Chebyshev nodes
Encyclopedia
In numerical analysis
, Chebyshev nodes are the roots of the Chebyshev polynomial of the first kind
. They are often used as nodes in polynomial interpolation
because the resulting interpolation polynomial minimizes the Runge's phenomenon
.
For nodes over an arbitrary interval [a, b] an affine transformation
can be used:
because they form a particularly good set of nodes for polynomial interpolation
. Given a function ƒ on the interval
and 
points 
in that interval, the interpolation polynomial is that unique polynomial 
of degree 
which has value 
at each point 
. The interpolation error at 
is
for some
in [−1, 1]. So it is logical to try to minimize
This product Π is a monic polynomial of degree n. It may be shown that the maximum absolute value of any such polynomial is bounded below by 21−n. This bound is attained by the scaled Chebyshev polynomials 21−n Tn, which are also monic. (Recall that |Tn(x)| ≤ 1 for x ∈ [−1, 1].). When interpolation nodes xi are the roots of the Tn, the interpolation error satisfies
therefore
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
, Chebyshev nodes are the roots of the Chebyshev polynomial of the first kind
Chebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
. They are often used as nodes in polynomial interpolation
Polynomial interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial: given some points, find a polynomial which goes exactly through these points.- Applications :...
because the resulting interpolation polynomial minimizes the Runge's phenomenon
Runge's phenomenon
In the mathematical field of numerical analysis, Runge's phenomenon is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree...
.
Definition
For a given natural number n, Chebyshev nodes in the interval [−1, 1] areFor nodes over an arbitrary interval [a, b] an affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
can be used:
Approximation using Chebyshev nodes
The Chebyshev nodes are important in approximation theoryApproximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby...
because they form a particularly good set of nodes for polynomial interpolation
Polynomial interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial: given some points, find a polynomial which goes exactly through these points.- Applications :...
. Given a function ƒ on the interval
and points in that interval, the interpolation polynomial is that unique polynomial of degree which has value at each point . The interpolation error at isfor some
in [−1, 1]. So it is logical to try to minimizeThis product Π is a monic polynomial of degree n. It may be shown that the maximum absolute value of any such polynomial is bounded below by 21−n. This bound is attained by the scaled Chebyshev polynomials 21−n Tn, which are also monic. (Recall that |Tn(x)| ≤ 1 for x ∈ [−1, 1].). When interpolation nodes xi are the roots of the Tn, the interpolation error satisfies
therefore