Computed Clenshaw–Curtis from an even-extension DFT

Input real numbers $a<b$, a natural $n\ge 0$, sampled values $y_0,\dots ,y_n$, and the phase/FFT controls for Arbitrary-length mixed-radix and Bluestein DFT: a phase tolerance $0<\tau \le 1\mathrm{/}4$ (required to be exact rational when $n\ge 1$), integer dyadic exponent limits $e_{\min }\le e_{\max }$, and integers $p_0\ge 2$, $p_{\max }\ge p_0$, $J\ge 1$, and $d_{\max }\ge 0$. Set $m=(a+b)\mathrm{/}2$ and $d=(b-a)\mathrm{/}2$. For an underlying $f\in C[a,b]$, admissible samples satisfy $y_0=f(m)$ when $n=0$ and $y_j=f(m+d\cos (j\pi \mathrm{/}n))$ for $0\le j\le n$ when $n\ge 1$, using the canonical real images of The canonical natural $\iota (n)=n\cdot 1_F$ of a field. When $n\ge 1$, also require every $y_j$ to lie in the representable real set $\mathbb{F}$ of the same floating-point model instance used by the called DFT; this ensures its complex input coordinates satisfy that procedure's input condition. No representability condition is needed for $n=0$, when no DFT call is made. The output is an approximate real quadrature value or the first status returned by the called DFT procedure.

1. If $n=0$, return $(b-a)y_0$, the midpoint convention of Clenshaw–Curtis quadrature with the midpoint zero-degree case.
2. If $n\ge 1$, form $z\in {\mathbb{C}}^{2n}$ by $z_j:=y_j+0i$ for $0\le j\le n$ and $z_{2n-j}:=y_j+0i$ for $1\le j<n$. This is the even extension $(y_0,\dots ,y_n,y_{n-1},\dots ,y_1)$, with real samples embedded as complex numbers with zero imaginary part. Call Arbitrary-length mixed-radix and Bluestein DFT on $z$ with length $2n$ and the same controls $(\tau ,e_{\min },e_{\max },p_0,p_{\max },J,d_{\max })$. If it returns a status instead of a vector, return that status.
3. Otherwise, take the real parts $X_k$ of output entries $0\le k\le n$, form the DCT-I coefficients $a_0=X_0\mathrm{/}(2n)$, $a_n=X_n\mathrm{/}(2n)$, and $a_k=X_k\mathrm{/}n$ for $1\le k<n$ as in Type-I cosine transform for Chebyshev–Lobatto data, and return $d{\sum }_{k=0}^n{a}_k{\mu }_k$ with the exact moments ${\mu }_k$ in Clenshaw–Curtis quadrature with the midpoint zero-degree case.

The postprocessing is exact real arithmetic applied to the approximate DFT output. The procedure makes no end-to-end floating-point error or stability claim; it only propagates the DFT routine's declared failure statuses.