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.

For a given natural number n, 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:

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 ${\displaystyle [-1,+1]}$ and ${\displaystyle n}$ points ${\displaystyle x_{1},x_{2},\ldots ,x_{n},}$ in that interval, the interpolation polynomial is that unique polynomial ${\displaystyle P_{n-1}}$ of degree at most ${\displaystyle n-1}$ which has value ${\displaystyle f(x_{i})}$ at each point ${\displaystyle x_{i}}$. The interpolation error at ${\displaystyle x}$ is

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