![Page 1: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/1.jpg)
Solving the Rational Interpolation Problem
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
FoCM 2017 Barcelona
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 2: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/2.jpg)
Lagrange Interpolation
Given (x1, y1), . . . , (xN , yN) ∈ K2
compute P ∈ K[x ] of minimal degreesuch that P(xi) = yi , i = 1, . . . ,N
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 3: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/3.jpg)
Lagrange Interpolation
Given (x1, y1), . . . , (xN , yN) ∈ K2
compute P ∈ K[x ] of minimal degreesuch that P(xi) = yi , i = 1, . . . ,N
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 4: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/4.jpg)
Lagrange Interpolation
Given (x1, y1), . . . , (xN , yN) ∈ K2
compute P ∈ K[x ] of minimal degreesuch that P(xi) = yi , i = 1, . . . ,N
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 5: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/5.jpg)
“Easy” Solution
1 x1 x2
1 . . . xN−11
1 x2 x22 . . . xN−1
2... ... ... . . ....
1 xN x2N . . . xN−1
N
a0
...aN−1
=
y1...yN
P =N−1∑j=0
ajxj =
N∑j=1
yj
∏i 6=j
x − xixj − xi
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 6: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/6.jpg)
“Easy” Solution
1 x1 x2
1 . . . xN−11
1 x2 x22 . . . xN−1
2... ... ... . . ....
1 xN x2N . . . xN−1
N
a0
...aN−1
=
y1...yN
P =N−1∑j=0
ajxj =
N∑j=1
yj
∏i 6=j
x − xixj − xi
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 7: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/7.jpg)
“Easy” Solution
1 x1 x2
1 . . . xN−11
1 x2 x22 . . . xN−1
2... ... ... . . ....
1 xN x2N . . . xN−1
N
a0
...aN−1
=
y1...yN
P =N−1∑j=0
ajxj
=N∑j=1
yj
∏i 6=j
x − xixj − xi
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 8: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/8.jpg)
“Easy” Solution
1 x1 x2
1 . . . xN−11
1 x2 x22 . . . xN−1
2... ... ... . . ....
1 xN x2N . . . xN−1
N
a0
...aN−1
=
y1...yN
P =N−1∑j=0
ajxj =
N∑j=1
yj
∏i 6=j
x − xixj − xi
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 9: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/9.jpg)
Hermite Interpolation
Given x1, . . . , xL,y10, . . . , y1N1−1, . . . , yL0, . . . , yLNL−1
compute P ∈ K[x ] of minimal degreesuch that P (j)(xi) = yij
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 10: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/10.jpg)
Hermite Interpolation
Given x1, . . . , xL,y10, . . . , y1N1−1, . . . , yL0, . . . , yLNL−1
compute P ∈ K[x ] of minimal degreesuch that P (j)(xi) = yij
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 11: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/11.jpg)
Hermite Interpolation
Given x1, . . . , xL,y10, . . . , y1N1−1, . . . , yL0, . . . , yLNL−1
compute P ∈ K[x ] of minimal degreesuch that P (j)(xi) = yij
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 12: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/12.jpg)
How to solve it?
1 x1 x2
1 . . . xN−11
0 1 2x1 . . . (N − 1)xN−21
......
... . . ....
0 0 0 . . . ∗xN−NL−1L
a0
...aN−1
=
y10...
yLNL−1
P =N−1∑j=0
ajxj =
L∑j=0
Pj(x)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 13: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/13.jpg)
How to solve it?
1 x1 x2
1 . . . xN−11
0 1 2x1 . . . (N − 1)xN−21
......
... . . ....
0 0 0 . . . ∗xN−NL−1L
a0
...aN−1
=
y10...
yLNL−1
P =N−1∑j=0
ajxj =
L∑j=0
Pj(x)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 14: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/14.jpg)
How to solve it?
1 x1 x2
1 . . . xN−11
0 1 2x1 . . . (N − 1)xN−21
......
... . . ....
0 0 0 . . . ∗xN−NL−1L
a0
...aN−1
=
y10...
yLNL−1
P =N−1∑j=0
ajxj
=L∑
j=0
Pj(x)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 15: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/15.jpg)
How to solve it?
1 x1 x2
1 . . . xN−11
0 1 2x1 . . . (N − 1)xN−21
......
... . . ....
