Non-iterative computation of Gauss-Jacobi quadrature
This work offers a fast, non-iterative method for high-degree Gauss-Jacobi quadrature, which is useful for numerical integration in scientific computing.
The paper provides asymptotic approximations for computing nodes and weights of Gauss-Jacobi quadrature without iteration, achieving nearly double precision relative accuracy for n≥100 and -1<α,β≤5, and 10^{-12} relative accuracy for n≥20.
Asymptotic approximations to the zeros of Jacobi polynomials are given, with methods to obtain the coefficients in the expansions. These approximations can be used as standalone methods for the non-iterative computation of the nodes of Gauss--Jacobi quadratures of high degree ($n\ge 100$). We also provide asymptotic approximations for functions related to the first order derivative of Jacobi polynomials which are used for computing the weights of the Gauss--Jacobi quadrature. The performance of the asymptotic approximations is illustrated with numerical examples, and it is shown that nearly double precision relative accuracy is obtained both for the nodes and the weights when $n\ge 100$ and $-1< α, β\le 5$. For smaller degrees the approximations are also useful as they provide $10^{-12}$ relative accuracy for the nodes when $n\ge 20$, and just one Newton step would be sufficient to guarantee double precision accuracy in that cases.