Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Applications of linear barycentric rational
interpolation at equidistant points
Jean-Paul Berrut and Georges Klein
University of Fribourg (Switzerland)
[email protected]
[email protected]
Oxford, November 2010
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
1/54
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-Klein
Applications of LBR interpolation at equidistant nodes
2/54
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-Klein
Applications of LBR interpolation at equidistant nodes
3/54
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,
f (x0 ), f (x1 ), . . . , f (xn ),
n + 1 distinct nodes and
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
X
fj ℓj (x),
j=0
ℓj (x) :=
Y (x − xk )
.
(xj − xk )
k6=j
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
4/54
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,
f (x0 ), f (x1 ), . . . , f (xn ),
n + 1 distinct nodes and
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
X
fj ℓj (x),
j=0
ℓj (x) :=
Y (x − xk )
.
(xj − xk )
k6=j
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
4/54
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
Y
νj :=
(xj − xk )−1 , j = 0, 1, . . . , n,
k6=j
the so–called weights, which may be computed in advance.
Rewrite the polynomial in its first barycentric form
pn [f ](x) = L(x)
n
X
j=0
where
L(x) :=
n
Y
νj
fj ,
x − xj
(x − xk ).
k=0
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
5/54
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
Y
νj :=
(xj − xk )−1 , j = 0, 1, . . . , n,
k6=j
the so–called weights, which may be computed in advance.
Rewrite the polynomial in its first barycentric form
pn [f ](x) = L(x)
n
X
j=0
where
L(x) :=
n
Y
νj
fj ,
x − xj
(x − xk ).
k=0
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
5/54
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-Klein
Applications of LBR interpolation at equidistant nodes
6/54
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-Klein
Applications of LBR interpolation at equidistant nodes
6/54
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-Klein
Applications of LBR interpolation at equidistant nodes
6/54
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:
n
X
νj
= pn [1](x).
1 = L(x)
x − xj
j=0
Dividing pn [f ] by 1 and cancelling L(x) gives
the barycentric form of the polynomial interpolant
pn [f ](x) =
n
X
j=0
n
X
j=0
Berrut-Klein
νj
fj
x − xj
νj
x − xj
.
Applications of LBR interpolation at equidistant nodes
7/54
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:
n
X
νj
= pn [1](x).
1 = L(x)
x − xj
j=0
Dividing pn [f ] by 1 and cancelling L(x) gives
the barycentric form of the polynomial interpolant
pn [f ](x) =
n
X
j=0
n
X
j=0
Berrut-Klein
νj
fj
x − xj
νj
x − xj
.
Applications of LBR interpolation at equidistant nodes
7/54
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
X
νbj
fj
x − xj
j=0
n
X
j=0
νbj
x − xj
= fk .
Simplification of the weights:
Cancellation of common factor leads to simplified weights.
For equidistant nodes,
∗
j n
νj = (−1)
.
j
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
8/54
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
X
νbj
fj
x − xj
j=0
n
X
j=0
νbj
x − xj
= fk .
Simplification of the weights:
Cancellation of common factor leads to simplified weights.
For equidistant nodes,
∗
j n
νj = (−1)
.
j
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
8/54
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
X
νj
fj
x − xj
j=0
n
X
j=0
νj
x − xj
,
the weights are defined in such a way that
L(x)
n
X
j=0
Modification of these weights
Berrut-Klein
νj
= 1.
x − xj
rational interpolant.
Applications of LBR interpolation at equidistant nodes
9/54
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
X
νj
fj
x − xj
j=0
n
X
j=0
νj
x − xj
,
the weights are defined in such a way that
L(x)
n
X
j=0
Modification of these weights
Berrut-Klein
νj
= 1.
x − xj
rational interpolant.
Applications of LBR interpolation at equidistant nodes
9/54
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
X
vj
fj
x − xj
j=0
n
X
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-Klein
Applications of LBR interpolation at equidistant nodes
10/54
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 by B.:
(−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-Klein
Applications of LBR interpolation at equidistant nodes
11/54
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 by B.:
(−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-Klein
Applications of LBR interpolation at equidistant nodes
11/54
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
X
λj (x)pj (x)
j=0
n−d
X
,
where λj (x) =
λj (x)
(−1)j
.
(x − xj ) . . . (x − xj+d )
j=0
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
12/54
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
X
λj (x)pj (x)
j=0
n−d
X
,
where λj (x) =
λj (x)
(−1)j
.
(x − xj ) . . . (x − xj+d )
j=0
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
12/54
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
X
λj (x)pj (x)
j=0
n−d
X
,
where λj (x) =
λj (x)
(−1)j
.
(x − xj ) . . . (x − xj+d )
j=0
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
12/54
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
X
vj
fj
x − xj
j=0
n
X
j=0
vj
x − xj
.
For equidistant nodes, the weights vj oscillate in sign with absolute
values
1, 1, . . . , 1, 1,
d = 0,
(B.)
1
4,
1 4
8, 8,
1
2 , 1, 1, . . . , 1, 1,
3
4 , 1, 1, . . . , 1, 1,
7
8 , 1, 1, . . . , 1, 1,
1
2,
3
4,
7
8,
1
4,
4 1
8, 8,
Berrut-Klein
d = 1,
(B.)
d = 2,
(Floater − Hormann)
d = 3.
(Floater − Hormann)
Applications of LBR interpolation at equidistant nodes
13/54
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 (xi +1 − xi ), then
0≤i ≤n−1
the rational function rn [f ] has no poles in lR,
if n − d is odd, then
(d+2)
krn [f ] − f k ≤ hd+1 (b − a) kfd+2 k
′′ k
krn [f ] − f k ≤ h(1 + β)(b − a) kf2
if d ≥ 1,
if d = 0;
if n − d is even, then
(d+2)
krn [f ] − f k ≤ hd+1 (b − a) kf d+2 k +
kf (d+1) k d+1
if d ≥ 1,
+ kf ′ k
if d = 0.
krn [f ] − f k ≤ h(1 + β) (b −
o
n |x − x | |x
−
x
|
i +1
i
i
i +1
,
β := max min
1≤i ≤n−2
|xi − xi −1 | |xi +1 − xi +2 |
′′
a) kf2 k
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
14/54
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-Klein
Applications of LBR interpolation at equidistant nodes
15/54
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)
rn [f ](x) =
k!
n
X
j=0
vj
rn [f ][(x)k , xj ]
x − xj
n
X
j=0
1 (k)
rn [f ](xi ) = −
k!
n
X
j=0
j6=i
,
vj
x − xj
x not a node,
!
.
vj rn [f ][(xi ) , xj ]
vi ,
Berrut-Klein
k
i = 0, . . . , n.
Applications of LBR interpolation at equidistant nodes
16/54
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