0 0 0 . . . ∗xN−NL−1L
a0
...aN−1
=
y10...
yLNL−1
P =N−1∑j=0
ajxj =
L∑j=0
Pj(x)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 16: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/16.jpg)
Computing
Linear Algebra O(N3)
Chinese Remainder Theorem(Hermite)Divided Differences
O(N log2(N))
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 17: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/17.jpg)
Computing
Linear Algebra O(N3)
Chinese Remainder Theorem(Hermite)
Divided Differences
O(N log2(N))
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 18: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/18.jpg)
Computing
Linear Algebra O(N3)
Chinese Remainder Theorem(Hermite)Divided Differences
O(N log2(N))
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 19: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/19.jpg)
Computing
Linear Algebra O(N3)
Chinese Remainder Theorem(Hermite)Divided Differences
O(N log2(N))
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 20: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/20.jpg)
Rational Interpolation
Given x1, . . . , xL,y10, . . . , y1N1−1, . . . , yL0, . . . , yLNL−1
compute R ∈ K(x) such that
R (j)(xi) = yij
of minimal degree
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 21: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/21.jpg)
Rational Interpolation
Given x1, . . . , xL,y10, . . . , y1N1−1, . . . , yL0, . . . , yLNL−1
compute R ∈ K(x) such that
R (j)(xi) = yij
of minimal degree
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 22: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/22.jpg)
Rational Interpolation
Given x1, . . . , xL,y10, . . . , y1N1−1, . . . , yL0, . . . , yLNL−1
compute R ∈ K(x) such that
R (j)(xi) = yij
of minimal degree
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 23: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/23.jpg)
Rational Interpolation
Given x1, . . . , xL,y10, . . . , y1N1−1, . . . , yL0, . . . , yLNL−1
compute R ∈ K(x) such that
R (j)(xi) = yij
of minimal degree
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 24: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/24.jpg)
Sub-Problem (Cauchy/osculatory)
Given 0 ≤ n ≤ N − 1 computeA ∈ K[x ]≤n, B ∈ K[x ]≤N−n−1 suchthat R := A
B solves the interpolationproblem
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 25: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/25.jpg)
Sub-Problem (Cauchy/osculatory)
Given 0 ≤ n ≤ N − 1
computeA ∈ K[x ]≤n, B ∈ K[x ]≤N−n−1 suchthat R := A
B solves the interpolationproblem
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 26: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/26.jpg)
Sub-Problem (Cauchy/osculatory)
Given 0 ≤ n ≤ N − 1 computeA ∈ K[x ]≤n, B ∈ K[x ]≤N−n−1
suchthat R := A
B solves the interpolationproblem
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 27: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/27.jpg)
Sub-Problem (Cauchy/osculatory)
Given 0 ≤ n ≤ N − 1 computeA ∈ K[x ]≤n, B ∈ K[x ]≤N−n−1 suchthat R := A
B solves the interpolationproblem
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 28: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/28.jpg)
Problem with the Sub-Problem
It may be unsolvable!
N = 3, n = 1, x1 = 1, x2 = 2y1,0 = 1, y1,1 = 0, y2,0 = 0
������a0+a1x
b0+b1x
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 29: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/29.jpg)
Problem with the Sub-Problem
It may be unsolvable!
N = 3, n = 1, x1 = 1, x2 = 2y1,0 = 1, y1,1 = 0, y2,0 = 0
������a0+a1x
b0+b1x
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 30: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/30.jpg)
Problem with the Sub-Problem
It may be unsolvable!
N = 3, n = 1, x1 = 1, x2 = 2y1,0 = 1, y1,1 = 0, y2,0 = 0
������a0+a1x
b0+b1x
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 31: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/31.jpg)
Problem with the Sub-Problem
It may be unsolvable!
