In numerical analysis, Chebyshev nodes (also called Chebyshev points or a Chebyshev grid) are a set of specific algebraic numbers used as nodes for polynomial interpolation and numerical integration. They are the projection of a set of equispaced points on the unit circle onto the real interval
[
−
1
,
1
]
{\displaystyle [-1,1]}
, the circle's diameter.
There are two kinds of Chebyshev nodes. The
n
{\displaystyle n}
Chebyshev nodes of the first kind, also called the Chebyshev–Gauss nodes or Chebyshev zeros, are the zeros of a Chebyshev polynomial of the first kind,
T
n
{\displaystyle T_{n}}
. The corresponding
n
+
1
{\displaystyle n+1}
Chebyshev nodes of the second kind, also called the Chebyshev–Lobatto nodes or Chebyshev extrema, are the extrema of
T
n
{\displaystyle T_{n}}
, which are also the zeros of a Chebyshev polynomial of the second kind,
U
n
−
1
{\displaystyle U_{n-1}}
, along with the two endpoints of the interval. Both types of numbers are commonly referred to as Chebyshev nodes or Chebyshev points in literature. They are named after 19th century Russian mathematician Pafnuty Chebyshev, who first introduced Chebyshev polynomials.
Unlike some other interpolation nodes, the Chebyshev nodes "nest": the existing nodes are retained when doubling the number of nodes, reducing computation for each grid refinement by half. Polynomial interpolants constructed from Chebyshev nodes minimize the effect of Runge's phenomenon. They can be easily converted to a representation as a weighted sum of Chebyshev polynomials using the fast Fourier transform.
Contents
Definition
For a given positive integer
n
{\displaystyle n}
, the
n
{\displaystyle n}
Chebyshev nodes of the first kind are given by
x
k
=
cos
(
k
+
1
2
)
π
n
,
k
=
0
,
…
,
n
−
1.
{\displaystyle x_{k}=\cos {\frac {{\bigl (}k+{\tfrac {1}{2}}{\bigr )}\pi }{n}},\quad k=0,\ldots ,n-1.}
This is the projection of
2
Properties
Both kinds of nodes are always symmetric about zero, the midpoint of the interval.
Examples
The node sets for the first few integers
n
{\displaystyle n}
are:
roots
(
T
0
)
=
{
}
,
roots
(
U
0
)
=
{
}
,
extrema
(
T
1
)
=
{
−
1
,
+
1
}
,
roots
Approximation
The Chebyshev nodes are important in approximation theory because they form a particularly good set of nodes for polynomial interpolation. Given a function f on the interval
[
−
1
,
+
1
]
{\displaystyle [-1,+1]}
and
n
{\displaystyle n}
points
x
1
,
x
2
,
…
,
x
n
,
{\displaystyle x_{1},x_{2},\ldots ,x_{n},}
in that interval, the interpolation polynomial is that unique polynomial
P
n
−
1
{\displaystyle P_{n-1}}
Modified even-order nodes
Some applications for interpolation nodes, such as the design of equally terminated passive Chebyshev filters, cannot use even-order Chebyshev nodes directly due to the lack of a root at 0. Instead, the Chebyshev nodes can be moved toward zero, with a double root at zero directly, using a transformation:
x
~
k
=
sgn
(
x
k
)
x
k
2
−
x
n
/
2
2
1
−
x
n
/
2
2
{\displaystyle {\tilde {x}}_{k}=\operatorname {sgn} (x_{k}){\sqrt {\frac {x_{k}^{2}-x_{n/2}^{2}}{1-x_{n/2}^{2}}}}}
For example, Chebyshev nodes of the first kind of order 4 are