(1)
Dij :=





matrices D (1) and D (2) (Baltensperger-B.-Noël (1999)):

vj 1
1 (1)
(1)


,
, i 6= j,
D
−
2D

ii
ij

vi xi − xj
xi − xj

n
n
(2)
X
X
Dij :=
(1)
(2)

Dik ,
i = j.
− Dik ;
−




k=0
k=0
k6=i
k6=i
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-Klein
Applications of LBR interpolation at equidistant nodes
17/54
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






(1)
Dij :=





matrices D (1) and D (2) (Baltensperger-B.-Noël (1999)):

vj 1
1 (1)
(1)


,
, i 6= j,
D
−
2D

ii
ij

vi xi − xj
xi − xj

n
n
(2)
X
X
Dij :=
(1)
(2)

Dik ,
i = j.
− Dik ;
−




k=0
k=0
k6=i
k6=i
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-Klein
Applications of LBR interpolation at equidistant nodes
17/54
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-K.)
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 ,
Berrut-Klein
j = 0, 1, . . . , n.
Applications of LBR interpolation at equidistant nodes
18/54
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-K.)
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 ,
Berrut-Klein
j = 0, 1, . . . , n.
Applications of LBR interpolation at equidistant nodes
18/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Differentiation matrices
Convergence rates
Example
Theorem (B.-Floater-K.) (continued)
With the intermediate points, we have
if d ≥ 1 and if f ∈ C d+3 [a, b], then
ke ′ k ≤ Chd
if d ≥ 2,
ke ′ k ≤ C (β + 1)h
if d = 1;
if d ≥ 2 and if f ∈ C d+4 [a, b], then
ke ′′ k ≤ C (β + 1)hd−1
if d ≥ 3,
ke ′′ k
if d = 2.
≤
C (β 2
+ β + 1)h
Mesh ratio
β := max
n
|xi +1 − xi | o
|xi − xi +1 |
.
, max
1≤i ≤n−1 |xi − xi −1 | 0≤i ≤n−2 |xi +1 − xi +2 |
max
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
19/54
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-Klein
Applications of LBR interpolation at equidistant nodes
20/54
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-Klein
Applications of LBR interpolation at equidistant nodes
20/54
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-Klein
Applications of LBR interpolation at equidistant nodes
20/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Differentiation matrices
Convergence rates
Example
Proof for the first derivative at the nodes
The Newton error formula
f (x) − pi (x) = (x − xi ) · · · (x − xi +d )f [xi , xi +1 , . . . , xi +d , x]
leads to
e(x) =
Pn−d
i =0
where after cancellation
A(x) :=
λi (x)(f (x) − pi (x))
A(x)
,
=
Pn−d
B(x)
i =0 λi (x)
n−d
X
(−1)i f [xi , . . . , xi +d , x]
i =0
and
B(x) :=
n−d
X
λi (x).
i =0
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
21/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Differentiation matrices
Convergence rates
Example
Proof for the first derivative at the nodes
The Newton error formula
f (x) − pi (x) = (x − xi ) · · · (x − xi +d )f [xi , xi +1 , . . . , xi +d , x]
leads to
e(x) =
Pn−d
i =0
where after cancellation
A(x) :=
λi (x)(f (x) − pi (x))
A(x)
,
=
Pn−d
B(x)
i =0 λi (x)
n−d
X
(−1)i f [xi , . . . , xi +d , x]
i =0
and
B(x) :=
n−d
X
λi (x).
i =0
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
21/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Differentiation matrices
Convergence rates
Example
Proof for the first derivative at the nodes
The Newton error formula
f (x) − pi (x) = (x − xi ) · · · (x − xi +d )f [xi , xi +1 , . . . , xi +d , x]
leads to
e(x) =
Pn−d
i =0
where after cancellation
A(x) :=
λi (x)(f (x) − pi (x))
A(x)
,
=
Pn−d
B(x)
i =0 λi (x)
n−d
X
(−1)i f [xi , . . . , xi +d , x]
i =0
and
B(x) :=
n−d
X
λi (x).
i =0
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
21/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Differentiation matrices
Convergence rates
Example
By definition and from the fact that e(xj ) = 0, we have
e ′ (xj ) = lim
x→xj
e(x)
.
x − xj
Defining the functions
Bj (x) :=
iY
+d
X
(−1)i
i ∈Ij
Cj (x) :=
X
i ∈I \Ij
where
k=i
k6=j
(−1)i
iY
+d
k=i
1
,
x − xk
1
,
x − xk
