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<title>Chebyshev nodes</title>

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Chebyshev nodes

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In <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a>, Chebyshev nodes (also called Chebyshev points or a Chebyshev grid) are a set of specific <a href="Algebraic_number" title="Algebraic number">algebraic numbers</a> used as nodes for <a href="Polynomial_interpolation" title="Polynomial interpolation">polynomial interpolation</a> and <a href="Numerical_integration" title="Numerical integration">numerical integration</a>. They are the <a href="Projection_(linear_algebra)" title="Projection (linear algebra)">projection</a> of a set of equispaced points on the <a href="Unit_circle" title="Unit circle">unit circle</a> onto the <a href="Real_interval" class="mw-redirect" title="Real interval">real interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}">

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<annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation>

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</math><img src="./51e3b7f14a6f70e614728c583409a0b9a8b9de01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}" loading="lazy">, the circle's <a href="Diameter" title="Diameter">diameter</a>.

There are two kinds of Chebyshev nodes. The ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">⁠ Chebyshev nodes of the first kind, also called the Chebyshev–Gauss nodes<a href="#cite_note-1">[1]</a> or Chebyshev zeros, are the <a href="Zero_of_a_function" title="Zero of a function">zeros</a> of a <a href="Chebyshev_polynomial" class="mw-redirect" title="Chebyshev polynomial">Chebyshev polynomial</a> of the first kind, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation>

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</math><img src="./4d9241493be76739f2400f258f32c24f9689161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}" loading="lazy">⁠. The corresponding ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}">

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<annotation encoding="application/x-tex">{\displaystyle n+1}</annotation>

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</math><img src="./2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy">⁠ Chebyshev nodes of the second kind, also called the Chebyshev–Lobatto nodes<a href="#cite_note-2">[2]</a> or Chebyshev extrema, are the <a href="Maximum_and_minimum" title="Maximum and minimum">extrema</a> of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation>

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</math><img src="./4d9241493be76739f2400f258f32c24f9689161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}" loading="lazy">⁠, which are also the zeros of a Chebyshev polynomial of the second kind, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{n-1}}">

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<annotation encoding="application/x-tex">{\displaystyle U_{n-1}}</annotation>

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</math><img src="./62721cdfea6ee951fdf6db099baeab7c91045db3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.906ex; height:2.509ex;" alt="{\displaystyle U_{n-1}}" loading="lazy">⁠, along with the two endpoints of the interval. Both types of numbers are commonly referred to as Chebyshev nodes or Chebyshev points in literature.<a href="#cite_note-3">[3]</a> They are named after 19th century Russian mathematician <a href="Pafnuty_Chebyshev" title="Pafnuty Chebyshev">Pafnuty Chebyshev</a>, who first introduced Chebyshev polynomials.

Unlike some other interpolation nodes, the Chebyshev nodes "nest": the existing nodes are retained when doubling the number of nodes, reducing computation for each grid refinement by half. Polynomial interpolants constructed from Chebyshev nodes minimize the effect of <a href="Runge's_phenomenon" title="Runge's phenomenon">Runge's phenomenon</a>.<a href="#cite_note-4">[4]</a> They can be easily converted to a representation as a weighted sum of Chebyshev polynomials using the <a href="Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

For a given positive integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, the ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">⁠ Chebyshev nodes of the first kind are given by

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<annotation encoding="application/x-tex">{\displaystyle x_{k}=\cos {\frac {{\bigl (}k+{\tfrac {1}{2}}{\bigr )}\pi }{n}},\quad k=0,\ldots ,n-1.}</annotation>

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This is the projection of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}">

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<annotation encoding="application/x-tex">{\displaystyle 2n}</annotation>

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</math><img src="./134afa8ff09fdddd24b06f289e92e3a045092bd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}" loading="lazy">⁠ equispaced points on the unit circle onto the interval ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}">

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<annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation>

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</math><img src="./51e3b7f14a6f70e614728c583409a0b9a8b9de01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}" loading="lazy">⁠, the circle's diameter. These points are also the roots of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation>

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</math><img src="./4d9241493be76739f2400f258f32c24f9689161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}" loading="lazy">⁠, the Chebyshev polynomial of the first kind with degree ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">⁠.

The ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}">

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<annotation encoding="application/x-tex">{\displaystyle n+1}</annotation>

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</math><img src="./2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy">⁠ Chebyshev nodes of the second kind are given by

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle x_{k}=\cos {\frac {k\pi }{n}},\quad k=0,\ldots ,n.}</annotation>

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This is also the projection of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}">

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<annotation encoding="application/x-tex">{\displaystyle 2n}</annotation>

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</math><img src="./134afa8ff09fdddd24b06f289e92e3a045092bd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}" loading="lazy">⁠ equispaced points on the unit circle onto ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}">

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<annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation>

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</math><img src="./51e3b7f14a6f70e614728c583409a0b9a8b9de01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}" loading="lazy">⁠, this time including the endpoints of the interval, each of which is only the projection of one point on the circle rather than two. These points are also the extrema of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation>

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</math><img src="./4d9241493be76739f2400f258f32c24f9689161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}" loading="lazy">⁠ in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}">

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<annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation>

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</math><img src="./51e3b7f14a6f70e614728c583409a0b9a8b9de01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}" loading="lazy">⁠, the places where it takes the value ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1}">

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<annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation>

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</math><img src="./0bfeaa85da53ad1947d8000926cfea33827ef1e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}" loading="lazy">⁠.<a href="#cite_note-5">[5]</a> The interior points among the nodes, not including the endpoints, are also the zeros of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{n-1}}">

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<annotation encoding="application/x-tex">{\displaystyle U_{n-1}}</annotation>

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</math><img src="./62721cdfea6ee951fdf6db099baeab7c91045db3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.906ex; height:2.509ex;" alt="{\displaystyle U_{n-1}}" loading="lazy">⁠, a Chebyshev polynomial of the second kind, a rescaling of the derivative of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}}">

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation>

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</math><img src="./4d9241493be76739f2400f258f32c24f9689161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}" loading="lazy">⁠.

For nodes over an arbitrary interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

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<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

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</math><img src="./9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"> an <a href="Affine_transformation" title="Affine transformation">affine transformation</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}">

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<annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation>

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</math><img src="./51e3b7f14a6f70e614728c583409a0b9a8b9de01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}" loading="lazy"> can be used:

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}_{k}={\tfrac {1}{2}}(a+b)+{\tfrac {1}{2}}(b-a)x_{k}.}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

Both kinds of nodes are always symmetric about zero, the midpoint of the interval.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

The node sets for the first few integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> are:

<mtr>

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<mtext>roots</mtext>

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<msub>

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<mtd>

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<mtd>

<mtext>roots</mtext>

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<msub>

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</mtd>

<mtd>

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<mtd>

<mtext>extrema</mtext>

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<msub>

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<mtd>

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<mtr>

<mtd>

<mtext>roots</mtext>

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</mtd>

<mtd>

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<mtd>

<mtext>roots</mtext>

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</mtd>

<mtd>

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<mtd>

<mtext>extrema</mtext>

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</mtd>

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<mtr>

<mtd>

<mtext>roots</mtext>

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<msub>

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</mtd>

<mtd>

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<msqrt>

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<msqrt>

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<mtext>roots</mtext>

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<msub>

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<mtd>

<mtext>extrema</mtext>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\text{roots}}(T_{0})&=\{\},&{\text{roots}}(U_{0})&=\{\},&{\text{extrema}}(T_{1})&=\{-1,+1\},\\{\text{roots}}(T_{1})&=\{0\},&{\text{roots}}(U_{1})&=\{0\},&{\text{extrema}}(T_{2})&=\{-1,0,+1\},\\{\text{roots}}(T_{2})&=\{-1/{\sqrt {2}},+1/{\sqrt {2}}\},&{\text{roots}}(U_{2})&=\{-1/2,+1/2\},&{\text{extrema}}(T_{3})&=\{-1,-1/2,+1/2,+1\}\\\end{aligned}}}</annotation>

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While these sets are sorted by ascending values, the defining formulas given above generate the Chebyshev nodes in reverse order from the greatest to the smallest.

<div class="mw-heading mw-heading2"><h2 id="Approximation">Approximation</h2></div>

The Chebyshev nodes are important in <a href="Approximation_theory" title="Approximation theory">approximation theory</a> because they form a particularly good set of nodes for <a href="Polynomial_interpolation" title="Polynomial interpolation">polynomial interpolation</a>. Given a function f on the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,+1]}">

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<annotation encoding="application/x-tex">{\displaystyle [-1,+1]}</annotation>

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</math><img src="./daa72f1a806823ec94fda7922597b19cbda684f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.269ex; height:2.843ex;" alt="{\displaystyle [-1,+1]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},x_{2},\ldots ,x_{n},}">

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<annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},\ldots ,x_{n},}</annotation>

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</math><img src="./cdce849f807aba5436eb6f37077398af863b312f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.175ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},\ldots ,x_{n},}" loading="lazy"> in that interval, the interpolation polynomial is that unique polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n-1}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{n-1}}</annotation>

