pracma (version 1.8.8)

simpadpt: Adaptive Simpson Quadrature

Description

Numerically evaluate an integral using adaptive Simpson's rule.

Usage

simpadpt(f, a, b, tol = 1e-6, ...)

Arguments

f

univariate function, the integrand.

a, b

lower limits of integration; must be finite.

tol

relative tolerance

...

additional arguments to be passed to f.

Value

• A numerical value or vector, the computed integral.

Details

Approximates the integral of the function f from a to b to within an error of tol using recursive adaptive Simpson quadrature.

References

Quarteroni, A., R. Sacco, and F. Saleri (2007). Numerical Mathematics. Second Edition, Springer-Verlag, Berlin Heidelberg.

See Also

quad, simpson2d

Examples

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

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()