pick the right method and don't need to worry about the domain

This project provide and interface for solving integration problems using guassian quadrature and monte carlo integratioon no matter the problem is univariable or multivarialbe. Furthermore, based on interest, I also implement the function for importance samplingin quasi way through halton sequences.

For Gaussian Quadrature method I implement:

1. Gauss Hermite
2. Gauss Laguerre
3. Gauss Legendre

note: rely on this project FastGaussQuadrature.jl to generate the nodes and weights required by the gaussian quadrature method

As for monte carlo method I implement:

1. naive monte carlo
2. quasi monte carlo
3. importance samplng since the quais montecarlo method rely on the halton sequences with the prime = 2, it's better to let the number of nodes = $2^n−1$

note: both the quasi monte carlo and importance sampling use HaltonSequences.jl to implment

• GHermite(), GLaguerre(), GLegendre(), GQ()
  • GQ() takes advantage of gausshermite and gausslaguerre to deal with univariable problem with domain [-Inf, 0]
  • GQ() will automatically apply different Gauss Quadraturer rules based on diffferent domain in multivariable problem
    • when domain = [-Inf, Inf], use Gauss-Hermite
    • when domain = [a, Inf], use Gauss-Laguerre
    • when domain = [-Inf, b], use Gauss-legendre(it might not be the best though)
    • when the domain is proper, use Gauss-legendre
  • all the others can estimate multivariable problem with the vector input
• MCM(), quaMCM(), IShalton()
  • IShalton() can only be used for univariable problem just now

# univariable
g = x -> exp(-x^2 / 3) * sqrt(1 + x^2)
a, b = -Inf, Inf  # domain
GHermite(g, a, b, 30)  # the last parameter is the n-point Gauss Quadrature nodes and weights
GLaguerre(g, a, b, 30) 

m(x, c=1e-9, k=2) = c * x^(-k-1) * (1-x)^(k+1)
a, b = -1e-5, 1  # domain
# note that I first change variable to the [0, 1] then do the importance sampling
d = truncated(Normal(1e-3, 1e-3), 0, 1)  # distribution(after change variable, domain=[0, 1])
nodesNum = [524_287, 1_048_575, 2_097_151, 8_388_607, 67_108_863]
IShalton(m, a, b, d, nodesNum)


# multivariable
g = x -> 1 / (x[1] + 1) + sqrt(x[2]) + 2 * (x[3]^2) + sqrt(2 * x[4]) + cbrt(x[5])
A, B = [0, 0, 0, 0, 0], [1, 1, 1, 1, 1]  # domain
nodesNum = [524_287, 1_048_575, 2_097_151, 8_388_607, 67_108_863]
MCM(g, A, B, nodesNum, seed=1234)
quaMCM(g, A, B, nodesNum)
GLegendre(g, A, B, 30)