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</math><img src="./9cd7d877c0c7616c7602990e94de76b02a81ad99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.811ex; height:2.509ex;" alt="{\displaystyle P_{n-1}}" loading="lazy"> of degree at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">

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<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>

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</math><img src="./fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy"> which has value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{i})}">

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<annotation encoding="application/x-tex">{\displaystyle f(x_{i})}</annotation>

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</math><img src="./77bf16c9153b454dbdec165ad8160083202d7f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.217ex; height:2.843ex;" alt="{\displaystyle f(x_{i})}" loading="lazy"> at each point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

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</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy">. The interpolation error at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is

<msub>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle f(x)-P_{n-1}(x)={\frac {f^{(n)}(\xi )}{n!}}\prod _{i=1}^{n}(x-x_{i})}</annotation>

</semantics>

</math>

for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi }</annotation>

</semantics>

</math><img src="./e0b461aaf61091abd5d2c808931c48b8ff9647db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }" loading="lazy"> (depending on x) in [−1, 1].<a href="#cite_note-6">[6]</a> So it is logical to try to minimize

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<annotation encoding="application/x-tex">{\displaystyle \max _{x\in [-1,1]}{\biggl |}\prod _{i=1}^{n}(x-x_{i}){\biggr |}.}</annotation>

</semantics>

</math>

This product is a <a href="Monic_polynomial" title="Monic polynomial">monic</a> 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 Tn, which are also monic. (Recall that |Tn(x)| ≤ 1 for x ∈ [−1, 1].<a href="#cite_note-7">[7]</a>) Therefore, when the interpolation nodes xi are the roots of Tn, the error satisfies

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<annotation encoding="application/x-tex">{\displaystyle \left|f(x)-P_{n-1}(x)\right|\leq {\frac {1}{2^{n-1}n!}}\max _{\xi \in [-1,1]}\left|f^{(n)}(\xi )\right|.}</annotation>

</semantics>

</math>

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

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<annotation encoding="application/x-tex">{\displaystyle \left|f(x)-P_{n-1}(x)\right|\leq {\frac {1}{2^{n-1}n!}}\left({\frac {b-a}{2}}\right)^{n}\max _{\xi \in [a,b]}\left|f^{(n)}(\xi )\right|.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Modified_even-order_nodes">Modified even-order nodes</h2></div>

Some applications for interpolation nodes, such as the design of equally terminated passive <a href="Chebyshev_filter" title="Chebyshev filter">Chebyshev filters</a>, 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:<a href="#cite_note-:02-8">[8]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}_{k}=\operatorname {sgn} (x_{k}){\sqrt {\frac {x_{k}^{2}-x_{n/2}^{2}}{1-x_{n/2}^{2}}}}}</annotation>

</semantics>

</math>

For example, Chebyshev nodes of the first kind of order 4 are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {0.9239,0.3827,-0.3827,-0.9239}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {0.9239,0.3827,-0.3827,-0.9239}}</annotation>

</semantics>

</math><img src="./b638f56fa657e625097c17a774636a370f00b6ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.555ex; height:2.509ex;" alt="{\displaystyle {0.9239,0.3827,-0.3827,-0.9239}}" loading="lazy">, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n/2}=0.382683}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{n/2}=0.382683}</annotation>

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</math><img src="./e33b585606d6612333f7430e6bd2c1644482edf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.075ex; height:3.009ex;" alt="{\displaystyle x_{n/2}=0.382683}" loading="lazy">. Applying the transformation yields new nodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {0.910180,0,0,-0.910180}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {0.910180,0,0,-0.910180}}</annotation>

</semantics>

</math><img src="./af0c10a3a778b22c145fd58ed36199fcfa557a04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.803ex; height:2.509ex;" alt="{\displaystyle {0.910180,0,0,-0.910180}}" loading="lazy">. The modified even-order nodes now include zero twice.

<ul><li><a href="Chebfun" title="Chebfun">Chebfun</a></li>

<li><a href="Chebyshev%E2%80%93Gauss_quadrature" title="Chebyshev–Gauss quadrature">Chebyshev–Gauss quadrature</a></li>

<li><a href="Lebesgue_constant" title="Lebesgue constant">Lebesgue constant</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;">

<li id="cite_note-1"><a href="#cite_ref-1">^</a> The name Chebyshev–Gauss nodes comes from the use of Chebyshev zeros in numerical integration, which can be seen as a variant of <a href="Gaussian_quadrature" title="Gaussian quadrature">Gaussian quadrature</a>.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> The name Chebyshev–Lobatto nodes comes from <a href="Rehuel_Lobatto" title="Rehuel Lobatto">Rehuel Lobatto</a>, who made a variant of Gaussian quadrature, known as <a href="Lobatto_quadrature" class="mw-redirect" title="Lobatto quadrature">Lobatto quadrature</a>, whose nodes included the ends of the interval, a feature shared by the Chebyshev extrema.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFTrefethen2013">Trefethen 2013</a>, pp. 7