N = 3, n = 1, x1 = 1, x2 = 2y1,0 = 1, y1,1 = 0, y2,0 = 0
������a0+a1x
b0+b1x
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 32: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/32.jpg)
The “Problem” is the denominator
One needs B(xi) 6= 0∀i = 0, . . . , LFor “generic” input, there is a unique
solutionDescription of the set of “bad points”
given in Cortadellas-D-MontoroarXiv:1704.08563
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 33: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/33.jpg)
The “Problem” is the denominator
One needs B(xi) 6= 0∀i = 0, . . . , L
For “generic” input, there is a uniquesolution
Description of the set of “bad points”given in Cortadellas-D-Montoro
arXiv:1704.08563
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 34: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/34.jpg)
The “Problem” is the denominator
One needs B(xi) 6= 0∀i = 0, . . . , LFor “generic” input, there is a unique
solution
Description of the set of “bad points”given in Cortadellas-D-Montoro
arXiv:1704.08563
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 35: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/35.jpg)
The “Problem” is the denominator
One needs B(xi) 6= 0∀i = 0, . . . , LFor “generic” input, there is a unique
solutionDescription of the set of “bad points”
given in Cortadellas-D-MontoroarXiv:1704.08563
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 36: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/36.jpg)
The Weak Sub-Problem
Given the data xi , yij computeA ∈ K[x ]≤n, B ∈ K[x ]≤N−n−1 such
that A(xi) = yiB(xi)
A(j)(xi) =
j∑s=0
(j)syisB(j−s)(xi)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 37: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/37.jpg)
The Weak Sub-Problem
Given the data xi , yij
computeA ∈ K[x ]≤n, B ∈ K[x ]≤N−n−1 such
that A(xi) = yiB(xi)
A(j)(xi) =
j∑s=0
(j)syisB(j−s)(xi)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 38: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/38.jpg)
The Weak Sub-Problem
Given the data xi , yij computeA ∈ K[x ]≤n, B ∈ K[x ]≤N−n−1
suchthat A(xi) = yiB(xi)
A(j)(xi) =
j∑s=0
(j)syisB(j−s)(xi)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 39: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/39.jpg)
The Weak Sub-Problem
Given the data xi , yij computeA ∈ K[x ]≤n, B ∈ K[x ]≤N−n−1 such
that A(xi) = yiB(xi)
A(j)(xi) =
j∑s=0
(j)syisB(j−s)(xi)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 40: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/40.jpg)
The Weak Sub-Problem
Given the data xi , yij computeA ∈ K[x ]≤n, B ∈ K[x ]≤N−n−1 such
that A(xi) = yiB(xi)
A(j)(xi) =
j∑s=0
(j)syisB(j−s)(xi)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 41: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/41.jpg)
Linear Algebra solves it!
The Weak Problem is alwayssolvableThe solution is “unique”The denominator of “the” solutiondoes not vanish in the xi ’s ⇐⇒the sub-problem can be solved
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 42: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/42.jpg)
Linear Algebra solves it!
The Weak Problem is alwayssolvable
The solution is “unique”The denominator of “the” solutiondoes not vanish in the xi ’s ⇐⇒the sub-problem can be solved
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 43: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/43.jpg)
Linear Algebra solves it!
The Weak Problem is alwayssolvableThe solution is “unique”
The denominator of “the” solutiondoes not vanish in the xi ’s ⇐⇒the sub-problem can be solved
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 44: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/44.jpg)
Linear Algebra solves it!
The Weak Problem is alwayssolvableThe solution is “unique”The denominator of “the” solutiondoes not vanish in the xi ’s ⇐⇒
the sub-problem can be solved
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 45: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/45.jpg)
Linear Algebra solves it!
The Weak Problem is alwayssolvableThe solution is “unique”The denominator of “the” solutiondoes not vanish in the xi ’s ⇐⇒the sub-problem can be solved
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 46: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/46.jpg)
How to solve it
Linear AlgebraSymmetric functions/SubresultantsEuclidean AlgorithmBarycentric CoordinatesJacobi’s Method. . .
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 47: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/47.jpg)
How to solve it
Linear Algebra
Symmetric functions/SubresultantsEuclidean AlgorithmBarycentric CoordinatesJacobi’s Method. . .
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 48: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/48.jpg)
How to solve it
Linear AlgebraSymmetric functions/Subresultants
Euclidean AlgorithmBarycentric CoordinatesJacobi’s Method. . .
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 49: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/49.jpg)
How to solve it
Linear AlgebraSymmetric functions/SubresultantsEuclidean Algorithm
Barycentric CoordinatesJacobi’s Method. . .
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 50: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/50.jpg)
How to solve it
Linear AlgebraSymmetric functions/SubresultantsEuclidean AlgorithmBarycentric Coordinates
Jacobi’s Method. . .
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 51: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/51.jpg)
How to solve it
Linear AlgebraSymmetric functions/SubresultantsEuclidean AlgorithmBarycentric CoordinatesJacobi’s Method
. . .
