Adaptive Simpson Quadrature
Numerically evaluate an integral using adaptive Simpson's rule.
simpadpt(f, a, b, tol = 1e-6, ...)
f |
univariate function, the integrand. |
a, b |
lower limits of integration; must be finite. |
tol |
relative tolerance |
... |
additional arguments to be passed to |
Approximates the integral of the function f from a to b to within
an error of tol using recursive adaptive Simpson quadrature.
A numerical value or vector, the computed integral.
Based on code from the book by Quarteroni et al., with some tricks borrowed from Matlab and Octave.
Quarteroni, A., R. Sacco, and F. Saleri (2007). Numerical Mathematics. Second Edition, Springer-Verlag, Berlin Heidelberg.
myf <- function(x, n) 1/(x+n) # 0.0953101798043249 , log((n+1)/n) for n=10
simpadpt(myf, 0, 1, n = 10) # 0.095310179804535
## Dilogarithm function
flog <- function(t) log(1-t) / t # singularity at t=1, almost at t=0
dilog <- function(x) simpadpt(flog, x, 0, tol = 1e-12)
dilog(1) # 1.64493406685615
# 1.64493406684823 = pi^2/6
## Not run:
N <- 51
xs <- seq(-5, 1, length.out = N)
ys <- numeric(N)
for (i in 1:N) ys[i] <- dilog(xs[i])
plot(xs, ys, type = "l", col = "blue",
main = "Dilogarithm function")
grid()
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.