I = {0, 1, . . . , n − d} and Ij = {i ∈ I : j − d ≤ i ≤ j},
we see that
(x − xj )B(x) = Bj (x) + (x − xj )Cj (x).
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
22/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Differentiation matrices
Convergence rates
Example
By definition and from the fact that e(xj ) = 0, we have
e ′ (xj ) = lim
x→xj
e(x)
.
x − xj
Defining the functions
Bj (x) :=
iY
+d
X
(−1)i
i ∈Ij
Cj (x) :=
X
i ∈I \Ij
where
k=i
k6=j
(−1)i
iY
+d
k=i
1
,
x − xk
1
,
x − xk
I = {0, 1, . . . , n − d} and Ij = {i ∈ I : j − d ≤ i ≤ j},
we see that
(x − xj )B(x) = Bj (x) + (x − xj )Cj (x).
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
22/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Differentiation matrices
Convergence rates
Example
By definition and from the fact that e(xj ) = 0, we have
e ′ (xj ) = lim
x→xj
e(x)
.
x − xj
Defining the functions
Bj (x) :=
iY
+d
X
(−1)i
i ∈Ij
Cj (x) :=
X
i ∈I \Ij
where
k=i
k6=j
(−1)i
iY
+d
k=i
1
,
x − xk
1
,
x − xk
I = {0, 1, . . . , n − d} and Ij = {i ∈ I : j − d ≤ i ≤ j},
we see that
(x − xj )B(x) = Bj (x) + (x − xj )Cj (x).
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
22/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Differentiation matrices
Convergence rates
Example
We have now
e(x)
A(x)
=
,
x − xj
Bj (x) + (x − xj )Cj (x)
and taking the limit of both sides as x → xj gives
e ′ (xj ) =
Berrut-Klein
A(xj )
.
Bj (xj )
Applications of LBR interpolation at equidistant nodes
23/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Differentiation matrices
Convergence rates
Example
We have now
e(x)
A(x)
=
,
x − xj
Bj (x) + (x − xj )Cj (x)
and taking the limit of both sides as x → xj gives
e ′ (xj ) =
Berrut-Klein
A(xj )
.
Bj (xj )
Applications of LBR interpolation at equidistant nodes
23/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Differentiation matrices
Convergence rates
Example
With x = xj in Bj (x), the products alternate in sign as (−1)i does,
so that all the terms in the sum have the same sign and
|Bj (xj )| =
+d
X iY
|xj − xk |−1 .
i ∈Ij k=i
k6=j
Therefore, by choosing any i ∈ Ij , we deduce that
i +d
Y
1
≤
|xj − xk | ≤ Chd ,
|Bj (xj )|
∀ i ∈ Ij .
k=i
k6=j
On the other hand Floater and Hormann had already shown that
|A(x)| ≤ C ,
whence the bound follows.
Berrut-Klein
x ∈ [a, b],
Applications of LBR interpolation at equidistant nodes
24/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Differentiation matrices
Convergence rates
Example
With x = xj in Bj (x), the products alternate in sign as (−1)i does,
so that all the terms in the sum have the same sign and
|Bj (xj )| =
+d
X iY
|xj − xk |−1 .
i ∈Ij k=i
k6=j
Therefore, by choosing any i ∈ Ij , we deduce that
i +d
Y
1
≤
|xj − xk | ≤ Chd ,
|Bj (xj )|
∀ i ∈ Ij .
k=i
k6=j
On the other hand Floater and Hormann had already shown that
|A(x)| ≤ C ,
whence the bound follows.
Berrut-Klein
x ∈ [a, b],
Applications of LBR interpolation at equidistant nodes
24/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Differentiation matrices
Convergence rates
Example
With x = xj in Bj (x), the products alternate in sign as (−1)i does,
so that all the terms in the sum have the same sign and
|Bj (xj )| =
+d
X iY
|xj − xk |−1 .
i ∈Ij k=i
k6=j
Therefore, by choosing any i ∈ Ij , we deduce that
i +d
Y
1
≤
|xj − xk | ≤ Chd ,
|Bj (xj )|
∀ i ∈ Ij .
k=i
k6=j
On the other hand Floater and Hormann had already shown that
|A(x)| ≤ C ,
whence the bound follows.
Berrut-Klein
x ∈ [a, b],
Applications of LBR interpolation at equidistant nodes
24/54
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 + x 2 ) in [−5, 5] for d = 3.
n
10
20
40
80
160
320
640
Interpolation
error
order
6.9e−02
2.8e−03
4.6
4.3e−06
9.4
5.1e−08
6.4
3.0e−09
4.1
1.8e−10
4.0
1.1e−11
4.0
First derivative
error
order
3.9e−01
3.1e−02
3.7
7.8e−05
8.6
1.2e−06
6.0
1.0e−07
3.6
1.2e−08
3.1
1.5e−09
3.0
Berrut-Klein
Second derivative
error
order
