9. ガウス求積：ぱらぱらめくる『A Short Course on Approximation Theory』
library(pracma) ## Dilogarithm function flog <- function(t) log(1-t)/t t<-seq(from=0,to=1,length=100) plot(t,flog(t)) quadgr(flog, 1, 0, tol = 1e-12)
library(pracma) # ２変数関数の場合 ## Example: f(x, y) = (y+1)*exp(x)*sin(16*y-4*(x+1)^2) f <- function(x, y) (y+1) * exp(x) * sin(16*y-4*(x+1)^2) # this is even faster than cubature::adaptIntegral(): quad2d(f, -1, 1, -1, 1) # 0.0179515583236958 # true value 0.01795155832370 ## Volume of the sphere: use polar coordinates f0 <- function(x, y) sqrt(1 - x^2 - y^2) # for x^2 + y^2 <= 1 fp <- function(x, y) y * f0(y*cos(x), y*sin(x)) quad2d(fp, 0, 2*pi, 0, 1, n = 101) # 2.09439597740074 2/3 * pi # 2.0943951023932
