Roots of the Chebyshev polynomials of the first kind
In numerical analysis, Chebyshev nodes are a set of specific real algebraic numbers, used as nodes for polynomial interpolation. They are the projection of equispaced points on the unit circle onto the real interval formula_1 the diameter of the circle.
The "Chebyshev nodes of the first kind", also called the Chebyshev zeros, are the zeros of the Chebyshev polynomials of the first kind. The "Chebyshev nodes of the second kind", also called the Chebyshev extrema, are the extrema of the Chebyshev polynomials of the first kind, which are also the zeros of the Chebyshev polynomials of the second kind. Both of these sets of numbers are commonly referred to as "Chebyshev nodes" in literature. Polynomial interpolants constructed from these nodes minimize the effect of Runge's phenomenon.
Definition.
For a given positive integer formula_2 the Chebyshev nodes of the first kind in the open interval formula_3 are
formula_4
These are the roots of the Chebyshev polynomials of the first kind with degree formula_2. For nodes over an arbitrary interval formula_6 an affine transformation can be used:
formula_7
Similarly, for a given positive integer formula_2 the Chebyshev nodes of the second kind in the closed interval formula_9 are
formula_10
These are the roots of the Chebyshev polynomials of the second kind with degree formula_2. For nodes over an arbitrary interval formula_12 an affine transformation can be used as above. The Chebyshev nodes of the second kind are also referred to as Chebyshev-Lobatto points or Chebyshev extreme points. Note that the Chebyshev nodes of the second kind include the end points of the interval while the Chebyshev nodes of the first kind do not include the end points. These formulas generate Chebyshev nodes which are sorted from greatest to least on the real interval.
Both kinds of nodes are always symmetric about the midpoint of the interval. Hence, for odd formula_2, both kinds of nodes will include the midpoint. Geometrically, for both kinds of nodes, we first place formula_2 points on the upper half of the unit circle with equal spacing between them. Then the points are projected down to the formula_15-axis. The projected points on the formula_15-axis are called Chebyshev nodes.
Approximation.
The Chebyshev nodes are important in approximation theory because they form a particularly good set of nodes for polynomial interpolation. Given a function "f" on the interval formula_17 and formula_2 points formula_19 in that interval, the interpolation polynomial is that unique polynomial formula_20 of degree at most formula_21 which has value formula_22 at each point formula_23. The interpolation error at formula_15 is
formula_25
for some formula_26 (depending on x) in [−1, 1]. So it is logical to try to minimize
formula_27
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].) Therefore, when the interpolation nodes "x""i" are the roots of "T""n", the error satisfies
formula_28
For an arbitrary interval ["a", "b"] a change of variable shows that
formula_29
Even order modified Chebyshev nodes.
Many applications for Chebyshev nodes, such as the design of equally terminated passive Chebyshev filters, cannot use Chebyshev nodes directly, due to the lack of a root at 0. However, the Chebyshev nodes may be modified into a usable form by translating the roots down such that the lowest roots are moved to zero, thereby creating two roots at zero of the modified Chebyshev nodes.
The even order modification translation is:
formula_30
The sign of the formula_31 function is chosen to be the same as the sign of formula_32.
For example, the Chebyshev nodes for a 4th order Chebyshev function are, {0.92388,0.382683,-0.382683,-0.92388}, and formula_33 is formula_34, or 0.146446. Running all the nodes through the translation yields formula_35 to be {0.910180, 0, 0, -0.910180}.
The modified even order Chebyshev nodes now contains two nodes of zero, and is suitable for use in designing even order Chebyshev filters with equally terminated passive element networks.
Notes.
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References.
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