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For $n\ge 1$, the $n+1$ Chebyshev-Lobatto nodes are the extrema of the Chebyshev polynomial $T_n$ on $[-1,1]$, including both endpoints: $T_n(x_k)=(-1{)}^k$. Imagine $n+1$ equally spaced angles on a semicircle and project their points onto the horizontal axis. The interpolation nodes crowd near $-1$ and $1$ because equal angular steps have smaller horizontal projections there. Reversing the order gives the same set. These are distinct from the roots of a polynomial $\cos ((2k+1)\pi /(2n+2))$ of $T_{n+1}$, which omit the endpoints. Driscoll and Braun call this endpoint-including family Chebyshev points of the second kind or Chebyshev extreme points.^[ref]

On an interval $[a,b]$, use $x_k=(a+b)/2+(b-a)\cos (k\pi /n)/2$. The Runge phenomenon example compares polynomial interpolation at eleven Chebyshev-Lobatto nodes with eleven equally spaced interpolation nodes. The distinction matters whenever sampling locations are a choice rather than fixed observations.

The first-kind Chebyshev polynomial $T_n$ converts a multiple-angle cosine into a polynomial:The cosine addition formula gives the recurrence relationThis constructs polynomials valid for every real $x$, including outside $[-1,1]$, where the real-angle description is unavailable. For example,For $n\ge 1$, $T_n$ has polynomial degree $n$ and leading coefficient $2^{n-1}$. On $[-1,1]$ its uniform norm is one, even as its oscillations become more numerous.^[ref][ref]

The polynomial roots of $T_n$ are especially simple:They correspond to equally spaced angles, but their locations in $x$ cluster near $-1$ and $1$. For $T_3$, the polynomial roots are $\sqrt{3}/2,0,-\sqrt{3}/2$. For $T_8$, the positive polynomial roots are approximately $0.980785,0.831470,0.555570,0.195090$, with their negatives completing the set. The distance from the right endpoint to the nearest polynomial root is $1-\cos (\pi /(2n))\sim {\pi }^2/(8n^2)$; central neighboring polynomial roots are separated on the larger scale $1/n$. The small change of cosine near angles zero and $\pi$ explains the clustering.

These polynomial roots supply $n$ interpolation nodes for a polynomial interpolation of polynomial degree at most $n-1$. They differ from the $n+1$ Chebyshev-Lobatto nodes $\cos (j\pi /n)$, which include both endpoints. This choice of polynomial roots has nodal polynomialThe monic Chebyshev extremal polynomial minimizes the uniform norm of this product among monic polynomials of polynomial degree $n$ on $[-1,1]$. Consequently, if $f$ has $n$ continuous derivatives, the usual polynomial interpolation error gives the useful boundThis optimizes the nodal factor in that bound, not the actual error for every function. Endpoint clustering helps control the large endpoint oscillations of equally spaced interpolation associated with Runge's phenomenon. The two node conventions and their geometric construction are discussed in Trefethen's authored text.^[ref]

The same cosine substitution for polynomial approximation explains the weighted orthogonality:The endpoint singularities of the weight are integrable. Thus the Chebyshev polynomials are orthogonal polynomials for this weight, making coefficients in a least-squares polynomial in an orthogonal-polynomial basis separate cleanly. Their polynomial roots also give Chebyshev–Gauss quadrature, connecting polynomial interpolation to Gaussian quadrature.^[ref]

Accuracy still depends on the target function. The absolute value function $|x|$ has a corner at zero, so the hypothesis requiring continuous derivatives in the displayed polynomial interpolation error bound fails. Endpoint clustering cannot remove that corner. A jump poses a stronger obstruction: polynomials are continuous functions, so they cannot converge in the uniform norm to a discontinuous function on the whole interval. Chebfun's documented examples show persistent overshoot for such jumps; these are a limitation of global polynomial interpolation, not a universal promise of rapid accuracy from Chebyshev polynomials.^[ref]

The Lagrange interpolation polynomial isThe interpolation nodes are distinct, so these weights are well defined. Applying the same Lagrange interpolation to the constant polynomial one gives $1={\sum }_r{w}_r{\pi }_{n,r}(x)$. Dividing this identity and the formula for $p$ by ${\pi }_n(x)$ gives the barycentric interpolation formulaThis displayed quotient is used away from the interpolation nodes; at $x=x_r$ its removable singularity is evaluated by setting $p(x_r)=u_r$. Multiplying every weight by the same nonzero constant leaves the quotient unchanged.

