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Title: Nodes and weights of Gauss–Kronrod quadrature Definition: For $n\geq 1$ the Gauss–Kronrod rule with $2n+1$ points, $\int_{-1}^{1}- f(x)\,dx \approx \sum_{k=1}^{2n+1} w_k f(x_k)$, extends the Gauss–Legendre rule- with $n$ points by $n+1$ nodes so as to be exact for every polynomial of degree+ f(x)\,\mathrm{d}x \approx \sum_{k=1}^{2n+1} w_k f(x_k)$, extends the Gauss–Legendre+ rule with $n$ points by $n+1$ nodes so as to be exact for every polynomial of degree at most $3n+1$ CITE{Wiki}. Listed are its nodes $x_1<\cdots<x_{2n+1}$ and its weights $w_k$.
the HREF{Legendre_polynomials}[Legendre polynomial] of degree $n$ and $E_{n+1}$ is the Stieltjes polynomial CITE{Wiki-Stieltjes}: the monic polynomial of degree- $n+1$ with $\int_{-1}^{1}P_n(x)E_{n+1}(x)\,x^j\,dx=0$ for $0\leq j\leq n$.'+ $n+1$ with $\int_{-1}^{1}P_n(x)E_{n+1}(x)\,x^j\,\mathrm{d}x=0$ for $0\leq j\leq+ n$.' formula-stieltjes: $E_2=x^2-\tfrac35$, $E_3=x^3-\tfrac67x$, $E_4=x^4-\tfrac{10}{9}x^2+\tfrac{155}{891}$, $E_5=x^5-\tfrac{15}{11}x^3+\tfrac{615}{1573}x$, $E_6=x^6-\tfrac{21}{13}x^4+\tfrac{567}{845}x^2-\tfrac{8043}{186745}$. $E_{n+1}$ has the parity of $n+1$, so $x=0$ is a root of $E_{n+1}$ for even $n$ and of $P_n$ for odd $n$.- formula-weights: $w_k=\dfrac{q_{\omega_n}(x_k)}{\omega_n'(x_k)}$ with $q_{\omega_n}(x)=\int_{-1}^{1}\dfrac{\omega_n(t)-\omega_n(x)}{t-x}\,dt$,- and $q_{\omega_n}=c_n+E_{n+1}\,q_n$, where $c_n=\int_{-1}^{1}P_n(x)\,x^n\,dx=\dfrac{2^{n+1}(n!)^2}{(2n+1)!}$+ formula-weights: $w_k=\dfrac{q_{\omega_n}(x_k)}{\omega_n'(x_k)}$ with $q_{\omega_n}(x)=\int_{-1}^{1}\dfrac{\omega_n(t)-\omega_n(x)}{t-x}\,\mathrm{d}t$,+ and $q_{\omega_n}=c_n+E_{n+1}\,q_n$, where $c_n=\int_{-1}^{1}P_n(x)\,x^n\,\mathrm{d}x=\dfrac{2^{n+1}(n!)^2}{(2n+1)!}$ and $q_n$ is the HREF{Secondary_polynomials_of_the_Legendre_polynomials}[secondary polynomial of $P_n$]. Hence at a Gauss node $w_{2j}=\lambda_j+\dfrac{c_n}{P_n'(x_{2j})\,E_{n+1}(x_{2j})}$
to $\lambda_j$ is negative because $P_n'$ and $E_{n+1}$ take opposite signs at every root of $P_n$, by the interlacing.- formula-exactness: $\sum_{k=1}^{2n+1} w_k x_k^m=\int_{-1}^{1}x^m\,dx=\dfrac{1+(-1)^m}{m+1}$+ formula-exactness: $\sum_{k=1}^{2n+1} w_k x_k^m=\int_{-1}^{1}x^m\,\mathrm{d}x=\dfrac{1+(-1)^m}{m+1}$ for $0\leq m\leq 3n+1$ CITE{Laurie}, and by the symmetry also for $m=3n+2$ when $n$ is odd; it fails at the next even $m$. So the degree of exactness is $3n+1$
is rational: $\tfrac89$, $\tfrac{28}{45}$, $\tfrac{22016}{48825}$, $\tfrac{201344}{581175}$, $\tfrac{118298624}{418034925}$ for $n=1,\ldots,5$.'- formula-lagrange: $w_k=\int_{-1}^{1}\ell_k(x)\,dx$ with $\ell_k(x)=\prod_{j\neq+ formula-lagrange: $w_k=\int_{-1}^{1}\ell_k(x)\,\mathrm{d}x$ with $\ell_k(x)=\prod_{j\neq k}\frac{x-x_j}{x_k-x_j}$ the Lagrange basis polynomial of the $2n+1$ nodes, as for every interpolatory rule; on HREF{Lagrange_basis_polynomials_for_equally_spaced_nodes}[equally
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