Chebyshev Nodes

A Chebyshev node is one of a set of abscissas used for polynomial interpolation, obtained by projecting equally spaced points on a semicircle onto its diameter. The roots of the Chebyshev polynomial of the first kind give the node set

(1)

for , ..., . These lie in and cluster toward the ends of the interval. They minimize the maximum absolute value on of the monic polynomial , which appears in the error formula for polynomial interpolation.

Another common choice, often called Chebyshev-Lobatto points, includes both endpoints and uses the extrema of ,

(2)

for , ..., and . Both conventions are called Chebyshev nodes, so the node formula should be specified. Nodes on a general interval are obtained from either set by the affine transformation

(3)