Runge’s Example f (x) =
1
1+25x2
on [−1, 1]
uniform knot distribution; n=2
Chebyshev interpolation; n=2
1
1
uniform knots
2
f(x) = 1/(1+25 x )
uniform knots
f(x) = 1/(1+25 x2)
0.9
0.8
0.8
0.6
0.7
0.6
0.4
0.5
0.2
0.4
0.3
0
0.2
−0.2
0.1
0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−0.4
−1
0.5
0.9
0
0.8
−0.5
0.7
−1
0.6
−1.5
0.5
−2
0.4
−2.5
0.3
−3
0.2
−3.5
0.1
−0.6
−0.4
5
2.5
−0.2
0
0.2
0.4
0.6
0.8
1
−0.2
0
0.2
0.4
0.6
0.8
1
0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Chebyshev interpolation; n=42
uniform knot distribution; n=42
x 10
−0.4
uniform knots
2
f(x) = 1/(1+25 x )
uniform knots
f(x) = 1/(1+25 x2)
−0.8
−0.6
1
1
−4
−1
−0.8
Chebyshev interpolation; n=12
uniform knot distribution; n=12
1
uniform knots
2
f(x) = 1/(1+25 x )
uniform knots
f(x) = 1/(1+25 x2)
0.9
2
0.8
0.7
1.5
0.6
1
0.5
0.4
0.5
0.3
0.2
0
0.1
−0.5
−1
−0.5
0
0.5
1
0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Runge’s Example f (x) =
1
1+25x2
Interpolation in uniformly distributed knots fails:
equid., n=2
n=12
0.7
5
0.5
2.5
x 10
n=42
0
0.6
2
−0.5
0.5
−1
0.4
−1.5
0.3
−2
1.5
1
−2.5
0.5
0.2
−3
0
0.1
−3.5
0
−1
0
1
−4
−1
0
1
−0.5
−1
0
1
Interpolation in Chebyshev points works quite well:
Chebyshev, n=2
1
n=22
−4
0.01
6
x 10
n=42
0.8
0.005
4
0.6
0
0.4
2
0.2
−0.005
0
0
−0.01
−0.2
−0.4
−1
0
−0.015
1 −1
n
kf − πnCheb kC([−1,1]
0
10
1.1−1
1
20
1.5−2
−2
−1
30
2.1−3
0
40
2.9−4
1