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFFinkMathews1999">Fink & Mathews 1999</a>, pp. 236–238

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFTrefethen2013">Trefethen 2013</a>, pp. 14

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFStewart1996">Stewart 1996</a>, (20.3)

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFStewart1996">Stewart 1996</a>, Lecture 20, §14

</li>

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFSaal1979" class="citation book cs1 cs1-prop-foreign-lang-source">Saal, Rudolf (January 1979). <a rel="nofollow" class="external text" href="https://archive.org/details/handbuchzumfilte0000saal">Handbook of Filter Design</a> (in English and German) (1st ed.). Munich, Germany: Allgemeine Elektricitäts-Gesellschaft. pp. 25, 26, 56–61, 116, 117. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-87087-070-2</bdi>.</cite>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<ul><li><cite id="CITEREFFinkMathews1999" class="citation book cs1">Fink, Kurtis D.; Mathews, John H. (1999). Numerical Methods using MATLAB (3rd ed.). Upper Saddle River NJ: Prentice Hall.</cite></li>

<li><cite id="CITEREFStewart1996" class="citation book cs1">Stewart, Gilbert W. (1996). Afternotes on Numerical Analysis. <a href="Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">SIAM</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-89871-362-6</bdi>.</cite></li>

<li><cite id="CITEREFTrefethen2013" class="citation cs2">Trefethen, Lloyd N. (2013), <a rel="nofollow" class="external text" href="https://people.maths.ox.ac.uk/trefethen/ATAP/">Approximation Theory and Approximation Practice</a>, SIAM</cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>Burden, Richard L.; Faires, J. Douglas: Numerical Analysis, 8th ed., pages 503–512, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-534-39200-8</bdi>.</li></ul>

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</style><div id="Algebraic_numbers198" style="font-size:114%;margin:0 4em"><a href="Algebraic_number" title="Algebraic number">Algebraic numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algebraic_integer" title="Algebraic integer">Algebraic integer</a></li>

<li><a href="Constructible_number" title="Constructible number">Constructible number</a></li>

<li><a href="Look-and-say_sequence" title="Look-and-say sequence">Conway's constant</a></li>

<li><a href="Cyclotomic_field" title="Cyclotomic field">Cyclotomic field</a></li>

<li><a href="Doubling_the_cube" title="Doubling the cube">Doubling the cube</a></li>

<li><a href="Eisenstein_integer" title="Eisenstein integer">Eisenstein integer</a></li>

<li><a href="Gaussian_integer" title="Gaussian integer">Gaussian integer</a></li>

<li><a href="Golden_ratio" title="Golden ratio">Golden ratio (φ)</a></li>

<li><a href="Perron_number" title="Perron number">Perron number</a></li>

<li><a href="Pisot%E2%80%93Vijayaraghavan_number" title="Pisot–Vijayaraghavan number">Pisot–Vijayaraghavan number</a></li>

<li><a href="Plastic_ratio" title="Plastic ratio">Plastic ratio (ρ)</a></li>

<li><a href="Quadratic_irrational_number" title="Quadratic irrational number">Quadratic irrational number</a></li>

<li><a href="Rational_number" title="Rational number">Rational number</a></li>

<li><a href="Root_of_unity" title="Root of unity">Root of unity</a></li>

<li><a href="Salem_number" title="Salem number">Salem number</a></li>

<li><a href="Silver_ratio" title="Silver ratio">Silver ratio (σ)</a></li>

<li><a href="Square_root_of_2" title="Square root of 2">Square root of 2</a></li>

<li><a href="Square_root_of_3" title="Square root of 3">Square root of 3</a></li>

<li><a href="Square_root_of_5" title="Square root of 5">Square root of 5</a></li>

<li><a href="Square_root_of_6" title="Square root of 6">Square root of 6</a></li>

<li><a href="Square_root_of_7" title="Square root of 7">Square root of 7</a></li>

<li><a href="Square_root_of_10" title="Square root of 10">Square root of 10</a></li>

<li><a href="Supergolden_ratio" title="Supergolden ratio">Supergolden ratio (ψ)</a></li>

<li><a href="Supersilver_ratio" title="Supersilver ratio">Supersilver ratio (ς)</a></li>

<li><a href="Twelfth_root_of_2" class="mw-redirect" title="Twelfth root of 2">Twelfth root of 2</a></li></ul>

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