Chebyshev points are generally more reliable at high degree. The Chebyshev-Lobatto nodes, or the affine images of the roots of a Chebyshev polynomial, cluster near the interval endpoints and control the amplification measured by the Lebesgue constant of interpolation. Equally spaced interpolation nodes can produce large endpoint oscillations and the Runge phenomenon. This is a comparison of numerical robustness and general approximation behaviour, not a claim that one set minimizes the error for every individual function.

For uniformly spaced nodes,There are $n-s$ negative factors, coming from $j>s$. Thus the weights fixed by part (b) satisfy this formula exactly. The printed sign $(-1{)}^s$ agrees for even $n$; for odd $n$ it differs by the common factor $-1$. Rescaling all weights by $(-1{)}^n$ gives the printed convention without changing the barycentric interpolation formula.

For even $n$, the required magnitude isStirling's formula givesThus central weights are exponentially larger in magnitude than endpoint weights. Their alternating signs can cause severe cancellation in polynomial interpolation at equispaced nodes; they reflect the unfavorable high-degree interpolation geometry associated with Runge's phenomenon. A barycentric evaluation formula improves the implementation but does not remove the intrinsic sensitivity or guarantee convergence for every smooth function. The comparison with the bounded relative weights at Chebyshev-Lobatto nodes is decisive.

Use the Chebyshev polynomial identities $T_0=1$, $T_1=x$, and $T_{j+1}=2x{T}_j-T_{j-1}$. Induction shows that $T_j$ has degree $j$ with leading coefficient $2^{j-1}$ for $j\ge 1$. The trigonometric definition also givesThereforeThe leading coefficient is $2^{1-n}{n}^{-1}\cdot n{2}^{n-1}=1$, and the degree is $n+1$. The endpoints are roots through $x^2-1$; the other $n-1$ points $\cos (s\pi /n)$ are roots through $T_n^{\prime }$. They are distinct, so this monic polynomial is exactly ${\pi }_n$.

Differentiating the product ${\pi }_n(x)=(x-x_s){\pi }_{n,s}(x)$ at $x_s$ gives ${\pi }_n^{\prime }(x_s)={\pi }_{n,s}(x_s)=w_s^{-1}$. For an interior node, differentiate the trigonometric form using $d/dx=-(\sin \theta {)}^{-1}d/d\theta$:At ${\theta }_s=s\pi /n$, this is $2^{1-n}n(-1{)}^s$. At either endpoint, taking the limit makes the two terms equal, doubling the value: ${\pi }_n^{\prime }(x_s)=2^{2-n}n(-1{)}^s$ for $s=0,n$.

After removing the common factor $2^{n-1}/n$, the barycentric interpolation formula therefore usesThese Chebyshev-Lobatto node weights differ in magnitude by only a factor of two.

The Runge phenomenon is the failure of increasing-degree polynomial interpolation at equally spaced interpolation nodes to converge to some smooth target functions, with large oscillations near the ends of the interval. It is a property of the approximation problem, and can occur even with exact arithmetic. A numerically stable evaluation method cannot remove it. Endpoint clustering, as in Chebyshev-Lobatto nodes, changes the approximation problem and can greatly improve it.^[ref]

Take $f(x)=1/(1+25x^2)$ on $[-1,1]$ and polynomial degree at most $10$. Use either $x_k=-1+k/5$ or $x_k=\cos (k\pi /10)$ for $k=0,\dots ,10$. Each interpolation polynomial passes through all eleven of its own samples, yet the first has severe endpoint oscillations and the second stays much closer to the target function. Evaluating both by the barycentric interpolation formula on $20,001$ equally spaced points, including the endpoints, gives sampled maximum absolute errors of approximately $1.915659$ and $0.132197$, respectively: about a $14.49$-fold reduction. These are maxima over the stated grid, not certified maxima over the continuous interval. The generator separately checks exact reproduction of all stored nodal values.

The polynomial interpolation error formula explains why interpolation nodes enter the analysis. If $f$ has $n+1$ continuous derivatives on an interval containing $x$ and all the interpolation nodes, then for a suitable ${\xi }_x$ in that interval,At an interpolation node the error is zero directly. Away from the interpolation nodes, both the product and the high-order derivative matter. A short proof applies Rolle's theorem repeatedly to $f(t)-p_n(t)-[f(x)-p_n(x)]{\prod }_i(t-x_i)/{\prod }_i(x-x_i)$, which vanishes at $n+2$ distinct points.^[ref]

The successful comparison here does not assert that Chebyshev-Lobatto nodes solve every approximation problem for every function. The derivatives, the polynomial degree, and measurement errors still matter. The Lebesgue constant of interpolation measures worst-case amplification of nodal errors, while the polynomial reproduction error bound relates interpolation to the best uniform approximation available at that polynomial degree.