1.5e+00
2.6e−01
2.5
1.5e−03
7.4
6.1e−05
4.6
9.4e−06
2.7
1.2e−06
2.9
3.0e−07
2.0
Applications of LBR interpolation at equidistant nodes
25/54
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 + x 2 ) in [−5, 5] for d = 3.
n
10
20
40
80
160
320
640
Interpolation
error
order
6.9e−02
2.8e−03
4.6
4.3e−06
9.4
5.1e−08
6.4
3.0e−09
4.1
1.8e−10
4.0
1.1e−11
4.0
First derivative
error
order
3.9e−01
3.1e−02
3.7
7.8e−05
8.6
1.2e−06
6.0
1.0e−07
3.6
1.2e−08
3.1
1.5e−09
3.0
Berrut-Klein
Second derivative
error
order
1.5e+00
2.6e−01
2.5
1.5e−03
7.4
6.1e−05
4.6
9.4e−06
2.7
1.2e−06
2.9
3.0e−07
2.0
Applications of LBR interpolation at equidistant nodes
25/54
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 + x 2 ) in [−5, 5] for d = 3.
n
10
20
40
80
160
320
640
Interpolation
error
order
6.9e−02
2.8e−03
4.6
4.3e−06
9.4
5.1e−08
6.4
3.0e−09
4.1
1.8e−10
4.0
1.1e−11
4.0
First derivative
error
order
3.9e−01
3.1e−02
3.7
7.8e−05
8.6
1.2e−06
6.0
1.0e−07
3.6
1.2e−08
3.1
1.5e−09
3.0
Berrut-Klein
Second derivative
error
order
1.5e+00
2.6e−01
2.5
1.5e−03
7.4
6.1e−05
4.6
9.4e−06
2.7
1.2e−06
2.9
3.0e−07
2.0
Applications of LBR interpolation at equidistant nodes
25/54
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 + x 2 ) in [−5, 5] for d = 3.
n
10
20
40
80
160
320
640
Interpolation
error
order
6.9e−02
2.8e−03
4.6
4.3e−06
9.4
5.1e−08
6.4
3.0e−09
4.1
1.8e−10
4.0
1.1e−11
4.0
First derivative
error
order
3.9e−01
3.1e−02
3.7
7.8e−05
8.6
1.2e−06
6.0
1.0e−07
3.6
1.2e−08
3.1
1.5e−09
3.0
Berrut-Klein
Second derivative
error
order
1.5e+00
2.6e−01
2.5
1.5e−03
7.4
6.1e−05
4.6
9.4e−06
2.7
1.2e−06
2.9
3.0e−07
2.0
Applications of LBR interpolation at equidistant nodes
25/54
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
FH d=3, cubic spline (spline toolbox)
0
10
−2
10
−4
log(error)
10
−6
10
−8
10
FH interpolant
Cubic spline
1st derivative FH
1st derivative spline
2nd derivative FH
2nd derivative spline
−10
10
−12
10
1
2
10
10
log(n)
Berrut-Klein
3
10
Applications of LBR interpolation at equidistant nodes
26/54
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-Klein
Applications of LBR interpolation at equidistant nodes
27/54
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-equidistant nodes
Let us now investigate the convergence rate of the k-th derivative,
k = 1, . . . , d + 1, of rn [f ] at equidistant and quasi-equidistant
nodes. By quasi-equidistant nodes (Elling 2007) we shall mean
here points whose minimal spacing hmin satisfies
hmin ≥ ch,
where c is a constant.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
28/54
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 equidistant or
quasi-equidistant, then
|e (k) (xj )| ≤ Chd+1−k ,
0 ≤ j ≤ n,
where C only depends on d, k and derivatives of f .
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
29/54
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,
n
X
dk
d k f (k)
≈ k rn [f ]
=
Dij fj ,
dx k x=xi
dx
x=xi
j=0
(k)
where Dij is the k-th derivative of the j-th Lagrange fundamental
rational function at the node xi .
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
30/54
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,
n
X
dk
d k f (k)
≈ k rn [f ]
=
Dij fj ,
dx k x=xi
dx
x=xi
j=0
(k)
where Dij is the k-th derivative of the j-th Lagrange fundamental
rational function at the node xi .
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
30/54
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,
n
X
dk
d k f (k)
≈ k rn [f ]
=
Dij fj ,
dx k x=xi
dx
x=xi
j=0
(k)
where Dij is the k-th derivative of the j-th Lagrange fundamental
rational function at the node xi .
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
30/54
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)
(k)
In order to establish formulas for the RFD weights Dij , we use the
differentiation matrix D (1) defined earlier for the first order
derivative and the “hybrid formula” (Tee 2006),