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 52: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/52.jpg)
How to solve it
Linear AlgebraSymmetric functions/SubresultantsEuclidean AlgorithmBarycentric CoordinatesJacobi’s Method. . .
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 53: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/53.jpg)
Euclidean Algorithm revisited
f :=∏L
i=1(x − xi)Ni
g ∈ K[x ]≤N−1, g(j)(xi) = yij
γ(`)f + β(`)g = α(`), ` = 1, . . .
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 54: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/54.jpg)
Euclidean Algorithm revisited
f :=∏L
i=1(x − xi)Ni
g ∈ K[x ]≤N−1, g(j)(xi) = yij
γ(`)f + β(`)g = α(`), ` = 1, . . .
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 55: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/55.jpg)
Euclidean Algorithm revisited
f :=∏L
i=1(x − xi)Ni
g ∈ K[x ]≤N−1, g(j)(xi) = yij
γ(`)f + β(`)g = α(`), ` = 1, . . .
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 56: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/56.jpg)
Euclidean Algorithm revisited
f :=∏L
i=1(x − xi)Ni
g ∈ K[x ]≤N−1, g(j)(xi) = yij
γ(`)f + β(`)g = α(`), ` = 1, . . .
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 57: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/57.jpg)
Euclidean Algorithm revisited
f :=∏L
i=0(x − xi)Ni
g ∈ K[x ]≤N−1, g(j)(xi) = yij
γ(`)`−1f + β
(`)` g = α
(`)N−1−`, ` = 1, . . .
` = N − n − 1, A = α(`), B = β(`)
work!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 58: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/58.jpg)
Euclidean Algorithm revisited
f :=∏L
i=0(x − xi)Ni
g ∈ K[x ]≤N−1, g(j)(xi) = yij
γ(`)`−1f + β
(`)` g = α
(`)N−1−`, ` = 1, . . .
` = N − n − 1, A = α(`), B = β(`)
work!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 59: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/59.jpg)
But...
you may have α(`) = β(`) = 0 !
However picking “the closest” nonzero` to N − n − 1 works...
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 60: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/60.jpg)
But...
you may have α(`) = β(`) = 0 !
However picking “the closest” nonzero` to N − n − 1 works...
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 61: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/61.jpg)
But...
you may have α(`) = β(`) = 0 !
However
picking “the closest” nonzero` to N − n − 1 works...
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 62: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/62.jpg)
But...
you may have α(`) = β(`) = 0 !
However picking “the closest” nonzero` to N − n − 1 works...
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 63: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/63.jpg)
Complexity?
Computing f , g : O(N log2 N)
One single step of the EEA costs
O(M(N) log(N)) = O(N log2 N)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 64: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/64.jpg)
Complexity?
Computing f , g : O(N log2 N)
One single step of the EEA costs
O(M(N) log(N)) = O(N log2 N)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 65: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/65.jpg)
Complexity?
Computing f , g : O(N log2 N)
One single step of the EEA costs
O(M(N) log(N))
= O(N log2 N)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 66: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/66.jpg)
Complexity?
Computing f , g : O(N log2 N)
One single step of the EEA costs
O(M(N) log(N)) = O(N log2 N)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 67: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/67.jpg)
Can you improve this?
Linear Algebra based methods:O(N3)Barycentric Coordinates:more stable numerically but yetO(N3)Jacobi’s Method:single roots+ mild conditions O(N2)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 68: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/68.jpg)
Can you improve this?
Linear Algebra based methods:O(N3)
Barycentric Coordinates:more stable numerically but yetO(N3)Jacobi’s Method:single roots+ mild conditions O(N2)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 69: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/69.jpg)
Can you improve this?
Linear Algebra based methods:O(N3)Barycentric Coordinates:
more stable numerically but yetO(N3)Jacobi’s Method:single roots+ mild conditions O(N2)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 70: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/70.jpg)
Can you improve this?
Linear Algebra based methods:O(N3)Barycentric Coordinates:more stable numerically but yetO(N3)
Jacobi’s Method:single roots+ mild conditions O(N2)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 71: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/71.jpg)
Can you improve this?
