Function Norm
The fnorm function calculates several different types of function
norms for depending on the argument p.
fnorm(f, g, x1, x2, p = 2, npoints = 100)
f, g |
functions given by name or string. |
x1, x2 |
endpoints of the interval. |
p |
Numeric scalar or Inf, -Inf; default is 2. |
npoints |
number of points to be considered in the interval. |
fnorm returns a scalar that gives some measure of the distance
of two functions f and g on the interval [x1, x2].
It takes npoints equidistant points in the interval, computes the
function values for f and g and applies Norm to
their difference.
Especially p=Inf returns the maximum norm,
while fnorm(f, g, x1, x2, p = 1, npoints) / npoints
would return some estimate of the mean distance.
Numeric scalar (or Inf), or NA if one of these functions
returns NA.
Another kind of ‘mean’ distance could be calculated by integrating the
difference f-g and dividing through the length of the interval.
xp <- seq(-1, 1, length.out = 6) yp <- runge(xp) p5 <- polyfit(xp, yp, 5) f5 <- function(x) polyval(p5, x) fnorm(runge, f5, -1, 1, p = Inf) #=> 0.4303246 fnorm(runge, f5, -1, 1, p = Inf, npoints = 1000) #=> 0.4326690 # Compute mean distance using fnorm: fnorm(runge, f5, -1, 1, p = 1, 1000) / 1000 #=> 0.1094193 # Compute mean distance by integration: fn <- function(x) abs(runge(x) - f5(x)) integrate(fn, -1, 1)$value / 2 #=> 0.1095285
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.