Cheap Numerical Integration through Data Points

Given \((x_i, f_i)\) where \(f_i = f(x_i)\), compute a cheap approximation of \(\int_a^b f(x) dx\).

integrate.xy(x, fx, a, b, use.spline=TRUE, xtol=2e-08)

Arguments

x

abscissa values.

fx

corresponding values of \(f(x)\).

a,b

the boundaries of integration; these default to min(x) and max(x) respectively.

use.spline

logical; if TRUE use an interpolating spline.

xtol

tolerance factor, typically around sqrt(.Machine$double.eps) ......(fixme)....

Details

Note that this is really not good for noisy fx values; probably a smoothing spline should be used in that case.

Also, we are not yet using Romberg in order to improve the trapezoid rule. This would be quite an improvement in equidistant cases.

Value

the approximate integral.

Author

Martin Maechler, May 1994 (for S).

See also

integrate for numerical integration of functions.

Examples

 x <- 1:4
 integrate.xy(x, exp(x))
#> [1] 52.40261
 print(exp(4) - exp(1), digits = 10) # the true integral
#> [1] 51.8798682

 for(n in c(10, 20,50,100, 200)) {
   x <- seq(1,4, len = n)
   cat(formatC(n,wid=4), formatC(integrate.xy(x, exp(x)), dig = 9),"\n")
 }
#>   10 51.8822763 
#>   20 51.8799518 
#>   50 51.8799052 
#>  100 51.8799054 
#>  200 51.8799054