Linear Algebra based methods:O(N3)Barycentric Coordinates:more stable numerically but yetO(N3)Jacobi’s Method:single roots+ mild conditions O(N2)
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 72: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/72.jpg)
Barycentric coordinates
(Schneider - Werner 86)
Any choice of a1, . . . , aN 6= 0 makes
R =
∑Ni=1
aiyi(x−xi )∑N
i=1ai
(x−xi )
a “solution” of the WRIPRIP is solvable ⇐⇒ ai 6= 0
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 73: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/73.jpg)
Barycentric coordinates
(Schneider - Werner 86)
Any choice of a1, . . . , aN 6= 0 makes
R =
∑Ni=1
aiyi(x−xi )∑N
i=1ai
(x−xi )
a “solution” of the WRIP
RIP is solvable ⇐⇒ ai 6= 0
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 74: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/74.jpg)
Barycentric coordinates
(Schneider - Werner 86)
Any choice of a1, . . . , aN 6= 0 makes
R =
∑Ni=1
aiyi(x−xi )∑N
i=1ai
(x−xi )
a “solution” of the WRIPRIP is solvable ⇐⇒
ai 6= 0
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 75: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/75.jpg)
Barycentric coordinates
(Schneider - Werner 86)
Any choice of a1, . . . , aN 6= 0 makes
R =
∑Ni=1
aiyi(x−xi )∑N
i=1ai
(x−xi )
a “solution” of the WRIPRIP is solvable ⇐⇒ ai 6= 0
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 76: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/76.jpg)
Jacobi’s method
(Egecioglu-Koc 89)
From a Bézout type identity:C f + B g = A
we pass to xkC + xk Bf g = xkAf If
k + deg(A) ≤ N − 2, Res∞(xkAf ) = 0
Separated equations for thecoefficients of A and B!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 77: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/77.jpg)
Jacobi’s method
(Egecioglu-Koc 89)
From a Bézout type identity:C f + B g = A
we pass to xkC + xk Bf g = xkAf If
k + deg(A) ≤ N − 2, Res∞(xkAf ) = 0
Separated equations for thecoefficients of A and B!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 78: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/78.jpg)
Jacobi’s method
(Egecioglu-Koc 89)
From a Bézout type identity:C f + B g = A
we pass to xkC + xk Bf g = xkAf
Ifk + deg(A) ≤ N − 2, Res∞(x
kAf ) = 0
Separated equations for thecoefficients of A and B!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 79: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/79.jpg)
Jacobi’s method
(Egecioglu-Koc 89)
From a Bézout type identity:C f + B g = A
we pass to xkC + xk Bf g = xkAf If
k + deg(A) ≤ N − 2, Res∞(xkAf ) = 0
Separated equations for thecoefficients of A and B!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 80: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/80.jpg)
Jacobi’s method
(Egecioglu-Koc 89)
From a Bézout type identity:C f + B g = A
we pass to xkC + xk Bf g = xkAf If
k + deg(A) ≤ N − 2, Res∞(xkAf ) = 0
Separated equations for thecoefficients of A and B!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 81: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/81.jpg)
Our contribution
(Cortadellas-D-Montoro)
Extend Jacobi’s method to:the case with multipliciteseliminate the “conditions”improve the complexity
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 82: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/82.jpg)
Our contribution
(Cortadellas-D-Montoro)
Extend Jacobi’s method to:
the case with multipliciteseliminate the “conditions”improve the complexity
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 83: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/83.jpg)
Our contribution
(Cortadellas-D-Montoro)
Extend Jacobi’s method to:the case with multiplicites
eliminate the “conditions”improve the complexity
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 84: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/84.jpg)
Our contribution
(Cortadellas-D-Montoro)
Extend Jacobi’s method to:the case with multipliciteseliminate the “conditions”
improve the complexity
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 85: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/85.jpg)
Our contribution
(Cortadellas-D-Montoro)
Extend Jacobi’s method to:the case with multipliciteseliminate the “conditions”improve the complexity
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 86: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/86.jpg)
Moreover...
The matrix of residues iscomputable, of size n × n, ofHankel typeComputing an element in thekernel of this matrix can be done atcost O(n log2 n)
B can be interpolated a posteriori
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 87: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/87.jpg)
Moreover...
The matrix of residues iscomputable, of size n × n, ofHankel type
Computing an element in thekernel of this matrix can be done atcost O(n log2 n)
B can be interpolated a posteriori
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 88: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/88.jpg)
Moreover...
