Misc.
About the logo
- 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)
- Number of points :
- 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
FastGaussQuadrature.AIRY_ROOTS
— ConstantThe first 11 roots of the Airy function in Float64 precision https://mathworld.wolfram.com/AiryFunctionZeros.html
Examples
julia> zeros = airy.(FastGaussQuadrature.AIRY_ROOTS);
julia> all(zeros .< 1e-14)
true
FastGaussQuadrature.BESSELJ0_ROOTS
— ConstantFirst twenty roots of Bessel function $J_0$ in Float64. https://mathworld.wolfram.com/BesselFunctionZeros.html
Examples
julia> zeros = besselj0.(FastGaussQuadrature.BESSELJ0_ROOTS);
julia> all(zeros .< 1e-14)
true
FastGaussQuadrature.BESSELJ1_ON_BESSELJ0_ROOTS
— ConstantValues of Bessel function $J_1$ on first ten roots of Bessel function J_0
.
Examples
julia> roots = approx_besselroots(0,10);
julia> (besselj1.(roots)).^2 ≈ FastGaussQuadrature.BESSELJ1_ON_BESSELJ0_ROOTS
true
FastGaussQuadrature.CUMSUMMAT_10
— Constant10-point Chebyshev type 2 integration matrix computed using Chebfun.
Used for numerical integration in boundary asymptotics for Gauss-Jacobi.
FastGaussQuadrature.DIFFMAT_10
— Constant10-point Chebyshev type 2 differentiation matrix computed using Chebfun.
Used for numerical differentiation in boundary asymptotics for Gauss-Jacobi.
FastGaussQuadrature.PIESSENS_C
— ConstantCoefficients 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.
FastGaussQuadrature.evalLaguerreRec
— MethodEvaluate the orthonormal associated Laguerre polynomial with positive leading coefficient, as well as its derivative, in the point x using the recurrence relation.
FastGaussQuadrature.feval_asy1
— MethodEvaluate the interior asymptotic formula at x = cos(t). Assumption:
length(t) == n ÷ 2
FastGaussQuadrature.feval_asy2
— MethodEvaluate the boundary asymptotic formula at x = cos(t). Assumption:
length(t) == n ÷ 2
FastGaussQuadrature.gausslaguerre_GW
— MethodCalculate Gauss-Laguerre nodes and weights from the eigenvalue decomposition of the Jacobi matrix.
FastGaussQuadrature.gausslaguerre_asy
— MethodCompute the Gauss-Laguerre rule using explicit asymptotic expansions for the nodes and weights. Optional parameters are:
reduced
: compute a reduced quadrature rule, discarding all points and weights as soon as the weights underflowT
: the order of the expansion. SetT=-1
to determine the order adaptively depending on the size of the terms in the expansionrecompute
: if a crude measure of the error is larger than a tolerance, the point and weight are recomputed using the (slower) recursion+newton approach, yielding more reliable accurate results.
FastGaussQuadrature.gausslaguerre_rec
— MethodCompute Gauss-Laguerre rule based on the recurrence relation, using Newton iterations on an initial guess.