“Chebyshev nodes”的意思、由来-开放百科全书

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Chebyshev nodes

释义

Definition
Approximation
Notes
References
Further reading

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.^[2]

Definition

For a given positive integer n the Chebyshev nodes in the interval (−1, 1) are

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:

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 and points in that interval, the interpolation polynomial is that unique polynomial of degree at most which has value at each point . The interpolation error at is

for some (depending on x) in [−1, 1].^[3] So it is logical to try to minimize

This product is a monic polynomial of degree n. It may be shown that the maximum absolute value (maximum norm) of any such polynomial is bounded from 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].^[4]) Therefore, when the interpolation nodes x_i are the roots of T_n, the error satisfies

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

Notes

1. ^Lloyd N. Trefethen, Approximation Theory and Approximation Practice (SIAM, 2012). Online: https://people.maths.ox.ac.uk/trefethen/ATAP/
2. ^Fink, Kurtis D., and John H. Mathews. Numerical Methods using MATLAB. Upper Saddle River, NJ: Prentice Hall, 1999. 3rd ed. pp. 236-238.
3. ^{{harvtxt|Stewart|1996}}, (20.3)
4. ^{{harvtxt|Stewart|1996}}, Lecture 20, §14

References

{{Citation | last1=Stewart | first1=Gilbert W. | title=Afternotes on Numerical Analysis | publisher=SIAM | isbn=978-0-89871-362-6 | year=1996}}.

Definition

Approximation

Notes

References

Further reading