The matrix of residues iscomputable, of size n × n, ofHankel typeComputing an element in thekernel of this matrix can be done atcost O(n log2 n)
B can be interpolated a posteriori
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 89: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/89.jpg)
Moreover...
The matrix of residues iscomputable, of size n × n, ofHankel typeComputing an element in thekernel of this matrix can be done atcost O(n log2 n)
B can be interpolated a posterioriTeresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 90: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/90.jpg)
Cost of the Algorithm
(Cortadellas-D-Montoro)O(n log2(n) + N log(N))
Euclides approach: O(N log2(N))
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 91: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/91.jpg)
Cost of the Algorithm
(Cortadellas-D-Montoro)O(n log2(n) + N log(N))
Euclides approach: O(N log2(N))
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 92: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/92.jpg)
What about the minimal problem?
Given (x1, y1), . . . , (xN , yN) ∈ K2
compute a rational functionR ∈ K(x) such that R (j)(xi) = yij of
minimal degree
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 93: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/93.jpg)
What about the minimal problem?
Given (x1, y1), . . . , (xN , yN) ∈ K2
compute a rational functionR ∈ K(x) such that R (j)(xi) = yij of
minimal degree
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 94: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/94.jpg)
Weak Version (single root)
(Ravi 97)
Any pair (A,B) solves the weak RIP
⇐⇒
x − x1 0 . . . 0 1 −y1
0 x − x2 . . . 0 1 −y2...
......
......
0 0 . . . x − xN 1 −yN
∗...∗AB
=
0
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 95: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/95.jpg)
Weak Version (single root)
(Ravi 97)
Any pair (A,B) solves the weak RIP
⇐⇒
x − x1 0 . . . 0 1 −y1
0 x − x2 . . . 0 1 −y2...
......
......
0 0 . . . x − xN 1 −yN
∗...∗AB
=
0
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 96: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/96.jpg)
Weak Version (single root)
(Ravi 97)
Any pair (A,B) solves the weak RIP
⇐⇒
x − x1 0 . . . 0 1 −y1
0 x − x2 . . . 0 1 −y2...
......
......
0 0 . . . x − xN 1 −yN
∗...∗AB
=
0
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 97: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/97.jpg)
Hilbert’s Syzygy’s Theorem
The kernel of this matrix is free ofrank 2There are two generators ofdegrees µ ≤ N − µµ should be the minimal degree!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 98: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/98.jpg)
Hilbert’s Syzygy’s Theorem
The kernel of this matrix is free ofrank 2
There are two generators ofdegrees µ ≤ N − µµ should be the minimal degree!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 99: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/99.jpg)
Hilbert’s Syzygy’s Theorem
The kernel of this matrix is free ofrank 2There are two generators ofdegrees µ ≤ N − µ
µ should be the minimal degree!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 100: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/100.jpg)
Hilbert’s Syzygy’s Theorem
The kernel of this matrix is free ofrank 2There are two generators ofdegrees µ ≤ N − µµ should be the minimal degree!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 101: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/101.jpg)
Don’t bother computing that!
(Cortadellas-D-Montoro)
µ is closest degree of a remainderin the EEA to N
2The “µ-basis” are just twoconsecutive rows of the EEA!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 102: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/102.jpg)
Don’t bother computing that!
(Cortadellas-D-Montoro)
µ is closest degree of a remainderin the EEA to N
2The “µ-basis” are just twoconsecutive rows of the EEA!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 103: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/103.jpg)
From the Weak to the Strong...
(Cortadellas-D-Montoro)
One uses the stratification of “badpoints” given in
Cortadellas-D-MontoroarXiv:1704.08563
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 104: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/104.jpg)
From the Weak to the Strong...
(Cortadellas-D-Montoro)One uses the stratification of “bad
points” given inCortadellas-D-MontoroarXiv:1704.08563
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 105: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/105.jpg)
To conclude...
ThanksAgnes, Marta and Thorsten!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 106: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/106.jpg)
To conclude...
ThanksAgnes, Marta and Thorsten!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem
![Page 107: Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro - UB · Solving the Rational Interpolation Problem Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro FoCM 2017 Barcelona](https://reader031.vdocuments.us/reader031/viewer/2022022108/5bfd7b4a09d3f2fe708cbb56/html5/thumbnails/107.jpg)
And Thank you!
Teresa Cortadellas, Carlos D’Andrea, Eulàlia Montoro
Solving the Rational Interpolation Problem