut Applications of LBR interpolation at equidistant nodes 43/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Indirect rational quadrature (IRQ)

Set u = (u0, . . . , un)T, f = (f0, . . . , fn)T and solve the system

n

  • j=1

Dijuj = fi, i = 1, . . . , n. The approximation un of the integral and thus the indirect rational quadrature formula may be given by Cramer’s rule un = 1 det

  • (Dij)

1≤i,j≤n

  • n
  • k=1

det        (Dij)

1≤i≤n 1≤j≤n−1 . . .

1

. . .

       fk =:

n

  • k=1

wI

k fk.

Berrut Applications of LBR interpolation at equidistant nodes 44/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Indirect rational quadrature (IRQ)

Set u = (u0, . . . , un)T, f = (f0, . . . , fn)T and solve the system

n

  • j=1

Dijuj = fi, i = 1, . . . , n. The approximation un of the integral and thus the indirect rational quadrature formula may be given by Cramer’s rule un = 1 det

  • (Dij)

1≤i,j≤n

  • n
  • k=1

det        (Dij)

1≤i≤n 1≤j≤n−1 . . .

1

. . .

       fk =:

n

  • k=1

wI

k fk.

Berrut Applications of LBR interpolation at equidistant nodes 44/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Results for f (x) = sin(100x) + 100

Table: Error in the interpolation and the rational quadrature of f (x) = sin(100x) + 100 for d = 5 at equispaced points in [0, 1] (computing the w D

k by a Gauss-Legendre quadrature with 125 points).

Interpolation DRQ IRQ n error

  • rder

error

  • rder

error

  • rder

20 2.0e+00 6.8e−03 2.7e−03 40 1.8e+00 0.2 1.4e−03 2.3 5.5e−02 −4.3 80 2.8e−02 6.0 9.0e−05 4.0 7.7e−04 6.2 160 6.6e−04 5.4 1.8e−07 9.0 5.7e−05 3.7 320 9.6e−06 6.1 5.7e−09 5.0 1.6e−06 5.2 640 1.3e−07 6.3 4.8e−11 6.9 3.4e−08 5.5 1280 1.1e−09 6.9 3.0e−13 7.3 7.3e−10 5.6

Berrut Applications of LBR interpolation at equidistant nodes 45/49

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Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Results for f (x) = sin(100x) + 100

Table: Error in the interpolation and the rational quadrature of f (x) = sin(100x) + 100 for d = 5 at equispaced points in [0, 1] (computing the w D

k by a Gauss-Legendre quadrature with 125 points).

Interpolation DRQ IRQ n error

  • rder

error

  • rder

error

  • rder

20 2.0e+00 6.8e−03 2.7e−03 40 1.8e+00 0.2 1.4e−03 2.3 5.5e−02 −4.3 80 2.8e−02 6.0 9.0e−05 4.0 7.7e−04 6.2 160 6.6e−04 5.4 1.8e−07 9.0 5.7e−05 3.7 320 9.6e−06 6.1 5.7e−09 5.0 1.6e−06 5.2 640 1.3e−07 6.3 4.8e−11 6.9 3.4e−08 5.5 1280 1.1e−09 6.9 3.0e−13 7.3 7.3e−10 5.6

Berrut Applications of LBR interpolation at equidistant nodes 45/49

slide-77
SLIDE 77

Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Results for f (x) = sin(100x) + 100

Table: Error in the interpolation and the rational quadrature of f (x) = sin(100x) + 100 for d = 5 at equispaced points in [0, 1] (computing the w D

k by a Gauss-Legendre quadrature with 125 points).

Interpolation DRQ IRQ n error

  • rder

error

  • rder

error

  • rder

20 2.0e+00 6.8e−03 2.7e−03 40 1.8e+00 0.2 1.4e−03 2.3 5.5e−02 −4.3 80 2.8e−02 6.0 9.0e−05 4.0 7.7e−04 6.2 160 6.6e−04 5.4 1.8e−07 9.0 5.7e−05 3.7 320 9.6e−06 6.1 5.7e−09 5.0 1.6e−06 5.2 640 1.3e−07 6.3 4.8e−11 6.9 3.4e−08 5.5 1280 1.1e−09 6.9 3.0e−13 7.3 7.3e−10 5.6

Berrut Applications of LBR interpolation at equidistant nodes 45/49

slide-78
SLIDE 78

Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Results for f (x) = sin(100x) + 100

Table: Error in the interpolation and the rational quadrature of f (x) = sin(100x) + 100 for d = 5 at equispaced points in [0, 1] (computing the w D

k by a Gauss-Legendre quadrature with 125 points).

Interpolation DRQ IRQ n error

  • rder

error

  • rder

error

  • rder

20 2.0e+00 6.8e−03 2.7e−03 40 1.8e+00 0.2 1.4e−03 2.3 5.5e−02 −4.3 80 2.8e−02 6.0 9.0e−05 4.0 7.7e−04 6.2 160 6.6e−04 5.4 1.8e−07 9.0 5.7e−05 3.7 320 9.6e−06 6.1 5.7e−09 5.0 1.6e−06 5.2 640 1.3e−07 6.3 4.8e−11 6.9 3.4e−08 5.5 1280 1.1e−09 6.9 3.0e−13 7.3 7.3e−10 5.6

Berrut Applications of LBR interpolation at equidistant nodes 45/49

slide-79
SLIDE 79

Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Comparison for f (x) = sin(100x) + 100

10

1

10

2

10

3

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10 10

2

10

4

10

6

n error d=5 Newton−Cotes Indirect rational Composite Simpson Composite Boole Direct rational Berrut Applications of LBR interpolation at equidistant nodes 46/49

slide-80
SLIDE 80

Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Comparison for f (x) = sin(100x) + 100

10

2

10

3

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10 n error d=5,6,7 IRQ d=5 IRQ d=6 IRQ d=7 Composite Simpson Composite Boole DRQ d=5 DRQ d=6 DRQ d=7 Berrut Applications of LBR interpolation at equidistant nodes 47/49

slide-81
SLIDE 81

Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants Integration of rational interpolants DRQ IRQ Example

Note...

In contrast with DRQ, IRQ yields not only the value un approximating the integral, but also approximate values of the primitive x

a f (y)dy at x1, . . . , xn−1 as u1, . . . , un−1 and at all other

x ∈ [a, b] as the interpolant

n

  • j=0

vj x − xj uj

n

  • j=0

vj x − xj = rn[u](x) ≈ x

a

f (y)dy, x ∈ [a, b]. This approximate primitive is infinitely smooth.

Berrut Applications of LBR interpolation at equidistant nodes 48/49

slide-82
SLIDE 82

Interpolation Differentiation of barycentric rational interpolants Linear barycentric rational finite differences Integration of barycentric rational interpolants

Thank you for your attention!

Berrut Applications of LBR interpolation at equidistant nodes 49/49