Polynomial Interpolation Using Equispaced versus Chebyshev-Lobatto Points

interpolation method

equally spaced points

number of interior points

25

This Demonstration plots

u(x)=

1

1+16

2

x

in the interval

[-1,1]

as well as the function's polynomial interpolation for equally spaced points and for the Chebyshev–Lobatto points.

In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by

y

j

=cos(jπ/N)

. These points are the extremums of the Chebyshev polynomial of the first kind

T

N

(x)

.

You can change the degree of interpolation or the number of interior interpolation points,

N

. As

N

gets larger, the error (computed as the norm of vector

v=

u(i)-

u

intepolation

(i)

i

with

i

varying from

-1

to

1

with a spacing of 0.005), displayed with the red text in the figure, decreases exponentially for the Chebyshev–Lobatto points, while the same computed error increases exponentially in the equispaced case. This behavior is known as the Runge phenomenon.