Misc. • The center points of circles are generated from nodes(x-axis) and weights(y-axis) of Gaussian quadrature.
• Number of points : n = 15
• $\textcolor{#CB3C33}{\text{\textbf{Red}}}$ : gausslegendre(n)
• $\textcolor{#389826}{\text{\textbf{Green}}}$ : gausschebyshev(n, 3)
• $\textcolor{#9558B2}{\text{\textbf{Purple}}}$ : gaussjacobi(n, 5/2, 1/2)
• These colors are from julia-logo-graphics.
• The logo is generated with Luxor.jl, and its source code is here.

Other docstrings from private methods

The first 11 roots of the Airy function in Float64 precision https://mathworld.wolfram.com/AiryFunctionZeros.html

Examples

julia> all(zeros .< 1e-14)
true
source

First twenty roots of Bessel funcion $J_0$ in Float64. https://mathworld.wolfram.com/BesselFunctionZeros.html

Examples

`jldoctest julia> zeros = besselj0.(FastGaussQuadrature.BESSELJ0_ROOTS);

julia> all(zeros .< 1e-14) true

source

Values of Bessel function $J_1$ on first ten roots of Bessel function J_0.

Examples

julia> roots = approx_besselroots(0,10);

true
source

Coefficients of Chebyshev series approximations for the zeros of the Bessel functions

$$$j_{\nu, s} \approx \sum_{k}^{n} C_{k,s} T_k(\frac{\nu-2}{3})$$$

where $j_{\nu, s}$ is a $s$-th zero of Bessel function $J_\nu$, $\{T_k\}$ are Chebyshev polynomials and $C_{k, s}$ is the coefficients.

source