Chebyshev nodes

In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial minimizes the effect of Runge's phenomenon.^[1]

Definition

For a given natural number n, Chebyshev nodes in the interval (−1, 1) are

$x_k = \cos\left(\frac{2k-1}{2n}\pi\right) \mbox{ , } k=1,\ldots,n.$

These are the roots of the Chebyshev polynomial of the first kind of degree n. For nodes over an arbitrary interval [a, b] an affine transformation can be used:

${x}_k = \frac{1}{2} (a+b) + \frac{1}{2} (b-a) \cos\left(\frac{2k-1}{2n}\pi\right) \mbox{ , } k=1, \ldots, n.$

Approximation

The Chebyshev nodes are important in approximation theory because they form a particularly good set of nodes for polynomial interpolation. Given a function ƒ on the interval $[-1,+1]$ and $n$ points $x_1, x_2, \ldots , x_n,$ in that interval, the interpolation polynomial is that unique polynomial $P_{n-1}$ of degree at most $n-1$ which has value $f(x_i)$ at each point $x_i$. The interpolation error at $x$ is

$f(x) - P_{n-1}(x) = \frac{f^{(n)}(\xi)}{n!} \prod_{i=1}^n (x-x_i)$

for some $\xi$ in [−1, 1].^[2] So it is logical to try to minimize

$\max_{x \in [-1,1]} \left| \prod_{i=1}^n (x-x_i) \right|.$

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 T_n, which are also monic. (Recall that |T_n(x)| ≤ 1 for x ∈ [−1, 1].^[3]). Therefore, when interpolation nodes x_i are the roots of T_n, the interpolation error satisfies

$\left|f(x) - P_{n-1}(x)\right| \le \frac{1}{2^{n - 1}n!} \max_{\xi \in [-1,1]} \left|f^{(n)} (\xi)\right|.$

For an arbitrary interval [a, b] a change of variable shows that

$\left|f(x) - P_{n-1}(x)\right| \le \frac{1}{2^{n - 1}n!} \left(\frac{b-a}{2}\right)^n \max_{\xi \in [a,b]} \left|f^{(n)} (\xi)\right|.$

Notes

1. Fink, Kurtis D., and John H. Mathews. Numerical Methods using MATLAB. Upper Saddle River, NJ: Prentice Hall, 1999. 3rd ed. pp. 236-238.
2. Stewart (1996), (20.3)
3. Stewart (1996), Lecture 20, §14

References

• Stewart, Gilbert W. (1996), Afternotes on Numerical Analysis, SIAM, ISBN 978-0-89871-362-6.

Further reading

• Burden, Richard L.; Faires, J. Douglas: Numerical Analysis, 8th ed., pages 503–512, ISBN 0-534-39200-8.