k wj (k−1)
(k−1)


,
i 6= j,
−
D
D

ij

 xi − xj wi ii
n
(k)
X
Dij :=
(k)

−
i = j,
Di ℓ ,



 ℓ=0
ℓ6=i
for higher order derivatives.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
31/54
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
order derivatives at x = 0 on an equidistant grid.
−4
−3
−2
−1
0
1
2
4
3
11
7
15
8
− 52
1
− 12
− 1835
576
4
3
11
7
15
8
3
4
1
63
5
72
1
− 128
2nd derivative (order 3)
1
− 12
1
− 128
1
63
5
72
5
− 28
11
− 32
− 355
126
5
− 28
− 11
32
4th derivative (order 1)
1763
12288
− 109
441
− 2845
2304
1
−4
6
−4
1
365
147
17017
3072
− 1133
147
− 3415
256
4826
441
327787
18432
− 1133
147
− 3415
256
365
147
17017
3072
Berrut-Klein
− 109
441
2845
− 2304
1763
12288
Applications of LBR interpolation at equidistant nodes
32/54
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
order derivatives at x = 0 on an equidistant grid.
0
1
2
3
4
5
− 14
3
−13
11
12
61
12
− 56
− 161
9
− 809
42
− 1615
84
11
41
− 10
211
14
337
21
− 1903
210
6
7
25
36
293
84
429
56
− 1775
588
637
450
6901
882
40726213
2469600
− 3976513
576240
8
2nd derivative (order 3)
35
12
15
4
319
90
379
105
42143
11760
− 26
3
− 77
6
− 25
2
− 529
42
− 1055
84
19
2
107
6
77
4
8129
420
3245
168
− 1727
140
127
− 210
179
336
4th derivative (order 1)
1
−4
6
−4
1
3
−14
26
−24
11
−2
1774
1125
9701
4410
326620243
172872000
83
− 10
2827
150
33253
1470
17221193
823200
− 5383
225
451
25
2719
98
2892553
102900
− 5741
750
− 3127
294
− 785833
82320
− 26069
882
− 6868019
246960
Berrut-Klein
− 27577
1470
− 16757309
686000
2113
− 1470
2097749
1646400
Applications of LBR interpolation at equidistant nodes
33/54
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
Figure: Absolute values of the weights for d = 3 for the approximation of
the first order derivative at x = 0 on an equispaced grid.
1.00
0.90
0.80
0.70
0.60
0.50
0.40
n=40
0.30
n=34
n=28
0.20
n=22
n=16
0.10
n=10
n=4
0.00
-20
-18
-16
-14
-12
-10
-8
-6
Berrut-Klein
-4
-2
0
2
4
6
8
10
12
16
18
20
Applications of LBR interpolation at equidistant nodes
34/54
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
Figure: Absolute values of the weights for d = 3 for the approximation of
the first order derivative at x = 0 on an equispaced grid.
4.00
3.50
3.00
2.50
n=19
n=17
2.00
n=15
n=13
1.50
n=11
1.00
n=9
n=7
0.50
n=5
n=3
0.00
0
1
2
3
4
5
6
7
8
9
10
11
12
Berrut-Klein
13
14
15
16
17
18
19
20
Applications of LBR interpolation at equidistant nodes
35/54
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
Figure: Relative errors in the approximation at x = 0 of the second and
fourth order derivatives of 1/(1 + 25x 2 ) sampled in [−5, 5].
0
10
−2
10
log(relative error)
−4
10
−6
10
−8
10
−10
10
FD, k=2
RFD d=4, k=2
FD, k=4
RFD d=4, k=4
1
2
10
10
log(n)
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
36/54
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 one-sided FD, resp. RFD for d = 4
Figure: Relative errors in the approximation at x = 0 of the second and
fourth order derivatives of 1/(1 + x 2 ) sampled in [0, 5].
6
10
FD, k=2
RFD d=4, k=2
FD, k=4
RFD d=4, k=4
4
10
2
log(relative error)
10
0
10
−2
10
−4
10
−6
10
−8
10
0
10
1
2
10
10
3
10
log(n)
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
37/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Integration of barycentric
rational interpolants
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
38/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Quadrature from equidistant samples
Problem: Given a real integrable function f sampled at n + 1
points, approximate
Z b
f (x)dx
I :=
by a linear quadrature rule
given data.
Pn
a
k=0 wk fk ,
where f0 , . . . , fn are the
Two main situations:
We can choose the points
Gauss quadrature, Clenshaw-Curtis, ...
f is sampled at n + 1 equidistant points
Newton-Cotes: unstable as n → ∞.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
39/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Quadrature from equidistant samples
Problem: Given a real integrable function f sampled at n + 1
points, approximate
Z b
f (x)dx
I :=
by a linear quadrature rule
given data.
Pn
a
k=0 wk fk ,
where f0 , . . . , fn are the
Two main situations:
We can choose the points
Gauss quadrature, Clenshaw-Curtis, ...
f is sampled at n + 1 equidistant points
Newton-Cotes: unstable as n → ∞.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
39/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Quadrature from equidistant samples
Problem: Given a real integrable function f sampled at n + 1
points, approximate
Z b
f (x)dx
I :=
by a linear quadrature rule
given data.
Pn
a
k=0 wk fk ,
where f0 , . . . , fn are the
Two main situations:
We can choose the points
Gauss quadrature, Clenshaw-Curtis, ...
f is sampled at n + 1 equidistant points
Newton-Cotes: unstable as n → ∞.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
39/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Integration of rational interpolants
Every linear interpolation formula trivially leads to a linear
quadrature rule.
For a barycentric rational interpolant, we have:
Z b Pn
Z b
Z b
vk
k=0 x−xk fk
rn [f ](x)dx =
f (x)dx ≈
I =
dx
Pn
vj
a
a
a
=
n
X
j=0 x−xj
wk fk =: Qn ,
k=0
where
wk :=
Z
b
a
Berrut-Klein
vk
x−xk
Pn
vj
j=0 x−xj
dx.
Applications of LBR interpolation at equidistant nodes
40/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Integration of rational interpolants
Every linear interpolation formula trivially leads to a linear
quadrature rule.
For a barycentric rational interpolant, we have:
Z b Pn
Z b
Z b
vk
k=0 x−xk fk
rn [f ](x)dx =
f (x)dx ≈
I =
dx
Pn
vj
a
a
a
=
n
X
j=0 x−xj
wk fk =: Qn ,
k=0
where
wk :=
Z
b
a
Berrut-Klein
vk
x−xk
Pn
vj
j=0 x−xj
dx.
Applications of LBR interpolation at equidistant nodes
40/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Integration of rational interpolants
Every linear interpolation formula trivially leads to a linear
quadrature rule.
For a barycentric rational interpolant, we have:
Z b Pn
Z b
Z b
vk
k=0 x−xk fk
rn [f ](x)dx =
f (x)dx ≈
I =
dx
Pn
vj
a
a
a
=
n
X
j=0 x−xj
wk fk =: Qn ,
k=0
where
wk :=
Z
b
a
Berrut-Klein
vk
x−xk
Pn
vj
j=0 x−xj
dx.
Applications of LBR interpolation at equidistant nodes
40/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
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-Klein
Applications of LBR interpolation at equidistant nodes
41/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
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-Klein
Applications of LBR interpolation at equidistant nodes
41/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
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 wkD , k = 0, . . . , n, be some approximations of the weights wk
in Qn ; then the direct rational quadrature rule reads
I =
Z
b
f (x)dx ≈
a
n
X
wkD fk ,
k=0
instead of Qn .
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
42/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
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 wkD , k = 0, . . . , n, be some approximations of the weights wk
in Qn ; then the direct rational quadrature rule reads
I =
Z
b
f (x)dx ≈
a
n
X
wkD fk ,
k=0
instead of Qn .
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
42/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Convergence and degree of precision of DRQ in general
Error in interpolation: O(hp ),
Rb
error in the quadrature approximating a rn [f ](x)dx: O(hq ).
If q ≥ p, then
Z
b
f (x)dx −
a
n
X
k=0
wkD fk ≤
Z
b
|f (x) − rn [f ](x)|dx
a
Z
+
b
rn [f ](x)dx −
a
p
≤ Ch .
n
X
k=0
wkD fk Similar arguments hold for the degree of precision.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
43/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Convergence and degree of precision of DRQ in general
Error in interpolation: O(hp ),
Rb
error in the quadrature approximating a rn [f ](x)dx: O(hq ).
If q ≥ p, then
Z
b
f (x)dx −
a
n
X
k=0
wkD fk ≤
Z
b
|f (x) − rn [f ](x)|dx
a
Z
+
b
rn [f ](x)dx −
a
p
≤ Ch .
n
X
k=0
wkD fk Similar arguments hold for the degree of precision.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
43/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Convergence and degree of precision of DRQ in general
Error in interpolation: O(hp ),
Rb
error in the quadrature approximating a rn [f ](x)dx: O(hq ).
If q ≥ p, then
Z
b
f (x)dx −
a
n
X
k=0
wkD fk ≤
Z
b
|f (x) − rn [f ](x)|dx
a
Z
+
b
rn [f ](x)dx −
a
p
≤ Ch .
n
X
k=0
wkD fk Similar arguments hold for the degree of precision.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
43/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
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
Z
x
f (y )dy
rn [u](x) ≈
a
as an ODE
rn′ [u](x) ≈ f (x),
rn [u](a) = 0
and collocate at the interpolation points.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
44/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
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
Z
x
f (y )dy
rn [u](x) ≈
a
as an ODE
rn′ [u](x) ≈ f (x),
rn [u](a) = 0
and collocate at the interpolation points.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
44/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
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 f ′ of the
first derivative of rn [f ] at the interpolation points
f ′ = Df,
where
(1)
Dij := Dij

