Editing Chebyshev nodes (section)

== Modified even-order nodes ==
Some applications for interpolation nodes, such as the design of equally terminated passive [[Chebyshev filter]]s, cannot use even-order Chebyshev nodes directly due to the lack of a root at 0. Instead, the Chebyshev nodes can be moved toward zero, with a double root at zero directly, using a transformation:<ref name=":02">{{Cite book |last=Saal |first=Rudolf |url=https://archive.org/details/handbuchzumfilte0000saal |title=Handbook of Filter Design |publisher=Allgemeine Elektricitäts-Gesellschaft |date=January 1979 |isbn=3-87087-070-2 |edition=1st |location=Munich, Germany |pages=25, 26, 56-61, 116, 117 |language=English, German}}</ref>

<math display=block>\tilde{x}_k = \operatorname{sgn}(x_k)\sqrt{\frac{x_k^2-x_{n/2}^2}{1-x_{n/2}^2}}</math>

For example, Chebyshev nodes of the first kind of order 4 are <math>{0.9239,0.3827,-0.3827,-0.9239}</math>, with <math>x_{n/2} = 0.382683</math>. Applying the transformation yields new nodes <math>{0.910180, 0, 0, -0.910180}</math>. The modified even-order nodes now include zero twice.