vj 1


, i 6= j,


 vi xi − xj
n
X
=
(1)

− Dik , i = j.



 k=0
k6=i
Remark: The matrix D is centro-skew symmetric.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
45/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
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 f ′ of the
first derivative of rn [f ] at the interpolation points
f ′ = Df,
where
(1)
Dij := Dij

vj 1


, i 6= j,


 vi xi − xj
n
X
=
(1)

− Dik , i = j.



 k=0
k6=i
Remark: The matrix D is centro-skew symmetric.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
45/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Indirect rational quadrature (IRQ)
Set u = (u0 , . . . , un )T , f = (f0 , . . . , fn )T and solve the system
n
X
Dij uj = fi ,
i = 1, . . . , n.
j=1
The approximation un of the integral and thus the indirect rational
quadrature formula may be given by Cramer’s rule


0


1

det  (Dij ) 1≤i ≤n
un =

1≤j≤n−1
det (Dij ) 1≤i ,j≤n k=1

n
X
=:
n
X
.
.
.
0
1
0
.
.
.
0



 fk


wkI fk .
k=1
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
46/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Indirect rational quadrature (IRQ)
Set u = (u0 , . . . , un )T , f = (f0 , . . . , fn )T and solve the system
n
X
Dij uj = fi ,
i = 1, . . . , n.
j=1
The approximation un of the integral and thus the indirect rational
quadrature formula may be given by Cramer’s rule


0


1

det  (Dij ) 1≤i ≤n
un =

1≤j≤n−1
det (Dij ) 1≤i ,j≤n k=1

n
X
=:
n
X
.
.
.
0
1
0
.
.
.
0



 fk


wkI fk .
k=1
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
46/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
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 equidistant
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
Z
b
f (x)dx −
a
n
X
k=0
wkD fk ≤ Chd+2 ,
where C is a constant depending only on d, on derivatives of f and
on the interval length b − a.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
47/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Degree of precision of DRQ in a particular case
Theorem
Suppose n and d, d ≤ n, are nonnegative integers, the rational
interpolant rn in the DRQ rule belongs to the family of
interpolants presented by Floater and Hormann and the nodes xi
are allocated symmetrically about the midpoint of the interval
[a, b]. Let the linear quadrature rule Q approximating the integral
of rn be symmetric and have degree of precision at least d + 2.
Then the resulting DRQ rule has degree of precision
d + 2,
d + 1,
d + 1,
d,
if
if
if
if
n
n
n
n
is even and d is odd,
and d are even,
is odd and d is even,
and d are odd.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
48/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Degree of precision of DRQ in a particular case
Idea of the proof:
The last two claims are trivial.
Let P(x) be either x d+2 or x d+1 .
The total quadrature error of the considered DRQ rule reads
Z
b
P(x)dx −Q[rn [P]] =
a
Z
a
b
P(x)dx −Q[P] + Q[P]−Q[rn [P]] .
Since Q is supposed to be symmetric and the interpolation error
P(x) − rn [P](x) is anti-symmetric, the above error vanishes.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
49/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
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 equidistant points in [0, 1]
(computing the wkD by a Gauss-Legendre quadrature with 125 points).
n
20
40
80
160
320
640
1280
Interpolation
error
order
2.0e+00
1.8e+00
0.2
2.8e−02
6.0
6.6e−04
5.4
9.6e−06
6.1
1.3e−07
6.3
1.1e−09
6.9
DRQ
error
order
6.8e−03
1.4e−03
2.3
9.0e−05
4.0
1.8e−07
9.0
5.7e−09
5.0
4.8e−11
6.9
3.0e−13
7.3
Berrut-Klein
IRQ
error
2.7e−03
5.5e−02
7.7e−04
5.7e−05
1.6e−06
3.4e−08
7.3e−10
order
−4.3
6.2
3.7
5.2
5.5
5.6
Applications of LBR interpolation at equidistant nodes
50/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
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 equidistant points in [0, 1]
(computing the wkD by a Gauss-Legendre quadrature with 125 points).
n
20
40
80
160
320
640
1280
Interpolation
error
order
2.0e+00
1.8e+00
0.2
2.8e−02
6.0
6.6e−04
5.4
9.6e−06
6.1
1.3e−07
6.3
1.1e−09
6.9
DRQ
error
order
6.8e−03
1.4e−03
2.3
9.0e−05
4.0
1.8e−07
9.0
5.7e−09
5.0
4.8e−11
6.9
3.0e−13
7.3
Berrut-Klein
IRQ
error
2.7e−03
5.5e−02
7.7e−04
5.7e−05
1.6e−06
3.4e−08
7.3e−10
order
−4.3
6.2
3.7
5.2
5.5
5.6
Applications of LBR interpolation at equidistant nodes
50/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
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 equidistant points in [0, 1]
(computing the wkD by a Gauss-Legendre quadrature with 125 points).
n
20
40
80
160
320
640
1280
Interpolation
error
order
2.0e+00
1.8e+00
0.2
2.8e−02
6.0
6.6e−04
5.4
9.6e−06
6.1
1.3e−07
6.3
1.1e−09
6.9
DRQ
error
order
6.8e−03
1.4e−03
2.3
9.0e−05
4.0
1.8e−07
9.0
5.7e−09
5.0
4.8e−11
6.9
3.0e−13
7.3
Berrut-Klein
IRQ
error
2.7e−03
5.5e−02
7.7e−04
5.7e−05
1.6e−06
3.4e−08
7.3e−10
order
−4.3
6.2
3.7
5.2
5.5
5.6
Applications of LBR interpolation at equidistant nodes
50/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
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 equidistant points in [0, 1]
(computing the wkD by a Gauss-Legendre quadrature with 125 points).
n
20
40
80
160
320
640
1280
Interpolation
error
order
2.0e+00
1.8e+00
0.2
2.8e−02
6.0
6.6e−04
5.4
9.6e−06
6.1
1.3e−07
6.3
1.1e−09
6.9
DRQ
error
order
6.8e−03
1.4e−03
2.3
9.0e−05
4.0
1.8e−07
9.0
5.7e−09
5.0
4.8e−11
6.9
3.0e−13
7.3
Berrut-Klein
IRQ
error
2.7e−03
5.5e−02
7.7e−04
5.7e−05
1.6e−06
3.4e−08
7.3e−10
order
−4.3
6.2
3.7
5.2
5.5
5.6
Applications of LBR interpolation at equidistant nodes
50/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Comparison for f (x) = sin(100x) + 100
d=5
Newton−Cotes
Indirect rational
Composite Simpson
Composite Boole
Direct rational
6
10
4
10
2
10
0
log(error)
10
−2
10
−4
10
−6
10
−8
10
−10
10
−12
10
1
10
2
3
10
10
log(n)
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
51/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Comparison for f (x) = sin(100x) + 100
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
0
10
−2
10
−4
log(error)
10
−6
10
−8
10
−10
10
−12
10
2
3
10
10
log(n)
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
52/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Integration of rational interpolants
DRQ and IRQ
Convergence rates and degree of precision
Example
Remember...
In contrast with DRQ, IRQ yields not only the value un
approximating
R x the integral, but also approximate values of the
primitive a f (y )dy at x1 , . . . , xn−1 as u1 , . . . , un−1 and at all other
x ∈ [a, b] as the interpolant
n
X
vj
uj
x − xj
j=0
n
X
j=0
vj
x − xj
= rn [u](x) ≈
Z
x
f (y )dy ,
x ∈ [a, b].
a
This approximate primitive is infinitely smooth.
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
53/54
Interpolation
Differentiation of barycentric rational interpolants
Linear barycentric rational finite differences
Integration of barycentric rational interpolants
Thank you for your attention!
Berrut-Klein
Applications of LBR interpolation at equidistant nodes
54/54